= 0.0, but there's still several extra instructions in the fast path. As for its speed, it at least outperforms the std::sin() function by an average of 0.3 microseconds per call. If you're writing audio plugins or doing digital signal processing, Chebyshev polynomials give you a cheap and predictable dithering effect "for free.". An approximation for the sine function that preserves the derivatives at multiples of 90 degrees is given by this formula. How about dectecting the CPU and using a native instruction on a modern processor with a lookup table or other optimized code on older machines. My custom x86 Math library code is for standard MSVC++ 2005 and forward. The cosine of 0 is well-defined, and is 1. gcc's code for the NaN case seems way over-complicated; it doesn't even use the sqrtf return value! Lastly, when you've finished all your fancy benchmarking and micro-optimization, make sure that your "fast" version is actually faster than the library version. The, Unless you bottleneck on uop throughput rather than sqrt latency, in which case using plain, What is the purpose of taking the address of. Also, I've benchmarked it and it twice slower than milianw answers. Here's a possible speedup which depends highly on your application. That's strictly bush league. Check this one for example . Making statements based on opinion; back them up with references or personal experience. This is a delightful representation of the Taylor Series. What makes an argument objectively more "compelling"? @roliu My mistake, I though I have to calculate Factorial several times, but I missed that I can use precomputed constant. (Negative a), Graphing and Writing Equations of Hyperbolas, Conic Sections: The Hyperbola part 1 of 2 Conic Sections: The Hyperbola part 2 of 2 Ex 1: Conic Section - Graph a Hyperbola with Center at the Origin (Horizontal) Ex 2: Conic Section - Graph a Hyperbola with Center at the Origin (Vertical) Ex 3: Conic Section - Graph a Hyperbola with Center NOT at the Origin (Horizontal) Ex 4: Conic Section - Graph a Hyperbola with Center NOT at the Origin (Vertical) Ex: Find the Equation of a Hyperbola Given the Center, Focus, and Vertex Determining the Type of Conic Section from General Form, Introduction to Set Theory Introduction to Subsets Set Operations and Venn Diagrams - Part 1 of 2 Set Operations and Venn Diagrams - Part 2 of 2 Solving Problems Using Venn Diagrams, Introduction to Regression Analysis Linear Regression – Example 1, Example 2 Quadratic Regression – Example 1, Example 2 Perform Quadratic Regression and Make Predictions Using Desmos Interpret a Quadratic Function Model: Fuel Consumption Ex 1: Cubic Regression on the TI84 – Natural Gas Consumption Ex 2: Cubic Regression on the TI84 - Total Sales Exponential Regression – Example 1, Example 2 Ex: Exponential Decay Regression Model (Declining Population) Ex: Exponential Growth Regression Model (Investment Account) Logarithmic Regression Logistic Regression, Financial Mathematics (See more under Math for Liberal Arts library), Simple Interest Formula Compounded and Continuous Interest Effective Yield Effective Yield on the TI84 Derive the Value of an Annuity Formula (Compounded Interest) Determining the Value of an Annuity Determining the Value of an Annuity Using the TI84 Determining the Monthly Saving Required to Reach a Financial Goal Determining the Monthly Saving Required to Reach a Financial Goal on the TI84 Ex 1: Find a Monthly Mortgage Payment with a Down Payment Ex 2: Find a Monthly Mortgage Payment with a Down Payment and Points Ex: Comparing Two Installments Loans (Car Loans) Ex: Simple Interest Discounted Loan Determining the Monthly Payments for a Loan Determining the Monthly Payments for a Loan on the TI84, Graphing Calculator Basics: Evaluating Expressions and Determining Function Values, Graphing Calculator Basics The Table Feature of the Graphing Calculator Evaluating Radical Expressions on the TI83/84 Determine Function Values Using Function Notation on the TI84, Graphing Functions and Determining Key Components of Functions, Graphing Lines on the Graphing Calculator Determining the Intersection of Two Graph on the TI83/84 Determining Relative Extrema on the Graphing Calculator Determine Function Values Using Function Notation on the TI84 Determine the value of the derivative function on the graphing calculator Determining the value of a definite integral on the graphing calculator Determining the Intersection of Two Graph on the TI83/84 Determining the Zeros or Roots of a Polynomial Function on the TI83/84 From a Graph Determine Where a Quadratic Function is Increasing and Decreasing Determining When a Polynomial Function is Increasing and Decreasing Determine Max/Mins and Incr/Decr Intervals Using a Free Online Graph Calc (MathAS) Determining the Zeros or Roots of a Polynomial Function on the TI83/84 Ex: Quadratic Function Application Using a Graphing Calculator - Rocket Launch Ex 1: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 2: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 1: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 2: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Solving Equations on the Graphing Calculator, Solving Linear Equations Graphically Ex: Solve a Linear Equation in One Variable Graphically using the TI84 Ex: Solve a Linear Inequality in One Variable Graphically using the TI84 Ex: Solving Absolute Value Equations on the Graphing Calculator Ex 1: Solve a Quadratic Equation Graphically on Calculator Ex 2: Solve a Quadratic Equation Graphically on Calculator Ex: Determine How a Final Exam Score Affect a Course Grade using the TI84 (Weighted Averages) Solving Polynomial Equations Graphically, Ex: Evaluate Common Logarithms on a Calculator Ex: Evaluate Common Logarithms on the Calculator Ex: Evaluate Natural Logarithms on the Calculator Ex: Change of Base Formula to Evaluate Logarithmic Expressions Ex: Change of Base Formula to Solve Basic Exponential Equations Ex: Evaluate Logarithmic Functions Using the Change of Base Formula Ex: Solve an Exponential Equation Graphically on the TI84 Ex: Perform Exponential Regression on a Graphing Calculator Ex: Comparing Linear and Exponential Regression Sequences and Series on the TI84 Sequences and Series on the TI84, Augmented Matrices on the Graphing Calculator Matrix Multiplication on the Graphing Calculator Inverse Matrices on the Graphing Calculator Ex 1: Solve a System of Two Equations Using a Matrix Equation Ex 2: Solve a System of Two Equations Using a Matrix Equation Ex: Solve a System of Three Equations Using a Matrix Equation Determinants on the Graphing Calculator Ex: Solve a System of Three Equations Using Cramer's Rule, Ex 1: Create a Scatter Plot and then Perform Linear Regression on the Calculator Ex 2: Creating a Scatter Plot and Performing Linear Regression on the Calculator Linear Regression – Example 1, Example 2 Ex: Matching Correlation Coefficients to Scatter Plots Quadratic Regression – Example 1, Example 2 Perform Quadratic Regression and Make Predictions Using Desmos Ex: Quadratic Regression on the TI84 - Stopping Distance Exponential Regression – Example 1, Example 2 Perform Exponential Regression and Make Predictions Using Desmos Logarithmic Regression Logistic Regression Regression and Systems of Equations: Application, Financial Mathematics on the Graphing Calculator, Loan Information on the TI83/84 Effective Yield on the TI84 Determining the Value of an Annuity Using the TI84 Determining the Monthly Payments for a Loan on the TI84 Determining the Monthly Saving Required to Reach a Financial Goal on the TI84, Determine if a Function is a Polynomial Function, Degree, Leading Term, and Leading Coefficient of a Polynomial Function, Ex: Information about a Given Polynomial Function, Turning Points and X Intercepts of a Polynomial Function, Summary of End Behavior or Long Run Behavior of Polynomial Functions, Determine the End Behavior of Power Functions, Ex: End Behavior or Long Run Behavior of Functions, Ex: End Behavior of a Polynomial Function in Factored Form, Ex: Find the Intercepts of a Polynomial Function in Factored Form, Ex: Determine the Least Possible Degree of a Polynomial From the Graph, Ex: Increasing / Decreasing / Relative Extrema from Analyzing a Graph, Analyze a Graph Using Desmos to Determine Key Components of a Quadratic (Incr / Decr / Extrema), Analyze a Graph Using Desmos to Determine Key Components of a Cubic (Incr / Decr / Extrema), Ex 1: Determine the Local / Relative Extrema of a Cubic Function Using Desmos, Ex 2: Determine the Local / Relative Extrema of a Cubic Function Using Desmos (Challenging), Determine the Maximum Volume of an Open Top Box Using a Graph Only, Ex: Concavity / Points of Inflection by Analyzing a Graph (Algebra Topic), Ex: Concavity / Increasing / Decreasing Functions as Tables (Algebra Topic), Ex: Find the Intercepts of a Polynomial Function (Factorable), Ex: Find the Intercepts of a Polynomial Function (Real Zero), Ex 1: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 2: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 3: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 1: Solve a Cubic Function Graphically (One Solution), Ex 2: Solve a Cubic Function Graphically (Two Solutions), Ex 1: Find a Degree 3 Polynomial Function Given Integer Zeros, Ex 2: Find a Degree 3 Polynomial Function Given Fractional Zeros, Ex 3: Find a Degree 3 Polynomial Function Given Imaginary Zeros, Ex 4: Find a Degree 3 Polynomial Function Given Complex Zeros, Ex 1: Find a Polynomial Function Given the Zeros or Roots with Multiplicity and a Point, Ex 2: Find a Polynomial Function Given the Zeros or Roots with Multiplicity and a Point, Find a Polynomial Function Given the Zeros and Leading Coefficient (Degree 3), Find a Polynomial Function Given the Zeros, Multiplicity, and (0,a) (Degree 3), Determine a Degree 3 Polynomial Function Given the Zeros and Intercept (2 complex), Ex 1: Find a Degree 4 Polynomial Function Given Integer and Complex Zeros, Ex 2: Find a Degree 4 Polynomial Function Given Integer and Complex Zeros, Ex 3: Find a Degree 4 Polynomial Function Given Fractional and Complex Zeros, Ex1: Find an Equation of a Degree 4 Polynomial Function From the Graph of the Function, Ex2: Find an Equation of a Degree 5 Polynomial Function From the Graph of the Function, Ex3: Find an Equation of a Degree 6 Polynomial Function From the Graph of the Function, Ex 1: Cubic Regression on the TI84 – Natural Gas Consumption, Ex 2: Cubic Regression on the TI84 - Total Sales, Real Zeros, Factors, and Graphs of Polynomial Functions, Find the Zeros of a Cubic Function Using Factor by Grouping (1 rational, 2 irrational), Find the Zeros of a Cubic Function Using Factor by Grouping (1 rational, 2 complex), Use the Remainder Theorem to Determine if a Binomial is a Factor of a Polynomial, Determine if a Binomial is a Factor of a Polynomial Graphically, Determine the Zeros/Roots and Multiplicity From a Graph of a Polynomial, Polynomial Function - Complex Factorization Theorem, Ex: Factor and Solve a Polynomial Equation, Ex: Solve a Basic Cubic Equation Using a Cube Root and Rational Exponent, Find the Intercepts of a Degree 4 Polynomial Function (Factorable), Find Zeros, Multiplicity, Degree, and End Behavior of a Factored Polynomial (Degree 11), Find Zeros, Multiplicity, Degree, and End Behavior of a Factored Polynomial (Degree 6), Ex 1: Find the Zeros of a Polynomial Function - Integer Zeros, Ex 2: Find the Zeros of a Polynomial Function - Real Rational Zeros, Ex 3: Find the Zeros of a Polynomial Function with Irrational Zeros, Ex 4: Find the Zeros of a Polynomial Function with Imaginary Zeros, Ex 5: Find the Zeros of a Polynomial Function with Complex Zeros, Ex 6: Find the Zeros of a Degree 4 Polynomial Function, Ex 7: Find the Zeros of a Degree 5 Polynomial Function, Ex 1: Write a Degree 3 Polynomial Function as a Product of Linear Factors (2 Imaginary), Ex 2: Write a Degree 3 Polynomial Function as a Product of Linear Factors (2 Irrational), Ex 3: Write a Degree 5 Polynomial Function as a Product of Linear Factors, Ex 1: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 2: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 1: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 2: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Determine Local (Relative) Extrema of a Polynomial Function Using the TI-84, Determine Local (Relative) Extrema of a Polynomial Function Using Desmos, Ex 1: Solve a Polynomial Inequality in Factored Form, Ex 2: Solve a Polynomial Inequality in Factored Form, Ex: Solve a Polynomial Equation Using a Graphing Calculator (Approximate Solutions), Ex: Solve a Polynomial Inequality Using Factor By Grouping (Degree 3), Ex: Solve a Polynomial Inequality Using Factor Using GCF (Degree 3), Ex: Solve a Polynomial Inequality Using Factor of a Trinomial (Degree 4), Ex 1: Determine if a Function is Odd, Even, or Neither, Ex 2: Determine if a Function is Odd, Even, or Neither, Determine if a Function is Even, Odd, or Neither Using a Graph (1), Determine if a Function is Even, Odd, or Neither Using a Graph (2), Introduction to Square Root and Perfect Squares, Approximate a Square Root to Two Decimal Places Using Trial and Error, Simplify Square Roots (Perfect Square Radicands), Simplify a Variety of Square Expressions (Simplify Perfectly), Simplify Square Roots with Variables (perfect squares), Simplify Square Roots of Decimals (Perfect Square Decimals), Simplify Square Roots (Not Perfect Square Radicands), Simplify Square Roots in the Form a*sqrt(b) (not perfect squares), Simplify Cube Roots (Perfect Cube Radicands), Simplify Cube Roots (Not Perfect Cube Radicands), Simplifying Radical Expressions Without Fractions, Simplifying Radical Expressions With Fractions, Order of Operations with a Fraction Containing a Square Root, Ex: Simplify Square Roots - Perfect Roots, Ex: Simplify Perfect Nth Roots - Radicals, Ex: Simplify Square Roots - Not Perfect Roots, Ex: Simplify Square Roots of Variable Expressions - Absolute Value Needed, Ex: Simplify a Radical Expression containing Square Roots in the Numerator and Denominator, Simplify Cube Roots with Variables (not perfect cubes), Ex: Simplify Radicals with the Same Radicand and Different Indexes, Ex: Simplify a Radical Containing a Fraction - Perfect Root, Ex: Simplify Radicals with Variables - Perfect Roots, Ex 1: Simplifying Perfect Nth Roots Containing Variables, Ex 2: Simplifying Perfect Nth Roots Containing Variables, Ex: Simplify Radicals Containing Variables With Large Exponents - Not Perfect Roots, Ex: Simplify Radicals with Negative Radicands and Odd Indexes, Simplify Radicals with Variables (Large Index and Exponents), Ex: Simplify Radicals with Variables - Not Perfect Roots, Evaluating Radical Expressions on the TI83/84, Approximating Square Roots Using Division (No Calculator Required), Multiplying Radical Expressions with Variables Using Distribution, Multiplying Binomial Radical Expressions with Variables, Multiplying Radicals Containing Variables, Multiplying Conjugates of Radical Expressions, Multiply Two Radicals with Variables (Index 4 and 5) Perfect Roots, Ex: Multiply Radical Conjugates - Square Roots, Square a Binomial with a Square Root of a Variable Expression, Multiply two Binomial Radical Expressions with Variables (Conjugates), Dividing Radicals without Variables (Basic with no rationalizing), Dividing Radicals with Variables (Basic with no rationalizing), Simplify the Square Root of Fraction: sqrt(a/b), a/sqrt(b), Rationalize the Denominator - Square Root with Variable, Rationalize the Denominator - Cube Root and 4th Root, Ex 1: Rationalize The Denominator of a Fraction (Basic), Ex 2: Rationalize The Denominator of a Fraction, Ex 3: Rationalize The Denominator of a Fraction, Ex 1: Rationalize the Denominator of a Radical Expression, Ex 2: Rationalize the Denominator of a Radical Expression, Ex: Rationalize the Denominator of a Radical Expression - Conjugate, Rationalize the Denominator - Conjugate with Variables, Ex: Write a Radical in Rational Exponent Form, Write Basic Expression in Radical Form and Using Rational Exponents, Write Expressions Using Radicals and Rational Exponents, Write Rational Exponents as Radicals and Radicals Using Rational Exponents (Variables), Simplify Radicals Using Rational Exponents, Ex: Simplify Exponential Expressions with Fraction Exponents (Power Property of Exponents), Ex: Simplify Exponential Expressions with Fraction Exponents (Quotient Property of Exponents), Ex: Simplify Expressions with Rational Exponents, Ex: Simplify an Expression with Rational Exponents and Write in Radical Form, Ex: Simplify an Expression with Negative Rational Exponents and Write in Radical Form, Ex 1: Simplify an Expression with a Negative Rational Exponent, Ex 2: Simplify an Expression with a Negative Rational Exponent, Simplify An Expression with Rational Exponents (Positive Only) Power/Quot, Simplify An Expression with Rational Exponents (Negative) Power/Quot, Simplify An Expression with Rational Exponents (Negative) Prod/Quot, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 1, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 2, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 3, TI-84: Comparing Radical Form and Rational Exponent Form, Ex: Multiply and Divide Radicals with Different Indexes Using Rational Exponents - Same Radicand, Ex: Multiply Radicals with Different Indexes Using Rational Exponents - Different Radicand, Ex: Evaluate an Expression with Rational Exponents Using Radicals, Simplify a Quotient with A Radical (Rational Exponents and Radical Form), Adding Radicals (Basic With No Simplifying), Adding Radicals That Requires Simplifying, Simplify and Add Square Roots: sqrt(a^2*c)+sqrt(b^2*c), Subtracting Radicals (Basic With No Simplifying), Subtracting Radicals That Requires Simplifying, Ex: Evaluate the Square Root of a Sum and the Sum of Square Roots on a Calculator, Ex: Add and Subtract Square Roots - No Simplifying, Ex: Add and Subtract Square Roots Containing Variables, Subtract Square Expressions with a Variable - Simplifying Required, Add Square Root Expressions with Variables, Add and Subtract Square Root Expressions with Variables (Adv), Solving Radical Equations with One Radical, Solve a Radical Equation with One Cube Root (Fraction Answer), Solve a Radical Equation with One Square Root (Fraction Answer), Solve a Basic Radical Equation Given Function Notation, Solve a Radical Equation Given Function Notation, Solving Radical Equations with Two Radicals, Ex 1: Solve a Basic Radical Equation - Square Roots, Ex 2: Solve Radical Equations - Square Roots, Ex 3: Solve Radical Equations - Square Roots, Ex 4: Solve Radical Equations - Square Roots, Ex 5: Solve Radical Equations - Two Square Roots, Ex 6: Solve Radical Equations - Two Square Roots, Ex 7: Solve Radical Equations - Two Square Roots, Ex: Solve a Radical Equation with One Radical: Factoring and Extraneous Solution, Ex: Solve a Radical Equation with Two Radicals: a*sqrt(b)=sqrt(d), Ex: Solve Radical Equations - Cube Roots / Fourth Roots, Solve Equations with Rational Exponents (One Solution), Solve Equations with Rational Exponents (Two Solutions), Solve an Equation with a Rational Exponent (x+b)^(c/d)=f (One Solution), Solve an Equation with a Rational Exponent (x+b)^(c/d)=f (Two Solutions), Ex: Solving an Equation with Rational Exponents Using Reciprocal Powers, Solve a Radical Equation Given Function Notation (Extraneous Sol), Radical Equation Application - Vehicle Speed from Skid Mark Length, Application: Evaluate a Square Function (Safe Speed), Application: Solve a Square Root Equation (Year of Obesity Percent), Ex: Radical Equation Application - Obesity Percentage, Ex: Radical Function Application Finding Inputs and Outputs (Speed and Length of Skids), Ex: Radical Function Outputs and Inputs Application - BMI, Ex: Radical Function Outputs and Inputs Application - Pendulum, Ex: Rational Function Outputs and Inputs Application - Average Cost, Kinetic Energy - Radical Equation Application, Volume of a Cone -Radical Equation Application, Volume of a Cone -Radical Equation Application (Special Formula), Ex: Determine a Real, Imaginary, and Complex Number, Simplify Square Roots to Imaginary Numbers, Write Number in the Form of Complex Numbers, Ex : Simplify Imaginary and Complex Numbers, Ex: Simplify, Add, and Subtract Imaginary and Complex Numbers, Ex 1: Adding and Subtracting Complex Numbers, Ex 2: Adding and Subtracting Complex Numbers, Ex 1: Simplify and Multiply Complex Numbers, Ex: Subtract and Multiply Complex Number, Simplifying Powers of i (Method of Dividing by 4), Ex: Raising the imaginary unit i to powers, Ex: Determine if a Triangle is a Right Triangle Given the Length of 3 Sides, Ex: Determine the Length of the Hypotenuse of a Right Triangle, Ex: Determine the Length of the Leg of a Right Triangle, Ex: Pythagorean Theorem App - Find the Width of a Laptop, Ex: Determine the Distance Between Two Points Using the Pythagorean Theorem, Ex: Find the Shortest Distance Between Two Locations North and West, Use The Pythagorean Theorem to Determine the Diagonal of a TV, Distance Formula App: Find the Closest Boat, Ex: Find the Endpoint of a Segment Given the Midpoint and One Endpoint, Ex: Determine the Distance Between Two Points (length of segment), Ex: Domain and Range of Square Root Functions, Ex: Domain and Range of Radical Functions, Ex: Domain of a Square Root Function with a Quadratic Radicand, Horizontal and Vertical Shifts of the Square Root Function, Horizontal and Vertical Stretches and Compressions of the Square Root Function, Ex: Match the Graph of a Horizontal and Vertical Shifted Graph to the Function, Ex: Match the Graph of a Reflected or Horizontally Compressed or Stretched Graph to a Function, Function Transformation Summary - The Square Root Function, Ex 1: Find the Equation of a Transformed Square Root Function from the Graph, Ex 2: Find the Equation of a Transformed Square Root Function From the Graph, Ex 3: Find the Equation of a Transformed Square Root Function From a Graph, Ex 4: Find the Equation of a Transformed Square Root Function From a Graph, Ex 1: Graphing a Transformation of the Square Root Function, Ex 2: Graphing a Transformation of the Square Root Function, Simplify and Give the Domain of Rational Expressions, Simplify Rational Expressions: (linear/quad) and (quad/quad), Ex 1: Simplifying Rational Expressions – Monomials, Ex: Find Values Where a Rational Expression is Undefined, Multiply Rational Expressions and Give the Domain, Divide Rational Expressions and Give the Domain, Ex 1: Multiply Rational Expressions – Monomials, Multiply Basic Rational Expressions: (ax/by)*(cy/dz) and (x/a)*(b/(x-c)), Multiply Rational Expressions: (quad/quad)*(quad/quad) a Not 1, Ex 1: Dividing Rational Expressions – Monomials, Divide Basic Rational Expressions: (Monomials and Linear Factors), Divide Rational Expressions: (quad/linear)/(quad/quad) a Not 1, Ex 1: Simplify a Complex Fraction (No Variables), Ex 2: Simplify a Complex Fraction (Variables), Ex 3: Simplify a Complex Fraction (Variables), Ex 4: Simplify a Complex Fraction (Variables), Ex 5: Simplify a Complex Fraction (Variables), Ex 6: Simplify a Complex Fraction (Variables), Ex 7: Simplify a Complex Fraction (Variables), Ex 1: Simplify a Complex Fraction with Variables (Basic), Ex 2: Simplify a Complex Fraction with Variables (Basic), Ex: Simplify a Complex Fraction with Variables (Factoring), Ex: Simplify a Complex Fraction with Addition and Subtraction and Constant Denominators, Ex: Simplify a Complex Fraction Subtraction and Variable Denominators, Ex: Simplify a Complex Fraction Subtraction and Variable Denominators with Factoring, Ex: Simplify a Complex Fraction with Addition and Constant and Variable Denominators, Ex: Simplify a Complex Fraction with Addition and Subtraction and Binomial Denominators, Complex Fraction Application: Simplify a Resistance Formula, Add and Subtract Rational Expressions with Like Denominators and Give the Domain, Add or Subtract Basic Rational Expressions with Like Denominators: (a and x+b), Add or Subtract Basic Rational Expressions with Like Denominators: (ax and bx^2), Add or Subtract Rational Expressions with Like Denominators: (x+a) and (x^2-bx+c), Ex 1: Add and Subtract Rational Expressions - Like Denominators, Ex 2: Add and Subtract Rational Expressions - Like Denominators, Ex 3: Add and Subtract Rational Expressions - Like Denominators, Add or Subtract Basic Rational Expressions with Unlike Denominators, Ex: Add and Subtract Rational Expressions - Opposite Denominators, Add Rational Expressions with Unlike Denominators and Give the Domain (Mono Denom), Subtract Rational Expressions with Unlike Denominators and Give the Domain, Subtract Rational Expressions with Unlike Denominators - 3 Expressions, Add and Subtract Rational Expressions with Unlike Denominators - 3 Expressions, Ex 1: Add and Subtract Rational Expressions - Unlike Denominators, Ex 2: Add and Subtract Rational Expressions - Unlike Denominators, Ex 3: Add and Subtract Rational Expressions - Unlike Denominators, Ex 4: Add and Subtract Rational Expressions - Unlike Denominators, Ex: Add and Subtract Rational Expressions with Unlike Monomial Denominators, Ex: Add Rational Expressions with Unlike Denominators, Ex: Subtract Rational Expressions with Unlike Denominators, Ex: Subtracting Rational Expressions (Factoring With A Not 1), Function Arithmetic (Add): f(x)+g(x) with Rational Functions, Function Arithmetic (Subtract): f(x)-g(x) with Rational Functions, Solve Rational Equations with Like Denominators, Solve a Rational Equation: (x-a)/x-b/c=d/x, Solve a Rational Equation: a/(x+b)=-c/(x+d) - No Cross Products, Ex 1: Solve Rational Equations Algebraically by Clearing Fractions and Solve Graphically, Ex 2: Solve Rational Equations Algebraically by Clearing Fractions and Solve Graphically, Ex: A Rational Equation with No Solution, Ex 1: Solve Equations with Fractions (Alternative Method), Ex 2: Solve Equations with Fractions (Alternative Method), Ex 1: Solve a Rational Equation (Alternative Method), Ex 2: Solve a Rational Equation (Alternative Method), Ex: Solve a Rational Equation - Alternative Method, Solve A Rational Equation (No Solution) x/(3x-9)-6=1/(x-3), Given f(x)=3/x-4/(3x), Solve f(x)=-7 (Rational Equation), Ex: Rational Function Outputs and Inputs Application - Time, Distance, Rate, Ex 1: Rational Equation Application - Painting Together, Ex 2: Rational Equation Application - Fill a Pool with Drain Open, Ex 3: Rational Equation Application - Plane and Car Travelling the Same Time, Ex 4: Rational Equation Application - Two Bikers Riding Different Distances, Ex: Rational Equation App - Find Individual Working Time Given Time Working Together, Ex: Rational Equation App - Find a Number Given the Sum of Reciprocals, Ex: Find How Faster Than the Speed Limit Is Needed Given a Travel Time, Rational Equation Application (Quadratic): Wind Speed, Rational Equation App (Quadratic): Find Individual Time Given Together Time, Rational Function Application: Function Value, Equation, End Behavior, Direct Variation Application: Currency Conversion, Ex: Direct Variation Application - Aluminum Can Usage, Inverse Variation Application: Water Temperature and Depth, Ex 2: Inverse Variation - Change of Variables, Ex 3: Inverse Variation - Fractional Variation Constant, Ex: Inverse Variation Application - Number of Workers and Job Time, Ex: Inverse Variation Application - Loudness and Distance, Joint Variation: Determine the Variation Constant (Volume of a Cone), Ex: Determine Rational Function Outputs and Inputs, Determining Vertical and Horizontal Asymptotes of Rational Functions, Determining Slant Asymptotes of Rational Functions, Ex 1: Determine Asymptotes and Graph a Rational Function, Ex 2: Determine Asymptotes and Graph a Rational Function, Ex 3: Determine Asymptotes and Graph a Rational Function, Ex 4: Determine Asymptotes and Graph a Rational Function (Slant), Ex: Determine Horizontal Asymptotes of Rational Functions, Ex: Find the Intercepts and Asymptotes of a Rational Function, Ex: Find the Intercepts, Asymptotes, and Hole of a Rational Function, Ex: Find a Rational Function Given the Vertical Asymptotes and Intercepts, Ex 1: Determine the Vertical and Slant Asymptotes of a Rational Function, Ex 2: Determine the Vertical and Slant Asymptotes of a Rational Function, Ex 1: Domain, Range, Asymptotes of a Basic Rational Function Using Translations, Ex 2: Domain, Range, Asymptotes of a Basic Rational Function Using Translations, Ex 1: Domain, Range, Asymptotes of a Basic Rational Function Using a Graph and Procedure, Ex 2: Domain, Range, Asymptotes of a Basic Rational Function Using a Graph and Procedure, Ex: Match Equations of Rational Functions to Graphs, Find the Intercepts and Asymptotes of a Rational Function (quad/quad with a not 1), Determine the Domain of Various Functions, Determine Vertical Intercepts of Various Functions, Determine Horizontal Intercepts of Various Functions (P1), Determine Horizontal Intercepts of Various Functions (P2), Determine Domain, Holes, and Asymptotes of a Rational Function (1), Determine Domain, Holes, and Asymptotes of a Rational Function (2), The Equation of a Rational Function from a Graph, Graph and Determine Key Components of a Rational Function (linear/linear) 1, Determine Key Components and Graph a Rational Function (linear/linear) 2, Match Rational Functions and Graphs: Translations, Graphing the Basic Rational Function f(x)=1/x, Graphing Reflections of the Basic Rational Function f(x)=1/x, Graphing Translations of the Basic Rational Function f(x)=1/x, Ex 1: Graph Two Translations of the Basic Rational Function f(x)=1/x, Ex 2: Graph Two Translations of the Basic Rational Function f(x)=1/x, Ex 1: Find the Equation of Rational Function From a Graph with a Hole, Ex 2: Find the Equation of Rational Function From a Graph with a Hole, Ex 3: Find the Equation of Rational Function From a Graph, Ex 4: Find the Equation of Rational Function From a Graph (Squared Intercept), Ex 5: Find the Equation of Rational Function From a Graph (Squared VA), Ex 6: Find the Equation of Rational Function From a Graph (Squared Intercept / VA), Rational Function Application - Concentration of a Mixture, Ex: Setting Up Partial Fraction Decomposition, Ex 1: Partial Fraction Decomposition (Linear Factors), Ex 2: Partial Fraction Decomposition (Linear Factors), Ex 3: Partial Fraction Decomposition (Repeated Linear Factors), Ex 4: Partial Fraction Decomposition (Repeated Linear Factors), Ex 5: Partial Fraction Decomposition (Linear and Quadratic Factors), Ex 6: Partial Fraction Decomposition (Repeating Quadratic Factors), Ex: Partial Fraction Decomposition - Degree 2 / Degree 3, Ex: Evaluate Functions and Composite Function in Context of a Story (Graphing Calculator), Ex: Intro Composite Function Notation Application Problem, Ex: Evaluate Composite Functions Using Tables of Values, Ex: Evaluate Composite Functions from Graphs, Ex 1: Determine Composite Function Values Using Table, Graph, and Function Rule, Ex 2: Determine Composite Function Values Using Table, Graph, and Function Rule, Determine f(g(x)) and g(f(x)) with Linear and Quadratic Functions, Determine f(f(x)) and g(g(x)) with Linear and Quadratic Functions, Ex: Find and Evaluate a Composition of Three Functions, Ex 2: Find Composite Function Values With Fractions, Ex: Find a Composition of Functions Involving Rational Functions, Ex: Domain of a Quotient and Composite Functions, Ex: Domain of Composite Function From Graphs, Ex: Inverse Function Notation and Reciprocal of a Function, Linear and Exponential Growth: Complete a Salary Table, Graphing by Plotting Points - Exponential, Graphing by Plotting Points - Exponential (6.3), Complete a Table of Values for an Exponential Equation in Two Variables, Evaluate Exponential Functions: Base 3 and 1/3, Write Exponential Equations Given Initial Values and Growth or Decay Rate, Write Exponential or Linear Equations to Model the Value of an Investment (Level 1), Write Exponential or Linear Equations to Model a Population (Level 2), Determine Exponential Function Values and Graph the Function, Graph a Basic Exponential Function Using a Table of Values, Graph an Exponential Function Using a Table of Values, Evaluate a Given Exponential Function to Predict a Future Population, Introduction to Exponential Equations in Two Variables, Determine the Initial Value and Percent Rate of Change from an Exponential Equation, Write an Exponential Equation that Models a Decreasing Population (Fox), Write an Exponential Equation to Model Wage Percent Increase over Years, Write an Exponential Equation to Model an Account Balance Over Years, Write an Exponential Equation to Model World Population Growth, Determine if Equations Are Linear or Exponential and Increasing or Decreasing, Interpret an Exponential Equation Modeling Depreciation, Interpret an Exponential Equation Modeling Rising Home Value, Compare Exponential Equations Modeling Account Values, Introduction to Exponential Functions in the Form f(x)=ab^x - Part 1, Introduction to Exponential Functions in the Form f(x)=ab^x - Part 2, Introduction to Exponential Functions in the Form f(x)=ae^(kx) - Part 1, Introduction to Exponential Functions in the Form f(x)=ae^(kx) - Part 2, Comparing Forms of Exponential Functions: y = ab^x and y = ae^(kx), Graphing Basic Exponential Functions: Growth and Decay, Interpret the Meaning of Ordered Pairs from a Graph (Exponential), Ex: Determine Exponential Graphs that Have Specific Characteristics - y = ab^x, Ex: Match Exponential Functions to Graphs, Match Exponential Growth and Decay Function with Graphs (Reflections), Describe an Exponential Function Transformation: y=e^(x)+3, Describe an Exponential Function Transformation: y=-2^x-1, Ex: Exponential Application Solved Using a Graphing Calculator, Determine if a Table Represents a Linear or Exponential Function, Ex: End (Long Run) Behavior of Exponential Functions, Ex: Find the Equation of a Transformed Exponential Function From a Graph, Ex: Match the Graphs of Translated Exponential Function to Equations, Ex: Match the Graphs of Reflected Exponential Functions to Equations, Ex 1: Determine if a Table of Value Represents a Linear or Exponential Function, Ex 2: Determine if a Table of Value Represents a Linear or Exponential Function, Ex 1: Determine if a Table of Value Represents a Linear or Exponential Function (Fractions/Decimals), Ex 2: Determine if a Table of Value Represents a Linear or Exponential Function (Fractions/Decimals), Ex 1: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Linear), Ex 2: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Linear), Ex 1: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Exponential), Ex 2: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Exponential), Determine if Two Linear Functions Are Inverses (1), Determine if Two Linear Functions Are Inverses (2), Determine if a Relation Given as a Table is a One-to-One Function, Ex 1: Determine if the Graph of a Relation is a One-to-One Function, Ex 2: Determine if the Graph of a Relation is a One-to-One Function, Ex 1: Determine if Two Functions Are Inverses, Ex 2: Determine if Two Functions Are Inverses, Ex: Find an Inverse Function From a Table, Ex: Function and Inverse Function Values Using a Table, Ex: Function and Inverse Function Values Using a Graph, Ex: Restrict the Domain to Make a Function 1 to 1, Then Find the Inverse, Ex: Find Inverse Function Values Without Finding the Inverse Function, Ex: Function and Inverse Function Values, Ex: Find the Inverse of a Square Root Function with Domain and Range, Ex: Find the Inverse of a Rational Function, Ex: Find the Inverse of a Rational Function and an Inverse Function Value, Ex 1: Determine If Two Functions Are Inverses, Ex 2: Determine If Two Functions Are Inverses, Ex: Find the Inverse Function of an Exponential Function, Ex: Write Exponential Equations as Logarithmic Equations, Ex: Write Logarithmic Equations as Exponential Equations, Ex: Write Exponential Equations as Logarithmic Equations - Variables, Ex: Write Logarithmic Equations As Exponential Equations - Variables, Ex: Write Exponential Equations with base 10 as Common Logarithmic Equations, Ex: Write Exponential Equations as Logarithmic Equations - Natural Logarithms, Use the Definition of a Logarithm to Show the Zero Exponent and Identity Property, Ex 1: Evaluate Logarithms Without a Calculator - Whole Numbers, Ex: Evaluate Logarithmic Expressions without a Calculator - Different Bases, Ex: Evaluate Logarithmic Expressions without a Calculator - Common Log, Ex: Solve a Basic Exponential Equation with Base Ten Using Logarithm Definition, Ex: Solve a Exponential Equation with Base Ten Using Logarithm Definition (Multiple Steps), Ex: Solve a Basic Exponential Equation with Base e Using Logarithm Definition, Ex: Solve a Exponential Equation with Base e Using Logarithm Definition (Multiple Steps), Ex 2: Evaluate Logarithms Without a Calculator - Fractions, Ex: Evaluate Common Logarithms on a Calculator, Ex: Evaluate Common Logarithms Without a Calculator, Ex: Evaluate Common Logarithms on the Calculator, Ex: Evaluate Natural Logarithms on the Calculator, Ex: Determine the Value of a Number on a Logarithmic Scale (Log Form), Ex: Determine the Value of a Number on a Logarithmic Scale (Exponential Form), Ex: Determine the Difference in Order of Magnitude to Two Quantities, Ex: Determine the Difference in Order of Magnitude to Two Quantities (Application), Ex: Graph an Exponential Function and Logarithmic Function, Ex: Properties and Characteristics of a Logarithmic Function, Ex 1: Match Graphs with Exponential and Logarithmic Functions, Ex 2: Match Graphs with Exponential and Logarithmic Functions - Base 10 and e, Ex: Vertical Asymptotes and Domain of Logarithmic Functions, Ex: Find the Domain of Logarithmic Functions, Determine the Domain, Range, and Asymptote of a Log Function y=-ln(x-6), Determine the Domain, Range, and Asymptote of a Log Function y=-log_3(x)+4, Graph Exponential and Logarithmic Functions on the TI-84 (Inverses), Graphing Logarithmic Functions Using Desmos.com, Graphing Log Functions on the TI-84: y=log_(1/2)(x), Graphing Log Functions on the TI-84: y=-log_(3)(x), Graphing Log Functions on the TI-84: y=log_(2)(x-3)+2, Graphing Log Functions on the TI-84: y=2log_(3)(x+1)-3, Graphing Log Functions by Hand: y=log_(1/2)(x), Graphing Log Functions by Hand: y=-log_(3)(x), Graphing Log Functions by Hand: y=log_(2)(x-3)+2, Graphing Log Functions by Hand: y=2log_(3)(x+1)-3, Graphing Log Functions by Hand: y=log_4(x), Graphing Log Functions by Hand: y=log_3(x)+2, Graphing Basic Logarithmic Functions Using Desmos, Expand Logarithms Using the Product Rule for Logs, Expand Logarithms Using Properties of Logarithms Rule and Factoring, Expand Logarithms Using Properties of Logarithms (Expressions), Ex: Expand a Logarithm Containing a Radical, Combine Logarithms Using Properties of Logarithms, Ex: Combine a Sum and Difference of Two Logarithms, Ex 1: Combine a Logarithmic Expression Into One Logarithm, Ex 2: Combine a Logarithmic Expression Into One Logarithm, Ex 1: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 2: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 3: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 4: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 5: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Using the Inverse Property of Logarithms and Exponentials to Evaluate Expressions, Ex 1: Evaluate a Logarithmic Expression Using the Properties of Logarithms, Ex 2: Evaluate a Logarithmic Expression Using the Properties of Logarithms, Ex: Simplify Log Expression with the Base and Base of the Number are the Same, Ex: Evaluate Exponential Expressions with Logarithmic Exponents, Ex: Change of Base Formula to Evaluate Logarithmic Expressions, Ex: Change of Base Formula to Solve Basic Exponential Equations, Ex: Evaluate Logarithmic Functions Using the Change of Base Formula, Evaluate Logarithms of any Base on the TI-84 (Not using Change of Base), Determine the Human Age of a Dog And the Dog Age, Given the Human Age, Solve Logarithmic Equations for the Base (No Calculator), Solve a Log Equation with Two Logs Equal to Each Other, Ex: Solve a Logarithmic Equations - One Step by Dividing with Rational Solution, Ex: Solve a Logarithmic Equations - One Step by Dividing with Irrational Solution, Ex: Solve a Logarithmic Equations - One Step by Dividing with Log of a Quantity, Ex: Solving a Variety of Logarithmic Equations, Ex: Solve a Basic Logarithmic Equation - Linear and Quadratic, Ex: Solve Basic Logarithmic Equation - Radicals, Ex: Solve Logarithmic Equations Containing Only Logarithms, Ex: Solve a Logarithmic Equation With a Difference - Linear, Ex: Solve a Logarithmic Equation With a Sum - Quadratic Formula, Ex: Solve a Logarithmic Equation With a Difference - Quadratic Formula, Ex: Solve a Logarithmic Equation Requiring the Quadratic Formula, Ex 3: Solve Logarithmic Equations - Base and Number are the Same, Ex: Solve a Logarithmic Equation - Composite Log Expression, Ex: Solve a Logarithmic Equation in Terms of Other Variables, Ex: Solve a Logarithmic Equation with a Fractional Exponent, Solve a Log Equation by Factoring (Trinomial), Logarithm Application: Magnitude of an Earthquake, Logarithm Application: Intensity of Two Sounds (Decibels), Ex: Logarithmic Function Application - Test Scores, Ex: Exponential Function Application with Logarithms, Ex: Logarithmic Function Application - pH (Outputs and Inputs), Ex: Logarithmic Function Application - Preston Curve (Outputs and Inputs), Solving Exponential Equations by Obtaining a Common Base, Ex 1: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 2: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 3: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 4: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 5: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 6: Solve Exponential Equations Using Like Bases - No Logarithms, Ex: Solve an Exponential Equation Graphically on the TI84, Solving Exponential Equations Using Logarithms, Ex 1: Solve a Basic Exponential Equation Using the Definition of a Logarithm, Ex 2: Solve a Basic Exponential Equation Using the Definition of a Logarithm, Ex 1: Solve Exponential Equations Using Logarithms, Ex 2: Solve Exponential Equations Using Logarithms, Ex 3: Solve Exponential Equations Using Logarithms, Ex 4: Solve Exponential Equations Using Logarithms, Ex 5: Solve Exponential Equations with Two Exponential Parts Using Logarithms, Ex 6: Solve an Exponential Equation Using Common Log, Ex 7: Solve an Exponential Equation Using Natural Log (Factoring), Ex: Solve an Exponential Equation Using Logarithms, Ex: Solve an Exponential Equation with Logarithms - Variable on Both Sides, Ex: Rewrite Exponential Functions: y = ab^t to y = ae^(kt), Ex: Rewrite Exponential Functions: y = ae^(kt) to y = ab^t, Determine a Continuous Exponential Decay Function and Make a Prediction, Ex: Write Linear and Exponential Functions Based Upon Given Information, Write an Exponential Function for Growth over Different Time Intervals, Ex: Find a Linear and Exponential Model for Population Growth, Ex: Use a Given Exponential Growth Function, Ex: Exponential Growth of Bacteria (Intro Question), Ex: Solve an Exponential Growth Equation Graphically Using the TI84 (Application), Ex: Compare Simple Interest and Annual Compounded Interest, Ex: Solve an Exponential Growth Equation Graphically Using the Desmos (Application), Ex: Solve an Exponential Decay Equation Graphically Using the TI84 (Application), Ex: Solve an Exponential Decay Equation Graphically Using the Desmos (Application), Ex: Determine the Doubling Time of an Investment Account Graphically (TI84), Exponential Growth App (y=ab^t) - Given Doubling Time, Ex: Determine the Half-Life of a Fish Population Graphically (TI84), Ex: Exponential Function Applications - Increasing Investment Value (Change of Base Used), Ex: Exponential Function Applications - Decreasing Water Level (Change of Base Used), Ex: Exponential Growth Function - Population, Ex: Exponential Function Application Using Logs - Export Values, Ex: Exponential Growth Function - Bacterial Growth, Ex: Exponential Decay Function with Logarithms, Ex: Exponential Growth Application - Predicting World Population, Ex: Basic Example of Exponential Decay Model, Ex: Exponential Decay Function - Half Life, Ex: Find the Initial Value and Exponential Growth or Decay Rate Given an Exponential Function, Ex: Exponential Functions: Growth Rate and Growth Factor, Ex: Exponential Functions: Decay Rate and Decay Factor, Ex: Determine Growth Exponential Functions Given Growth Rate and Initial Value (y=ab^x), Ex: Determine Exponential Decay Functions Given Decay Rate and Initial Value (y=ab^x), Exponential Decay App (y=ab^x) - Given Half Life, Exponential Decay App (y=ae^(kt)) - Given Half Life, Exponential Decay App (y=ab^t) - Find Initial Amount Given Half Life, Exponential Decay App with Logs (y=ae^(kt)) - Find Half Life, Ex: Exponential Model - Determine Age Using Carbon-14 Given Half Life, Find Initial Amount and Amount After Time From Half-Life (y=ae^(kt)), Exponential Growth App (y=ab^t) - Find Initial Amount Given Doubling Time, Ex: Practice Writing Exponential Equations - Doubling Equation and Halving Equation, Exponential Growth App with Logs (y=ae^(kt)) - Find Initial Amount Given Doubling Time, Ex: Find an Exponential Growth Function Given Two Points - Initial Value Given, Ex: Find an Exponential Decay Function Given Two Points - Initial Value Given, Ex: Find an Exponential Function Given Two Points - Initial Value Not Given, Exponential Function Application (y=ab^x) - Population Growth of India, Exponential Function Application (y=ab^x) - Population Decline of Chicago, Exponential Function Application (y=ab^x) - Depreciation of a Car, Annual Depreciation of a New Car: Find the Future Value, Exponential Function Application (y=ab^x) - Declining Computer Value, Determine an Exponential Decay Function P(t)=a(b)^t (No Logs), Exponential Function Application (y=ab^x) - Home Values, Exponential Function Application (y=ae^(kt)) - Bacteria Growth, Ex: Application - Cyclical Around Linear Growth, Ex: Application - Cyclical Around Exponential Growth, Ex: Find Annual Interest Rate Given f(t)=ae^(kt), Ex: Find Annual Depreciation Rate Given f(t)=ae^(kt), Ex: Find Continuous Interest Rate Given f(t)=ab^t, Ex: Find Continuous Depreciation Rate Given f(t)=ab^t, Determine an Exponential Function on the TI-84 Given Two Points (Growth), Determine an Exponential Function on the TI-84 Given Two Points (Decay), Determine the Total Return of an Investment as Percent, Compounded Interest: Solve for Principal (Present Value), Ex 2: Compounded Interest with Logarithms, Ex: Find Exponential Growth Rate and Make Prediction of a Future Home Value, Compounded Interest (Slightly Different Form of Equation): Find Future Value, Effective Interest Rate (Effective Yield), Ex 2: Continuous Interest with Logarithms, Ex 3: Continuous Interest with Logarithms and Doubling Time, Continuous Interest - Find the Initial Investment Needed, Ex: Perform Exponential Regression on a Graphing Calculator, Perform Exponential Regression and Make Predictions Using Desmos, Ex: Exponential Regression Application on the TI84 (Decreasing Polio Case), Ex: Exponential Decay Regression Model (Declining Population), Ex: Exponential Growth Regression Model (Investment Account), Ex: Comparing Linear and Exponential Regression, Ex: Find an Exponential Function for a Semi-Log Graph, Ex: Newton's Law of Cooling - Exponential Function App, Exponential Function App. Pyrex® Red Lid For 8" Square Glass Baking Dish,
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= 0.0, but there's still several extra instructions in the fast path. As for its speed, it at least outperforms the std::sin() function by an average of 0.3 microseconds per call. If you're writing audio plugins or doing digital signal processing, Chebyshev polynomials give you a cheap and predictable dithering effect "for free.". An approximation for the sine function that preserves the derivatives at multiples of 90 degrees is given by this formula. How about dectecting the CPU and using a native instruction on a modern processor with a lookup table or other optimized code on older machines. My custom x86 Math library code is for standard MSVC++ 2005 and forward. The cosine of 0 is well-defined, and is 1. gcc's code for the NaN case seems way over-complicated; it doesn't even use the sqrtf return value! Lastly, when you've finished all your fancy benchmarking and micro-optimization, make sure that your "fast" version is actually faster than the library version. The, Unless you bottleneck on uop throughput rather than sqrt latency, in which case using plain, What is the purpose of taking the address of. Also, I've benchmarked it and it twice slower than milianw answers. Here's a possible speedup which depends highly on your application. That's strictly bush league. Check this one for example . Making statements based on opinion; back them up with references or personal experience. This is a delightful representation of the Taylor Series. What makes an argument objectively more "compelling"? @roliu My mistake, I though I have to calculate Factorial several times, but I missed that I can use precomputed constant. (Negative a), Graphing and Writing Equations of Hyperbolas, Conic Sections: The Hyperbola part 1 of 2 Conic Sections: The Hyperbola part 2 of 2 Ex 1: Conic Section - Graph a Hyperbola with Center at the Origin (Horizontal) Ex 2: Conic Section - Graph a Hyperbola with Center at the Origin (Vertical) Ex 3: Conic Section - Graph a Hyperbola with Center NOT at the Origin (Horizontal) Ex 4: Conic Section - Graph a Hyperbola with Center NOT at the Origin (Vertical) Ex: Find the Equation of a Hyperbola Given the Center, Focus, and Vertex Determining the Type of Conic Section from General Form, Introduction to Set Theory Introduction to Subsets Set Operations and Venn Diagrams - Part 1 of 2 Set Operations and Venn Diagrams - Part 2 of 2 Solving Problems Using Venn Diagrams, Introduction to Regression Analysis Linear Regression – Example 1, Example 2 Quadratic Regression – Example 1, Example 2 Perform Quadratic Regression and Make Predictions Using Desmos Interpret a Quadratic Function Model: Fuel Consumption Ex 1: Cubic Regression on the TI84 – Natural Gas Consumption Ex 2: Cubic Regression on the TI84 - Total Sales Exponential Regression – Example 1, Example 2 Ex: Exponential Decay Regression Model (Declining Population) Ex: Exponential Growth Regression Model (Investment Account) Logarithmic Regression Logistic Regression, Financial Mathematics (See more under Math for Liberal Arts library), Simple Interest Formula Compounded and Continuous Interest Effective Yield Effective Yield on the TI84 Derive the Value of an Annuity Formula (Compounded Interest) Determining the Value of an Annuity Determining the Value of an Annuity Using the TI84 Determining the Monthly Saving Required to Reach a Financial Goal Determining the Monthly Saving Required to Reach a Financial Goal on the TI84 Ex 1: Find a Monthly Mortgage Payment with a Down Payment Ex 2: Find a Monthly Mortgage Payment with a Down Payment and Points Ex: Comparing Two Installments Loans (Car Loans) Ex: Simple Interest Discounted Loan Determining the Monthly Payments for a Loan Determining the Monthly Payments for a Loan on the TI84, Graphing Calculator Basics: Evaluating Expressions and Determining Function Values, Graphing Calculator Basics The Table Feature of the Graphing Calculator Evaluating Radical Expressions on the TI83/84 Determine Function Values Using Function Notation on the TI84, Graphing Functions and Determining Key Components of Functions, Graphing Lines on the Graphing Calculator Determining the Intersection of Two Graph on the TI83/84 Determining Relative Extrema on the Graphing Calculator Determine Function Values Using Function Notation on the TI84 Determine the value of the derivative function on the graphing calculator Determining the value of a definite integral on the graphing calculator Determining the Intersection of Two Graph on the TI83/84 Determining the Zeros or Roots of a Polynomial Function on the TI83/84 From a Graph Determine Where a Quadratic Function is Increasing and Decreasing Determining When a Polynomial Function is Increasing and Decreasing Determine Max/Mins and Incr/Decr Intervals Using a Free Online Graph Calc (MathAS) Determining the Zeros or Roots of a Polynomial Function on the TI83/84 Ex: Quadratic Function Application Using a Graphing Calculator - Rocket Launch Ex 1: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 2: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 1: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 2: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Solving Equations on the Graphing Calculator, Solving Linear Equations Graphically Ex: Solve a Linear Equation in One Variable Graphically using the TI84 Ex: Solve a Linear Inequality in One Variable Graphically using the TI84 Ex: Solving Absolute Value Equations on the Graphing Calculator Ex 1: Solve a Quadratic Equation Graphically on Calculator Ex 2: Solve a Quadratic Equation Graphically on Calculator Ex: Determine How a Final Exam Score Affect a Course Grade using the TI84 (Weighted Averages) Solving Polynomial Equations Graphically, Ex: Evaluate Common Logarithms on a Calculator Ex: Evaluate Common Logarithms on the Calculator Ex: Evaluate Natural Logarithms on the Calculator Ex: Change of Base Formula to Evaluate Logarithmic Expressions Ex: Change of Base Formula to Solve Basic Exponential Equations Ex: Evaluate Logarithmic Functions Using the Change of Base Formula Ex: Solve an Exponential Equation Graphically on the TI84 Ex: Perform Exponential Regression on a Graphing Calculator Ex: Comparing Linear and Exponential Regression Sequences and Series on the TI84 Sequences and Series on the TI84, Augmented Matrices on the Graphing Calculator Matrix Multiplication on the Graphing Calculator Inverse Matrices on the Graphing Calculator Ex 1: Solve a System of Two Equations Using a Matrix Equation Ex 2: Solve a System of Two Equations Using a Matrix Equation Ex: Solve a System of Three Equations Using a Matrix Equation Determinants on the Graphing Calculator Ex: Solve a System of Three Equations Using Cramer's Rule, Ex 1: Create a Scatter Plot and then Perform Linear Regression on the Calculator Ex 2: Creating a Scatter Plot and Performing Linear Regression on the Calculator Linear Regression – Example 1, Example 2 Ex: Matching Correlation Coefficients to Scatter Plots Quadratic Regression – Example 1, Example 2 Perform Quadratic Regression and Make Predictions Using Desmos Ex: Quadratic Regression on the TI84 - Stopping Distance Exponential Regression – Example 1, Example 2 Perform Exponential Regression and Make Predictions Using Desmos Logarithmic Regression Logistic Regression Regression and Systems of Equations: Application, Financial Mathematics on the Graphing Calculator, Loan Information on the TI83/84 Effective Yield on the TI84 Determining the Value of an Annuity Using the TI84 Determining the Monthly Payments for a Loan on the TI84 Determining the Monthly Saving Required to Reach a Financial Goal on the TI84, Determine if a Function is a Polynomial Function, Degree, Leading Term, and Leading Coefficient of a Polynomial Function, Ex: Information about a Given Polynomial Function, Turning Points and X Intercepts of a Polynomial Function, Summary of End Behavior or Long Run Behavior of Polynomial Functions, Determine the End Behavior of Power Functions, Ex: End Behavior or Long Run Behavior of Functions, Ex: End Behavior of a Polynomial Function in Factored Form, Ex: Find the Intercepts of a Polynomial Function in Factored Form, Ex: Determine the Least Possible Degree of a Polynomial From the Graph, Ex: Increasing / Decreasing / Relative Extrema from Analyzing a Graph, Analyze a Graph Using Desmos to Determine Key Components of a Quadratic (Incr / Decr / Extrema), Analyze a Graph Using Desmos to Determine Key Components of a Cubic (Incr / Decr / Extrema), Ex 1: Determine the Local / Relative Extrema of a Cubic Function Using Desmos, Ex 2: Determine the Local / Relative Extrema of a Cubic Function Using Desmos (Challenging), Determine the Maximum Volume of an Open Top Box Using a Graph Only, Ex: Concavity / Points of Inflection by Analyzing a Graph (Algebra Topic), Ex: Concavity / Increasing / Decreasing Functions as Tables (Algebra Topic), Ex: Find the Intercepts of a Polynomial Function (Factorable), Ex: Find the Intercepts of a Polynomial Function (Real Zero), Ex 1: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 2: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 3: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 1: Solve a Cubic Function Graphically (One Solution), Ex 2: Solve a Cubic Function Graphically (Two Solutions), Ex 1: Find a Degree 3 Polynomial Function Given Integer Zeros, Ex 2: Find a Degree 3 Polynomial Function Given Fractional Zeros, Ex 3: Find a Degree 3 Polynomial Function Given Imaginary Zeros, Ex 4: Find a Degree 3 Polynomial Function Given Complex Zeros, Ex 1: Find a Polynomial Function Given the Zeros or Roots with Multiplicity and a Point, Ex 2: Find a Polynomial Function Given the Zeros or Roots with Multiplicity and a Point, Find a Polynomial Function Given the Zeros and Leading Coefficient (Degree 3), Find a Polynomial Function Given the Zeros, Multiplicity, and (0,a) (Degree 3), Determine a Degree 3 Polynomial Function Given the Zeros and Intercept (2 complex), Ex 1: Find a Degree 4 Polynomial Function Given Integer and Complex Zeros, Ex 2: Find a Degree 4 Polynomial Function Given Integer and Complex Zeros, Ex 3: Find a Degree 4 Polynomial Function Given Fractional and Complex Zeros, Ex1: Find an Equation of a Degree 4 Polynomial Function From the Graph of the Function, Ex2: Find an Equation of a Degree 5 Polynomial Function From the Graph of the Function, Ex3: Find an Equation of a Degree 6 Polynomial Function From the Graph of the Function, Ex 1: Cubic Regression on the TI84 – Natural Gas Consumption, Ex 2: Cubic Regression on the TI84 - Total Sales, Real Zeros, Factors, and Graphs of Polynomial Functions, Find the Zeros of a Cubic Function Using Factor by Grouping (1 rational, 2 irrational), Find the Zeros of a Cubic Function Using Factor by Grouping (1 rational, 2 complex), Use the Remainder Theorem to Determine if a Binomial is a Factor of a Polynomial, Determine if a Binomial is a Factor of a Polynomial Graphically, Determine the Zeros/Roots and Multiplicity From a Graph of a Polynomial, Polynomial Function - Complex Factorization Theorem, Ex: Factor and Solve a Polynomial Equation, Ex: Solve a Basic Cubic Equation Using a Cube Root and Rational Exponent, Find the Intercepts of a Degree 4 Polynomial Function (Factorable), Find Zeros, Multiplicity, Degree, and End Behavior of a Factored Polynomial (Degree 11), Find Zeros, Multiplicity, Degree, and End Behavior of a Factored Polynomial (Degree 6), Ex 1: Find the Zeros of a Polynomial Function - Integer Zeros, Ex 2: Find the Zeros of a Polynomial Function - Real Rational Zeros, Ex 3: Find the Zeros of a Polynomial Function with Irrational Zeros, Ex 4: Find the Zeros of a Polynomial Function with Imaginary Zeros, Ex 5: Find the Zeros of a Polynomial Function with Complex Zeros, Ex 6: Find the Zeros of a Degree 4 Polynomial Function, Ex 7: Find the Zeros of a Degree 5 Polynomial Function, Ex 1: Write a Degree 3 Polynomial Function as a Product of Linear Factors (2 Imaginary), Ex 2: Write a Degree 3 Polynomial Function as a Product of Linear Factors (2 Irrational), Ex 3: Write a Degree 5 Polynomial Function as a Product of Linear Factors, Ex 1: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 2: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 1: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 2: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Determine Local (Relative) Extrema of a Polynomial Function Using the TI-84, Determine Local (Relative) Extrema of a Polynomial Function Using Desmos, Ex 1: Solve a Polynomial Inequality in Factored Form, Ex 2: Solve a Polynomial Inequality in Factored Form, Ex: Solve a Polynomial Equation Using a Graphing Calculator (Approximate Solutions), Ex: Solve a Polynomial Inequality Using Factor By Grouping (Degree 3), Ex: Solve a Polynomial Inequality Using Factor Using GCF (Degree 3), Ex: Solve a Polynomial Inequality Using Factor of a Trinomial (Degree 4), Ex 1: Determine if a Function is Odd, Even, or Neither, Ex 2: Determine if a Function is Odd, Even, or Neither, Determine if a Function is Even, Odd, or Neither Using a Graph (1), Determine if a Function is Even, Odd, or Neither Using a Graph (2), Introduction to Square Root and Perfect Squares, Approximate a Square Root to Two Decimal Places Using Trial and Error, Simplify Square Roots (Perfect Square Radicands), Simplify a Variety of Square Expressions (Simplify Perfectly), Simplify Square Roots with Variables (perfect squares), Simplify Square Roots of Decimals (Perfect Square Decimals), Simplify Square Roots (Not Perfect Square Radicands), Simplify Square Roots in the Form a*sqrt(b) (not perfect squares), Simplify Cube Roots (Perfect Cube Radicands), Simplify Cube Roots (Not Perfect Cube Radicands), Simplifying Radical Expressions Without Fractions, Simplifying Radical Expressions With Fractions, Order of Operations with a Fraction Containing a Square Root, Ex: Simplify Square Roots - Perfect Roots, Ex: Simplify Perfect Nth Roots - Radicals, Ex: Simplify Square Roots - Not Perfect Roots, Ex: Simplify Square Roots of Variable Expressions - Absolute Value Needed, Ex: Simplify a Radical Expression containing Square Roots in the Numerator and Denominator, Simplify Cube Roots with Variables (not perfect cubes), Ex: Simplify Radicals with the Same Radicand and Different Indexes, Ex: Simplify a Radical Containing a Fraction - Perfect Root, Ex: Simplify Radicals with Variables - Perfect Roots, Ex 1: Simplifying Perfect Nth Roots Containing Variables, Ex 2: Simplifying Perfect Nth Roots Containing Variables, Ex: Simplify Radicals Containing Variables With Large Exponents - Not Perfect Roots, Ex: Simplify Radicals with Negative Radicands and Odd Indexes, Simplify Radicals with Variables (Large Index and Exponents), Ex: Simplify Radicals with Variables - Not Perfect Roots, Evaluating Radical Expressions on the TI83/84, Approximating Square Roots Using Division (No Calculator Required), Multiplying Radical Expressions with Variables Using Distribution, Multiplying Binomial Radical Expressions with Variables, Multiplying Radicals Containing Variables, Multiplying Conjugates of Radical Expressions, Multiply Two Radicals with Variables (Index 4 and 5) Perfect Roots, Ex: Multiply Radical Conjugates - Square Roots, Square a Binomial with a Square Root of a Variable Expression, Multiply two Binomial Radical Expressions with Variables (Conjugates), Dividing Radicals without Variables (Basic with no rationalizing), Dividing Radicals with Variables (Basic with no rationalizing), Simplify the Square Root of Fraction: sqrt(a/b), a/sqrt(b), Rationalize the Denominator - Square Root with Variable, Rationalize the Denominator - Cube Root and 4th Root, Ex 1: Rationalize The Denominator of a Fraction (Basic), Ex 2: Rationalize The Denominator of a Fraction, Ex 3: Rationalize The Denominator of a Fraction, Ex 1: Rationalize the Denominator of a Radical Expression, Ex 2: Rationalize the Denominator of a Radical Expression, Ex: Rationalize the Denominator of a Radical Expression - Conjugate, Rationalize the Denominator - Conjugate with Variables, Ex: Write a Radical in Rational Exponent Form, Write Basic Expression in Radical Form and Using Rational Exponents, Write Expressions Using Radicals and Rational Exponents, Write Rational Exponents as Radicals and Radicals Using Rational Exponents (Variables), Simplify Radicals Using Rational Exponents, Ex: Simplify Exponential Expressions with Fraction Exponents (Power Property of Exponents), Ex: Simplify Exponential Expressions with Fraction Exponents (Quotient Property of Exponents), Ex: Simplify Expressions with Rational Exponents, Ex: Simplify an Expression with Rational Exponents and Write in Radical Form, Ex: Simplify an Expression with Negative Rational Exponents and Write in Radical Form, Ex 1: Simplify an Expression with a Negative Rational Exponent, Ex 2: Simplify an Expression with a Negative Rational Exponent, Simplify An Expression with Rational Exponents (Positive Only) Power/Quot, Simplify An Expression with Rational Exponents (Negative) Power/Quot, Simplify An Expression with Rational Exponents (Negative) Prod/Quot, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 1, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 2, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 3, TI-84: Comparing Radical Form and Rational Exponent Form, Ex: Multiply and Divide Radicals with Different Indexes Using Rational Exponents - Same Radicand, Ex: Multiply Radicals with Different Indexes Using Rational Exponents - Different Radicand, Ex: Evaluate an Expression with Rational Exponents Using Radicals, Simplify a Quotient with A Radical (Rational Exponents and Radical Form), Adding Radicals (Basic With No Simplifying), Adding Radicals That Requires Simplifying, Simplify and Add Square Roots: sqrt(a^2*c)+sqrt(b^2*c), Subtracting Radicals (Basic With No Simplifying), Subtracting Radicals That Requires Simplifying, Ex: Evaluate the Square Root of a Sum and the Sum of Square Roots on a Calculator, Ex: Add and Subtract Square Roots - No Simplifying, Ex: Add and Subtract Square Roots Containing Variables, Subtract Square Expressions with a Variable - Simplifying Required, Add Square Root Expressions with Variables, Add and Subtract Square Root Expressions with Variables (Adv), Solving Radical Equations with One Radical, Solve a Radical Equation with One Cube Root (Fraction Answer), Solve a Radical Equation with One Square Root (Fraction Answer), Solve a Basic Radical Equation Given Function Notation, Solve a Radical Equation Given Function Notation, Solving Radical Equations with Two Radicals, Ex 1: Solve a Basic Radical Equation - Square Roots, Ex 2: Solve Radical Equations - Square Roots, Ex 3: Solve Radical Equations - Square Roots, Ex 4: Solve Radical Equations - Square Roots, Ex 5: Solve Radical Equations - Two Square Roots, Ex 6: Solve Radical Equations - Two Square Roots, Ex 7: Solve Radical Equations - Two Square Roots, Ex: Solve a Radical Equation with One Radical: Factoring and Extraneous Solution, Ex: Solve a Radical Equation with Two Radicals: a*sqrt(b)=sqrt(d), Ex: Solve Radical Equations - Cube Roots / Fourth Roots, Solve Equations with Rational Exponents (One Solution), Solve Equations with Rational Exponents (Two Solutions), Solve an Equation with a Rational Exponent (x+b)^(c/d)=f (One Solution), Solve an Equation with a Rational Exponent (x+b)^(c/d)=f (Two Solutions), Ex: Solving an Equation with Rational Exponents Using Reciprocal Powers, Solve a Radical Equation Given Function Notation (Extraneous Sol), Radical Equation Application - Vehicle Speed from Skid Mark Length, Application: Evaluate a Square Function (Safe Speed), Application: Solve a Square Root Equation (Year of Obesity Percent), Ex: Radical Equation Application - Obesity Percentage, Ex: Radical Function Application Finding Inputs and Outputs (Speed and Length of Skids), Ex: Radical Function Outputs and Inputs Application - BMI, Ex: Radical Function Outputs and Inputs Application - Pendulum, Ex: Rational Function Outputs and Inputs Application - Average Cost, Kinetic Energy - Radical Equation Application, Volume of a Cone -Radical Equation Application, Volume of a Cone -Radical Equation Application (Special Formula), Ex: Determine a Real, Imaginary, and Complex Number, Simplify Square Roots to Imaginary Numbers, Write Number in the Form of Complex Numbers, Ex : Simplify Imaginary and Complex Numbers, Ex: Simplify, Add, and Subtract Imaginary and Complex Numbers, Ex 1: Adding and Subtracting Complex Numbers, Ex 2: Adding and Subtracting Complex Numbers, Ex 1: Simplify and Multiply Complex Numbers, Ex: Subtract and Multiply Complex Number, Simplifying Powers of i (Method of Dividing by 4), Ex: Raising the imaginary unit i to powers, Ex: Determine if a Triangle is a Right Triangle Given the Length of 3 Sides, Ex: Determine the Length of the Hypotenuse of a Right Triangle, Ex: Determine the Length of the Leg of a Right Triangle, Ex: Pythagorean Theorem App - Find the Width of a Laptop, Ex: Determine the Distance Between Two Points Using the Pythagorean Theorem, Ex: Find the Shortest Distance Between Two Locations North and West, Use The Pythagorean Theorem to Determine the Diagonal of a TV, Distance Formula App: Find the Closest Boat, Ex: Find the Endpoint of a Segment Given the Midpoint and One Endpoint, Ex: Determine the Distance Between Two Points (length of segment), Ex: Domain and Range of Square Root Functions, Ex: Domain and Range of Radical Functions, Ex: Domain of a Square Root Function with a Quadratic Radicand, Horizontal and Vertical Shifts of the Square Root Function, Horizontal and Vertical Stretches and Compressions of the Square Root Function, Ex: Match the Graph of a Horizontal and Vertical Shifted Graph to the Function, Ex: Match the Graph of a Reflected or Horizontally Compressed or Stretched Graph to a Function, Function Transformation Summary - The Square Root Function, Ex 1: Find the Equation of a Transformed Square Root Function from the Graph, Ex 2: Find the Equation of a Transformed Square Root Function From the Graph, Ex 3: Find the Equation of a Transformed Square Root Function From a Graph, Ex 4: Find the Equation of a Transformed Square Root Function From a Graph, Ex 1: Graphing a Transformation of the Square Root Function, Ex 2: Graphing a Transformation of the Square Root Function, Simplify and Give the Domain of Rational Expressions, Simplify Rational Expressions: (linear/quad) and (quad/quad), Ex 1: Simplifying Rational Expressions – Monomials, Ex: Find Values Where a Rational Expression is Undefined, Multiply Rational Expressions and Give the Domain, Divide Rational Expressions and Give the Domain, Ex 1: Multiply Rational Expressions – Monomials, Multiply Basic Rational Expressions: (ax/by)*(cy/dz) and (x/a)*(b/(x-c)), Multiply Rational Expressions: (quad/quad)*(quad/quad) a Not 1, Ex 1: Dividing Rational Expressions – Monomials, Divide Basic Rational Expressions: (Monomials and Linear Factors), Divide Rational Expressions: (quad/linear)/(quad/quad) a Not 1, Ex 1: Simplify a Complex Fraction (No Variables), Ex 2: Simplify a Complex Fraction (Variables), Ex 3: Simplify a Complex Fraction (Variables), Ex 4: Simplify a Complex Fraction (Variables), Ex 5: Simplify a Complex Fraction (Variables), Ex 6: Simplify a Complex Fraction (Variables), Ex 7: Simplify a Complex Fraction (Variables), Ex 1: Simplify a Complex Fraction with Variables (Basic), Ex 2: Simplify a Complex Fraction with Variables (Basic), Ex: Simplify a Complex Fraction with Variables (Factoring), Ex: Simplify a Complex Fraction with Addition and Subtraction and Constant Denominators, Ex: Simplify a Complex Fraction Subtraction and Variable Denominators, Ex: Simplify a Complex Fraction Subtraction and Variable Denominators with Factoring, Ex: Simplify a Complex Fraction with Addition and Constant and Variable Denominators, Ex: Simplify a Complex Fraction with Addition and Subtraction and Binomial Denominators, Complex Fraction Application: Simplify a Resistance Formula, Add and Subtract Rational Expressions with Like Denominators and Give the Domain, Add or Subtract Basic Rational Expressions with Like Denominators: (a and x+b), Add or Subtract Basic Rational Expressions with Like Denominators: (ax and bx^2), Add or Subtract Rational Expressions with Like Denominators: (x+a) and (x^2-bx+c), Ex 1: Add and Subtract Rational Expressions - Like Denominators, Ex 2: Add and Subtract Rational Expressions - Like Denominators, Ex 3: Add and Subtract Rational Expressions - Like Denominators, Add or Subtract Basic Rational Expressions with Unlike Denominators, Ex: Add and Subtract Rational Expressions - Opposite Denominators, Add Rational Expressions with Unlike Denominators and Give the Domain (Mono Denom), Subtract Rational Expressions with Unlike Denominators and Give the Domain, Subtract Rational Expressions with Unlike Denominators - 3 Expressions, Add and Subtract Rational Expressions with Unlike Denominators - 3 Expressions, Ex 1: Add and Subtract Rational Expressions - Unlike Denominators, Ex 2: Add and Subtract Rational Expressions - Unlike Denominators, Ex 3: Add and Subtract Rational Expressions - Unlike Denominators, Ex 4: Add and Subtract Rational Expressions - Unlike Denominators, Ex: Add and Subtract Rational Expressions with Unlike Monomial Denominators, Ex: Add Rational Expressions with Unlike Denominators, Ex: Subtract Rational Expressions with Unlike Denominators, Ex: Subtracting Rational Expressions (Factoring With A Not 1), Function Arithmetic (Add): f(x)+g(x) with Rational Functions, Function Arithmetic (Subtract): f(x)-g(x) with Rational Functions, Solve Rational Equations with Like Denominators, Solve a Rational Equation: (x-a)/x-b/c=d/x, Solve a Rational Equation: a/(x+b)=-c/(x+d) - No Cross Products, Ex 1: Solve Rational Equations Algebraically by Clearing Fractions and Solve Graphically, Ex 2: Solve Rational Equations Algebraically by Clearing Fractions and Solve Graphically, Ex: A Rational Equation with No Solution, Ex 1: Solve Equations with Fractions (Alternative Method), Ex 2: Solve Equations with Fractions (Alternative Method), Ex 1: Solve a Rational Equation (Alternative Method), Ex 2: Solve a Rational Equation (Alternative Method), Ex: Solve a Rational Equation - Alternative Method, Solve A Rational Equation (No Solution) x/(3x-9)-6=1/(x-3), Given f(x)=3/x-4/(3x), Solve f(x)=-7 (Rational Equation), Ex: Rational Function Outputs and Inputs Application - Time, Distance, Rate, Ex 1: Rational Equation Application - Painting Together, Ex 2: Rational Equation Application - Fill a Pool with Drain Open, Ex 3: Rational Equation Application - Plane and Car Travelling the Same Time, Ex 4: Rational Equation Application - Two Bikers Riding Different Distances, Ex: Rational Equation App - Find Individual Working Time Given Time Working Together, Ex: Rational Equation App - Find a Number Given the Sum of Reciprocals, Ex: Find How Faster Than the Speed Limit Is Needed Given a Travel Time, Rational Equation Application (Quadratic): Wind Speed, Rational Equation App (Quadratic): Find Individual Time Given Together Time, Rational Function Application: Function Value, Equation, End Behavior, Direct Variation Application: Currency Conversion, Ex: Direct Variation Application - Aluminum Can Usage, Inverse Variation Application: Water Temperature and Depth, Ex 2: Inverse Variation - Change of Variables, Ex 3: Inverse Variation - Fractional Variation Constant, Ex: Inverse Variation Application - Number of Workers and Job Time, Ex: Inverse Variation Application - Loudness and Distance, Joint Variation: Determine the Variation Constant (Volume of a Cone), Ex: Determine Rational Function Outputs and Inputs, Determining Vertical and Horizontal Asymptotes of Rational Functions, Determining Slant Asymptotes of Rational Functions, Ex 1: Determine Asymptotes and Graph a Rational Function, Ex 2: Determine Asymptotes and Graph a Rational Function, Ex 3: Determine Asymptotes and Graph a Rational Function, Ex 4: Determine Asymptotes and Graph a Rational Function (Slant), Ex: Determine Horizontal Asymptotes of Rational Functions, Ex: Find the Intercepts and Asymptotes of a Rational Function, Ex: Find the Intercepts, Asymptotes, and Hole of a Rational Function, Ex: Find a Rational Function Given the Vertical Asymptotes and Intercepts, Ex 1: Determine the Vertical and Slant Asymptotes of a Rational Function, Ex 2: Determine the Vertical and Slant Asymptotes of a Rational Function, Ex 1: Domain, Range, Asymptotes of a Basic Rational Function Using Translations, Ex 2: Domain, Range, Asymptotes of a Basic Rational Function Using Translations, Ex 1: Domain, Range, Asymptotes of a Basic Rational Function Using a Graph and Procedure, Ex 2: Domain, Range, Asymptotes of a Basic Rational Function Using a Graph and Procedure, Ex: Match Equations of Rational Functions to Graphs, Find the Intercepts and Asymptotes of a Rational Function (quad/quad with a not 1), Determine the Domain of Various Functions, Determine Vertical Intercepts of Various Functions, Determine Horizontal Intercepts of Various Functions (P1), Determine Horizontal Intercepts of Various Functions (P2), Determine Domain, Holes, and Asymptotes of a Rational Function (1), Determine Domain, Holes, and Asymptotes of a Rational Function (2), The Equation of a Rational Function from a Graph, Graph and Determine Key Components of a Rational Function (linear/linear) 1, Determine Key Components and Graph a Rational Function (linear/linear) 2, Match Rational Functions and Graphs: Translations, Graphing the Basic Rational Function f(x)=1/x, Graphing Reflections of the Basic Rational Function f(x)=1/x, Graphing Translations of the Basic Rational Function f(x)=1/x, Ex 1: Graph Two Translations of the Basic Rational Function f(x)=1/x, Ex 2: Graph Two Translations of the Basic Rational Function f(x)=1/x, Ex 1: Find the Equation of Rational Function From a Graph with a Hole, Ex 2: Find the Equation of Rational Function From a Graph with a Hole, Ex 3: Find the Equation of Rational Function From a Graph, Ex 4: Find the Equation of Rational Function From a Graph (Squared Intercept), Ex 5: Find the Equation of Rational Function From a Graph (Squared VA), Ex 6: Find the Equation of Rational Function From a Graph (Squared Intercept / VA), Rational Function Application - Concentration of a Mixture, Ex: Setting Up Partial Fraction Decomposition, Ex 1: Partial Fraction Decomposition (Linear Factors), Ex 2: Partial Fraction Decomposition (Linear Factors), Ex 3: Partial Fraction Decomposition (Repeated Linear Factors), Ex 4: Partial Fraction Decomposition (Repeated Linear Factors), Ex 5: Partial Fraction Decomposition (Linear and Quadratic Factors), Ex 6: Partial Fraction Decomposition (Repeating Quadratic Factors), Ex: Partial Fraction Decomposition - Degree 2 / Degree 3, Ex: Evaluate Functions and Composite Function in Context of a Story (Graphing Calculator), Ex: Intro Composite Function Notation Application Problem, Ex: Evaluate Composite Functions Using Tables of Values, Ex: Evaluate Composite Functions from Graphs, Ex 1: Determine Composite Function Values Using Table, Graph, and Function Rule, Ex 2: Determine Composite Function Values Using Table, Graph, and Function Rule, Determine f(g(x)) and g(f(x)) with Linear and Quadratic Functions, Determine f(f(x)) and g(g(x)) with Linear and Quadratic Functions, Ex: Find and Evaluate a Composition of Three Functions, Ex 2: Find Composite Function Values With Fractions, Ex: Find a Composition of Functions Involving Rational Functions, Ex: Domain of a Quotient and Composite Functions, Ex: Domain of Composite Function From Graphs, Ex: Inverse Function Notation and Reciprocal of a Function, Linear and Exponential Growth: Complete a Salary Table, Graphing by Plotting Points - Exponential, Graphing by Plotting Points - Exponential (6.3), Complete a Table of Values for an Exponential Equation in Two Variables, Evaluate Exponential Functions: Base 3 and 1/3, Write Exponential Equations Given Initial Values and Growth or Decay Rate, Write Exponential or Linear Equations to Model the Value of an Investment (Level 1), Write Exponential or Linear Equations to Model a Population (Level 2), Determine Exponential Function Values and Graph the Function, Graph a Basic Exponential Function Using a Table of Values, Graph an Exponential Function Using a Table of Values, Evaluate a Given Exponential Function to Predict a Future Population, Introduction to Exponential Equations in Two Variables, Determine the Initial Value and Percent Rate of Change from an Exponential Equation, Write an Exponential Equation that Models a Decreasing Population (Fox), Write an Exponential Equation to Model Wage Percent Increase over Years, Write an Exponential Equation to Model an Account Balance Over Years, Write an Exponential Equation to Model World Population Growth, Determine if Equations Are Linear or Exponential and Increasing or Decreasing, Interpret an Exponential Equation Modeling Depreciation, Interpret an Exponential Equation Modeling Rising Home Value, Compare Exponential Equations Modeling Account Values, Introduction to Exponential Functions in the Form f(x)=ab^x - Part 1, Introduction to Exponential Functions in the Form f(x)=ab^x - Part 2, Introduction to Exponential Functions in the Form f(x)=ae^(kx) - Part 1, Introduction to Exponential Functions in the Form f(x)=ae^(kx) - Part 2, Comparing Forms of Exponential Functions: y = ab^x and y = ae^(kx), Graphing Basic Exponential Functions: Growth and Decay, Interpret the Meaning of Ordered Pairs from a Graph (Exponential), Ex: Determine Exponential Graphs that Have Specific Characteristics - y = ab^x, Ex: Match Exponential Functions to Graphs, Match Exponential Growth and Decay Function with Graphs (Reflections), Describe an Exponential Function Transformation: y=e^(x)+3, Describe an Exponential Function Transformation: y=-2^x-1, Ex: Exponential Application Solved Using a Graphing Calculator, Determine if a Table Represents a Linear or Exponential Function, Ex: End (Long Run) Behavior of Exponential Functions, Ex: Find the Equation of a Transformed Exponential Function From a Graph, Ex: Match the Graphs of Translated Exponential Function to Equations, Ex: Match the Graphs of Reflected Exponential Functions to Equations, Ex 1: Determine if a Table of Value Represents a Linear or Exponential Function, Ex 2: Determine if a Table of Value Represents a Linear or Exponential Function, Ex 1: Determine if a Table of Value Represents a Linear or Exponential Function (Fractions/Decimals), Ex 2: Determine if a Table of Value Represents a Linear or Exponential Function (Fractions/Decimals), Ex 1: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Linear), Ex 2: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Linear), Ex 1: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Exponential), Ex 2: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Exponential), Determine if Two Linear Functions Are Inverses (1), Determine if Two Linear Functions Are Inverses (2), Determine if a Relation Given as a Table is a One-to-One Function, Ex 1: Determine if the Graph of a Relation is a One-to-One Function, Ex 2: Determine if the Graph of a Relation is a One-to-One Function, Ex 1: Determine if Two Functions Are Inverses, Ex 2: Determine if Two Functions Are Inverses, Ex: Find an Inverse Function From a Table, Ex: Function and Inverse Function Values Using a Table, Ex: Function and Inverse Function Values Using a Graph, Ex: Restrict the Domain to Make a Function 1 to 1, Then Find the Inverse, Ex: Find Inverse Function Values Without Finding the Inverse Function, Ex: Function and Inverse Function Values, Ex: Find the Inverse of a Square Root Function with Domain and Range, Ex: Find the Inverse of a Rational Function, Ex: Find the Inverse of a Rational Function and an Inverse Function Value, Ex 1: Determine If Two Functions Are Inverses, Ex 2: Determine If Two Functions Are Inverses, Ex: Find the Inverse Function of an Exponential Function, Ex: Write Exponential Equations as Logarithmic Equations, Ex: Write Logarithmic Equations as Exponential Equations, Ex: Write Exponential Equations as Logarithmic Equations - Variables, Ex: Write Logarithmic Equations As Exponential Equations - Variables, Ex: Write Exponential Equations with base 10 as Common Logarithmic Equations, Ex: Write Exponential Equations as Logarithmic Equations - Natural Logarithms, Use the Definition of a Logarithm to Show the Zero Exponent and Identity Property, Ex 1: Evaluate Logarithms Without a Calculator - Whole Numbers, Ex: Evaluate Logarithmic Expressions without a Calculator - Different Bases, Ex: Evaluate Logarithmic Expressions without a Calculator - Common Log, Ex: Solve a Basic Exponential Equation with Base Ten Using Logarithm Definition, Ex: Solve a Exponential Equation with Base Ten Using Logarithm Definition (Multiple Steps), Ex: Solve a Basic Exponential Equation with Base e Using Logarithm Definition, Ex: Solve a Exponential Equation with Base e Using Logarithm Definition (Multiple Steps), Ex 2: Evaluate Logarithms Without a Calculator - Fractions, Ex: Evaluate Common Logarithms on a Calculator, Ex: Evaluate Common Logarithms Without a Calculator, Ex: Evaluate Common Logarithms on the Calculator, Ex: Evaluate Natural Logarithms on the Calculator, Ex: Determine the Value of a Number on a Logarithmic Scale (Log Form), Ex: Determine the Value of a Number on a Logarithmic Scale (Exponential Form), Ex: Determine the Difference in Order of Magnitude to Two Quantities, Ex: Determine the Difference in Order of Magnitude to Two Quantities (Application), Ex: Graph an Exponential Function and Logarithmic Function, Ex: Properties and Characteristics of a Logarithmic Function, Ex 1: Match Graphs with Exponential and Logarithmic Functions, Ex 2: Match Graphs with Exponential and Logarithmic Functions - Base 10 and e, Ex: Vertical Asymptotes and Domain of Logarithmic Functions, Ex: Find the Domain of Logarithmic Functions, Determine the Domain, Range, and Asymptote of a Log Function y=-ln(x-6), Determine the Domain, Range, and Asymptote of a Log Function y=-log_3(x)+4, Graph Exponential and Logarithmic Functions on the TI-84 (Inverses), Graphing Logarithmic Functions Using Desmos.com, Graphing Log Functions on the TI-84: y=log_(1/2)(x), Graphing Log Functions on the TI-84: y=-log_(3)(x), Graphing Log Functions on the TI-84: y=log_(2)(x-3)+2, Graphing Log Functions on the TI-84: y=2log_(3)(x+1)-3, Graphing Log Functions by Hand: y=log_(1/2)(x), Graphing Log Functions by Hand: y=-log_(3)(x), Graphing Log Functions by Hand: y=log_(2)(x-3)+2, Graphing Log Functions by Hand: y=2log_(3)(x+1)-3, Graphing Log Functions by Hand: y=log_4(x), Graphing Log Functions by Hand: y=log_3(x)+2, Graphing Basic Logarithmic Functions Using Desmos, Expand Logarithms Using the Product Rule for Logs, Expand Logarithms Using Properties of Logarithms Rule and Factoring, Expand Logarithms Using Properties of Logarithms (Expressions), Ex: Expand a Logarithm Containing a Radical, Combine Logarithms Using Properties of Logarithms, Ex: Combine a Sum and Difference of Two Logarithms, Ex 1: Combine a Logarithmic Expression Into One Logarithm, Ex 2: Combine a Logarithmic Expression Into One Logarithm, Ex 1: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 2: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 3: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 4: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 5: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Using the Inverse Property of Logarithms and Exponentials to Evaluate Expressions, Ex 1: Evaluate a Logarithmic Expression Using the Properties of Logarithms, Ex 2: Evaluate a Logarithmic Expression Using the Properties of Logarithms, Ex: Simplify Log Expression with the Base and Base of the Number are the Same, Ex: Evaluate Exponential Expressions with Logarithmic Exponents, Ex: Change of Base Formula to Evaluate Logarithmic Expressions, Ex: Change of Base Formula to Solve Basic Exponential Equations, Ex: Evaluate Logarithmic Functions Using the Change of Base Formula, Evaluate Logarithms of any Base on the TI-84 (Not using Change of Base), Determine the Human Age of a Dog And the Dog Age, Given the Human Age, Solve Logarithmic Equations for the Base (No Calculator), Solve a Log Equation with Two Logs Equal to Each Other, Ex: Solve a Logarithmic Equations - One Step by Dividing with Rational Solution, Ex: Solve a Logarithmic Equations - One Step by Dividing with Irrational Solution, Ex: Solve a Logarithmic Equations - One Step by Dividing with Log of a Quantity, Ex: Solving a Variety of Logarithmic Equations, Ex: Solve a Basic Logarithmic Equation - Linear and Quadratic, Ex: Solve Basic Logarithmic Equation - Radicals, Ex: Solve Logarithmic Equations Containing Only Logarithms, Ex: Solve a Logarithmic Equation With a Difference - Linear, Ex: Solve a Logarithmic Equation With a Sum - Quadratic Formula, Ex: Solve a Logarithmic Equation With a Difference - Quadratic Formula, Ex: Solve a Logarithmic Equation Requiring the Quadratic Formula, Ex 3: Solve Logarithmic Equations - Base and Number are the Same, Ex: Solve a Logarithmic Equation - Composite Log Expression, Ex: Solve a Logarithmic Equation in Terms of Other Variables, Ex: Solve a Logarithmic Equation with a Fractional Exponent, Solve a Log Equation by Factoring (Trinomial), Logarithm Application: Magnitude of an Earthquake, Logarithm Application: Intensity of Two Sounds (Decibels), Ex: Logarithmic Function Application - Test Scores, Ex: Exponential Function Application with Logarithms, Ex: Logarithmic Function Application - pH (Outputs and Inputs), Ex: Logarithmic Function Application - Preston Curve (Outputs and Inputs), Solving Exponential Equations by Obtaining a Common Base, Ex 1: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 2: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 3: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 4: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 5: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 6: Solve Exponential Equations Using Like Bases - No Logarithms, Ex: Solve an Exponential Equation Graphically on the TI84, Solving Exponential Equations Using Logarithms, Ex 1: Solve a Basic Exponential Equation Using the Definition of a Logarithm, Ex 2: Solve a Basic Exponential Equation Using the Definition of a Logarithm, Ex 1: Solve Exponential Equations Using Logarithms, Ex 2: Solve Exponential Equations Using Logarithms, Ex 3: Solve Exponential Equations Using Logarithms, Ex 4: Solve Exponential Equations Using Logarithms, Ex 5: Solve Exponential Equations with Two Exponential Parts Using Logarithms, Ex 6: Solve an Exponential Equation Using Common Log, Ex 7: Solve an Exponential Equation Using Natural Log (Factoring), Ex: Solve an Exponential Equation Using Logarithms, Ex: Solve an Exponential Equation with Logarithms - Variable on Both Sides, Ex: Rewrite Exponential Functions: y = ab^t to y = ae^(kt), Ex: Rewrite Exponential Functions: y = ae^(kt) to y = ab^t, Determine a Continuous Exponential Decay Function and Make a Prediction, Ex: Write Linear and Exponential Functions Based Upon Given Information, Write an Exponential Function for Growth over Different Time Intervals, Ex: Find a Linear and Exponential Model for Population Growth, Ex: Use a Given Exponential Growth Function, Ex: Exponential Growth of Bacteria (Intro Question), Ex: Solve an Exponential Growth Equation Graphically Using the TI84 (Application), Ex: Compare Simple Interest and Annual Compounded Interest, Ex: Solve an Exponential Growth Equation Graphically Using the Desmos (Application), Ex: Solve an Exponential Decay Equation Graphically Using the TI84 (Application), Ex: Solve an Exponential Decay Equation Graphically Using the Desmos (Application), Ex: Determine the Doubling Time of an Investment Account Graphically (TI84), Exponential Growth App (y=ab^t) - Given Doubling Time, Ex: Determine the Half-Life of a Fish Population Graphically (TI84), Ex: Exponential Function Applications - Increasing Investment Value (Change of Base Used), Ex: Exponential Function Applications - Decreasing Water Level (Change of Base Used), Ex: Exponential Growth Function - Population, Ex: Exponential Function Application Using Logs - Export Values, Ex: Exponential Growth Function - Bacterial Growth, Ex: Exponential Decay Function with Logarithms, Ex: Exponential Growth Application - Predicting World Population, Ex: Basic Example of Exponential Decay Model, Ex: Exponential Decay Function - Half Life, Ex: Find the Initial Value and Exponential Growth or Decay Rate Given an Exponential Function, Ex: Exponential Functions: Growth Rate and Growth Factor, Ex: Exponential Functions: Decay Rate and Decay Factor, Ex: Determine Growth Exponential Functions Given Growth Rate and Initial Value (y=ab^x), Ex: Determine Exponential Decay Functions Given Decay Rate and Initial Value (y=ab^x), Exponential Decay App (y=ab^x) - Given Half Life, Exponential Decay App (y=ae^(kt)) - Given Half Life, Exponential Decay App (y=ab^t) - Find Initial Amount Given Half Life, Exponential Decay App with Logs (y=ae^(kt)) - Find Half Life, Ex: Exponential Model - Determine Age Using Carbon-14 Given Half Life, Find Initial Amount and Amount After Time From Half-Life (y=ae^(kt)), Exponential Growth App (y=ab^t) - Find Initial Amount Given Doubling Time, Ex: Practice Writing Exponential Equations - Doubling Equation and Halving Equation, Exponential Growth App with Logs (y=ae^(kt)) - Find Initial Amount Given Doubling Time, Ex: Find an Exponential Growth Function Given Two Points - Initial Value Given, Ex: Find an Exponential Decay Function Given Two Points - Initial Value Given, Ex: Find an Exponential Function Given Two Points - Initial Value Not Given, Exponential Function Application (y=ab^x) - Population Growth of India, Exponential Function Application (y=ab^x) - Population Decline of Chicago, Exponential Function Application (y=ab^x) - Depreciation of a Car, Annual Depreciation of a New Car: Find the Future Value, Exponential Function Application (y=ab^x) - Declining Computer Value, Determine an Exponential Decay Function P(t)=a(b)^t (No Logs), Exponential Function Application (y=ab^x) - Home Values, Exponential Function Application (y=ae^(kt)) - Bacteria Growth, Ex: Application - Cyclical Around Linear Growth, Ex: Application - Cyclical Around Exponential Growth, Ex: Find Annual Interest Rate Given f(t)=ae^(kt), Ex: Find Annual Depreciation Rate Given f(t)=ae^(kt), Ex: Find Continuous Interest Rate Given f(t)=ab^t, Ex: Find Continuous Depreciation Rate Given f(t)=ab^t, Determine an Exponential Function on the TI-84 Given Two Points (Growth), Determine an Exponential Function on the TI-84 Given Two Points (Decay), Determine the Total Return of an Investment as Percent, Compounded Interest: Solve for Principal (Present Value), Ex 2: Compounded Interest with Logarithms, Ex: Find Exponential Growth Rate and Make Prediction of a Future Home Value, Compounded Interest (Slightly Different Form of Equation): Find Future Value, Effective Interest Rate (Effective Yield), Ex 2: Continuous Interest with Logarithms, Ex 3: Continuous Interest with Logarithms and Doubling Time, Continuous Interest - Find the Initial Investment Needed, Ex: Perform Exponential Regression on a Graphing Calculator, Perform Exponential Regression and Make Predictions Using Desmos, Ex: Exponential Regression Application on the TI84 (Decreasing Polio Case), Ex: Exponential Decay Regression Model (Declining Population), Ex: Exponential Growth Regression Model (Investment Account), Ex: Comparing Linear and Exponential Regression, Ex: Find an Exponential Function for a Semi-Log Graph, Ex: Newton's Law of Cooling - Exponential Function App, Exponential Function App. Pyrex® Red Lid For 8" Square Glass Baking Dish,
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= 0.0, but there's still several extra instructions in the fast path. As for its speed, it at least outperforms the std::sin() function by an average of 0.3 microseconds per call. If you're writing audio plugins or doing digital signal processing, Chebyshev polynomials give you a cheap and predictable dithering effect "for free.". An approximation for the sine function that preserves the derivatives at multiples of 90 degrees is given by this formula. How about dectecting the CPU and using a native instruction on a modern processor with a lookup table or other optimized code on older machines. My custom x86 Math library code is for standard MSVC++ 2005 and forward. The cosine of 0 is well-defined, and is 1. gcc's code for the NaN case seems way over-complicated; it doesn't even use the sqrtf return value! Lastly, when you've finished all your fancy benchmarking and micro-optimization, make sure that your "fast" version is actually faster than the library version. The, Unless you bottleneck on uop throughput rather than sqrt latency, in which case using plain, What is the purpose of taking the address of. Also, I've benchmarked it and it twice slower than milianw answers. Here's a possible speedup which depends highly on your application. That's strictly bush league. Check this one for example . Making statements based on opinion; back them up with references or personal experience. This is a delightful representation of the Taylor Series. What makes an argument objectively more "compelling"? @roliu My mistake, I though I have to calculate Factorial several times, but I missed that I can use precomputed constant. (Negative a), Graphing and Writing Equations of Hyperbolas, Conic Sections: The Hyperbola part 1 of 2 Conic Sections: The Hyperbola part 2 of 2 Ex 1: Conic Section - Graph a Hyperbola with Center at the Origin (Horizontal) Ex 2: Conic Section - Graph a Hyperbola with Center at the Origin (Vertical) Ex 3: Conic Section - Graph a Hyperbola with Center NOT at the Origin (Horizontal) Ex 4: Conic Section - Graph a Hyperbola with Center NOT at the Origin (Vertical) Ex: Find the Equation of a Hyperbola Given the Center, Focus, and Vertex Determining the Type of Conic Section from General Form, Introduction to Set Theory Introduction to Subsets Set Operations and Venn Diagrams - Part 1 of 2 Set Operations and Venn Diagrams - Part 2 of 2 Solving Problems Using Venn Diagrams, Introduction to Regression Analysis Linear Regression – Example 1, Example 2 Quadratic Regression – Example 1, Example 2 Perform Quadratic Regression and Make Predictions Using Desmos Interpret a Quadratic Function Model: Fuel Consumption Ex 1: Cubic Regression on the TI84 – Natural Gas Consumption Ex 2: Cubic Regression on the TI84 - Total Sales Exponential Regression – Example 1, Example 2 Ex: Exponential Decay Regression Model (Declining Population) Ex: Exponential Growth Regression Model (Investment Account) Logarithmic Regression Logistic Regression, Financial Mathematics (See more under Math for Liberal Arts library), Simple Interest Formula Compounded and Continuous Interest Effective Yield Effective Yield on the TI84 Derive the Value of an Annuity Formula (Compounded Interest) Determining the Value of an Annuity Determining the Value of an Annuity Using the TI84 Determining the Monthly Saving Required to Reach a Financial Goal Determining the Monthly Saving Required to Reach a Financial Goal on the TI84 Ex 1: Find a Monthly Mortgage Payment with a Down Payment Ex 2: Find a Monthly Mortgage Payment with a Down Payment and Points Ex: Comparing Two Installments Loans (Car Loans) Ex: Simple Interest Discounted Loan Determining the Monthly Payments for a Loan Determining the Monthly Payments for a Loan on the TI84, Graphing Calculator Basics: Evaluating Expressions and Determining Function Values, Graphing Calculator Basics The Table Feature of the Graphing Calculator Evaluating Radical Expressions on the TI83/84 Determine Function Values Using Function Notation on the TI84, Graphing Functions and Determining Key Components of Functions, Graphing Lines on the Graphing Calculator Determining the Intersection of Two Graph on the TI83/84 Determining Relative Extrema on the Graphing Calculator Determine Function Values Using Function Notation on the TI84 Determine the value of the derivative function on the graphing calculator Determining the value of a definite integral on the graphing calculator Determining the Intersection of Two Graph on the TI83/84 Determining the Zeros or Roots of a Polynomial Function on the TI83/84 From a Graph Determine Where a Quadratic Function is Increasing and Decreasing Determining When a Polynomial Function is Increasing and Decreasing Determine Max/Mins and Incr/Decr Intervals Using a Free Online Graph Calc (MathAS) Determining the Zeros or Roots of a Polynomial Function on the TI83/84 Ex: Quadratic Function Application Using a Graphing Calculator - Rocket Launch Ex 1: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 2: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 1: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial Ex 2: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Solving Equations on the Graphing Calculator, Solving Linear Equations Graphically Ex: Solve a Linear Equation in One Variable Graphically using the TI84 Ex: Solve a Linear Inequality in One Variable Graphically using the TI84 Ex: Solving Absolute Value Equations on the Graphing Calculator Ex 1: Solve a Quadratic Equation Graphically on Calculator Ex 2: Solve a Quadratic Equation Graphically on Calculator Ex: Determine How a Final Exam Score Affect a Course Grade using the TI84 (Weighted Averages) Solving Polynomial Equations Graphically, Ex: Evaluate Common Logarithms on a Calculator Ex: Evaluate Common Logarithms on the Calculator Ex: Evaluate Natural Logarithms on the Calculator Ex: Change of Base Formula to Evaluate Logarithmic Expressions Ex: Change of Base Formula to Solve Basic Exponential Equations Ex: Evaluate Logarithmic Functions Using the Change of Base Formula Ex: Solve an Exponential Equation Graphically on the TI84 Ex: Perform Exponential Regression on a Graphing Calculator Ex: Comparing Linear and Exponential Regression Sequences and Series on the TI84 Sequences and Series on the TI84, Augmented Matrices on the Graphing Calculator Matrix Multiplication on the Graphing Calculator Inverse Matrices on the Graphing Calculator Ex 1: Solve a System of Two Equations Using a Matrix Equation Ex 2: Solve a System of Two Equations Using a Matrix Equation Ex: Solve a System of Three Equations Using a Matrix Equation Determinants on the Graphing Calculator Ex: Solve a System of Three Equations Using Cramer's Rule, Ex 1: Create a Scatter Plot and then Perform Linear Regression on the Calculator Ex 2: Creating a Scatter Plot and Performing Linear Regression on the Calculator Linear Regression – Example 1, Example 2 Ex: Matching Correlation Coefficients to Scatter Plots Quadratic Regression – Example 1, Example 2 Perform Quadratic Regression and Make Predictions Using Desmos Ex: Quadratic Regression on the TI84 - Stopping Distance Exponential Regression – Example 1, Example 2 Perform Exponential Regression and Make Predictions Using Desmos Logarithmic Regression Logistic Regression Regression and Systems of Equations: Application, Financial Mathematics on the Graphing Calculator, Loan Information on the TI83/84 Effective Yield on the TI84 Determining the Value of an Annuity Using the TI84 Determining the Monthly Payments for a Loan on the TI84 Determining the Monthly Saving Required to Reach a Financial Goal on the TI84, Determine if a Function is a Polynomial Function, Degree, Leading Term, and Leading Coefficient of a Polynomial Function, Ex: Information about a Given Polynomial Function, Turning Points and X Intercepts of a Polynomial Function, Summary of End Behavior or Long Run Behavior of Polynomial Functions, Determine the End Behavior of Power Functions, Ex: End Behavior or Long Run Behavior of Functions, Ex: End Behavior of a Polynomial Function in Factored Form, Ex: Find the Intercepts of a Polynomial Function in Factored Form, Ex: Determine the Least Possible Degree of a Polynomial From the Graph, Ex: Increasing / Decreasing / Relative Extrema from Analyzing a Graph, Analyze a Graph Using Desmos to Determine Key Components of a Quadratic (Incr / Decr / Extrema), Analyze a Graph Using Desmos to Determine Key Components of a Cubic (Incr / Decr / Extrema), Ex 1: Determine the Local / Relative Extrema of a Cubic Function Using Desmos, Ex 2: Determine the Local / Relative Extrema of a Cubic Function Using Desmos (Challenging), Determine the Maximum Volume of an Open Top Box Using a Graph Only, Ex: Concavity / Points of Inflection by Analyzing a Graph (Algebra Topic), Ex: Concavity / Increasing / Decreasing Functions as Tables (Algebra Topic), Ex: Find the Intercepts of a Polynomial Function (Factorable), Ex: Find the Intercepts of a Polynomial Function (Real Zero), Ex 1: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 2: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 3: Find the Intercepts and the End Behavior of a Polynomial Function, Ex 1: Solve a Cubic Function Graphically (One Solution), Ex 2: Solve a Cubic Function Graphically (Two Solutions), Ex 1: Find a Degree 3 Polynomial Function Given Integer Zeros, Ex 2: Find a Degree 3 Polynomial Function Given Fractional Zeros, Ex 3: Find a Degree 3 Polynomial Function Given Imaginary Zeros, Ex 4: Find a Degree 3 Polynomial Function Given Complex Zeros, Ex 1: Find a Polynomial Function Given the Zeros or Roots with Multiplicity and a Point, Ex 2: Find a Polynomial Function Given the Zeros or Roots with Multiplicity and a Point, Find a Polynomial Function Given the Zeros and Leading Coefficient (Degree 3), Find a Polynomial Function Given the Zeros, Multiplicity, and (0,a) (Degree 3), Determine a Degree 3 Polynomial Function Given the Zeros and Intercept (2 complex), Ex 1: Find a Degree 4 Polynomial Function Given Integer and Complex Zeros, Ex 2: Find a Degree 4 Polynomial Function Given Integer and Complex Zeros, Ex 3: Find a Degree 4 Polynomial Function Given Fractional and Complex Zeros, Ex1: Find an Equation of a Degree 4 Polynomial Function From the Graph of the Function, Ex2: Find an Equation of a Degree 5 Polynomial Function From the Graph of the Function, Ex3: Find an Equation of a Degree 6 Polynomial Function From the Graph of the Function, Ex 1: Cubic Regression on the TI84 – Natural Gas Consumption, Ex 2: Cubic Regression on the TI84 - Total Sales, Real Zeros, Factors, and Graphs of Polynomial Functions, Find the Zeros of a Cubic Function Using Factor by Grouping (1 rational, 2 irrational), Find the Zeros of a Cubic Function Using Factor by Grouping (1 rational, 2 complex), Use the Remainder Theorem to Determine if a Binomial is a Factor of a Polynomial, Determine if a Binomial is a Factor of a Polynomial Graphically, Determine the Zeros/Roots and Multiplicity From a Graph of a Polynomial, Polynomial Function - Complex Factorization Theorem, Ex: Factor and Solve a Polynomial Equation, Ex: Solve a Basic Cubic Equation Using a Cube Root and Rational Exponent, Find the Intercepts of a Degree 4 Polynomial Function (Factorable), Find Zeros, Multiplicity, Degree, and End Behavior of a Factored Polynomial (Degree 11), Find Zeros, Multiplicity, Degree, and End Behavior of a Factored Polynomial (Degree 6), Ex 1: Find the Zeros of a Polynomial Function - Integer Zeros, Ex 2: Find the Zeros of a Polynomial Function - Real Rational Zeros, Ex 3: Find the Zeros of a Polynomial Function with Irrational Zeros, Ex 4: Find the Zeros of a Polynomial Function with Imaginary Zeros, Ex 5: Find the Zeros of a Polynomial Function with Complex Zeros, Ex 6: Find the Zeros of a Degree 4 Polynomial Function, Ex 7: Find the Zeros of a Degree 5 Polynomial Function, Ex 1: Write a Degree 3 Polynomial Function as a Product of Linear Factors (2 Imaginary), Ex 2: Write a Degree 3 Polynomial Function as a Product of Linear Factors (2 Irrational), Ex 3: Write a Degree 5 Polynomial Function as a Product of Linear Factors, Ex 1: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 2: The Zero Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 1: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Ex 2: The Intersection Feature of the TI84 to Find Rational Zeros of a Polynomial, Determine Local (Relative) Extrema of a Polynomial Function Using the TI-84, Determine Local (Relative) Extrema of a Polynomial Function Using Desmos, Ex 1: Solve a Polynomial Inequality in Factored Form, Ex 2: Solve a Polynomial Inequality in Factored Form, Ex: Solve a Polynomial Equation Using a Graphing Calculator (Approximate Solutions), Ex: Solve a Polynomial Inequality Using Factor By Grouping (Degree 3), Ex: Solve a Polynomial Inequality Using Factor Using GCF (Degree 3), Ex: Solve a Polynomial Inequality Using Factor of a Trinomial (Degree 4), Ex 1: Determine if a Function is Odd, Even, or Neither, Ex 2: Determine if a Function is Odd, Even, or Neither, Determine if a Function is Even, Odd, or Neither Using a Graph (1), Determine if a Function is Even, Odd, or Neither Using a Graph (2), Introduction to Square Root and Perfect Squares, Approximate a Square Root to Two Decimal Places Using Trial and Error, Simplify Square Roots (Perfect Square Radicands), Simplify a Variety of Square Expressions (Simplify Perfectly), Simplify Square Roots with Variables (perfect squares), Simplify Square Roots of Decimals (Perfect Square Decimals), Simplify Square Roots (Not Perfect Square Radicands), Simplify Square Roots in the Form a*sqrt(b) (not perfect squares), Simplify Cube Roots (Perfect Cube Radicands), Simplify Cube Roots (Not Perfect Cube Radicands), Simplifying Radical Expressions Without Fractions, Simplifying Radical Expressions With Fractions, Order of Operations with a Fraction Containing a Square Root, Ex: Simplify Square Roots - Perfect Roots, Ex: Simplify Perfect Nth Roots - Radicals, Ex: Simplify Square Roots - Not Perfect Roots, Ex: Simplify Square Roots of Variable Expressions - Absolute Value Needed, Ex: Simplify a Radical Expression containing Square Roots in the Numerator and Denominator, Simplify Cube Roots with Variables (not perfect cubes), Ex: Simplify Radicals with the Same Radicand and Different Indexes, Ex: Simplify a Radical Containing a Fraction - Perfect Root, Ex: Simplify Radicals with Variables - Perfect Roots, Ex 1: Simplifying Perfect Nth Roots Containing Variables, Ex 2: Simplifying Perfect Nth Roots Containing Variables, Ex: Simplify Radicals Containing Variables With Large Exponents - Not Perfect Roots, Ex: Simplify Radicals with Negative Radicands and Odd Indexes, Simplify Radicals with Variables (Large Index and Exponents), Ex: Simplify Radicals with Variables - Not Perfect Roots, Evaluating Radical Expressions on the TI83/84, Approximating Square Roots Using Division (No Calculator Required), Multiplying Radical Expressions with Variables Using Distribution, Multiplying Binomial Radical Expressions with Variables, Multiplying Radicals Containing Variables, Multiplying Conjugates of Radical Expressions, Multiply Two Radicals with Variables (Index 4 and 5) Perfect Roots, Ex: Multiply Radical Conjugates - Square Roots, Square a Binomial with a Square Root of a Variable Expression, Multiply two Binomial Radical Expressions with Variables (Conjugates), Dividing Radicals without Variables (Basic with no rationalizing), Dividing Radicals with Variables (Basic with no rationalizing), Simplify the Square Root of Fraction: sqrt(a/b), a/sqrt(b), Rationalize the Denominator - Square Root with Variable, Rationalize the Denominator - Cube Root and 4th Root, Ex 1: Rationalize The Denominator of a Fraction (Basic), Ex 2: Rationalize The Denominator of a Fraction, Ex 3: Rationalize The Denominator of a Fraction, Ex 1: Rationalize the Denominator of a Radical Expression, Ex 2: Rationalize the Denominator of a Radical Expression, Ex: Rationalize the Denominator of a Radical Expression - Conjugate, Rationalize the Denominator - Conjugate with Variables, Ex: Write a Radical in Rational Exponent Form, Write Basic Expression in Radical Form and Using Rational Exponents, Write Expressions Using Radicals and Rational Exponents, Write Rational Exponents as Radicals and Radicals Using Rational Exponents (Variables), Simplify Radicals Using Rational Exponents, Ex: Simplify Exponential Expressions with Fraction Exponents (Power Property of Exponents), Ex: Simplify Exponential Expressions with Fraction Exponents (Quotient Property of Exponents), Ex: Simplify Expressions with Rational Exponents, Ex: Simplify an Expression with Rational Exponents and Write in Radical Form, Ex: Simplify an Expression with Negative Rational Exponents and Write in Radical Form, Ex 1: Simplify an Expression with a Negative Rational Exponent, Ex 2: Simplify an Expression with a Negative Rational Exponent, Simplify An Expression with Rational Exponents (Positive Only) Power/Quot, Simplify An Expression with Rational Exponents (Negative) Power/Quot, Simplify An Expression with Rational Exponents (Negative) Prod/Quot, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 1, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 2, Simplify An Expression with Rational Exponents (Negative) Power/Prod/Quot 3, TI-84: Comparing Radical Form and Rational Exponent Form, Ex: Multiply and Divide Radicals with Different Indexes Using Rational Exponents - Same Radicand, Ex: Multiply Radicals with Different Indexes Using Rational Exponents - Different Radicand, Ex: Evaluate an Expression with Rational Exponents Using Radicals, Simplify a Quotient with A Radical (Rational Exponents and Radical Form), Adding Radicals (Basic With No Simplifying), Adding Radicals That Requires Simplifying, Simplify and Add Square Roots: sqrt(a^2*c)+sqrt(b^2*c), Subtracting Radicals (Basic With No Simplifying), Subtracting Radicals That Requires Simplifying, Ex: Evaluate the Square Root of a Sum and the Sum of Square Roots on a Calculator, Ex: Add and Subtract Square Roots - No Simplifying, Ex: Add and Subtract Square Roots Containing Variables, Subtract Square Expressions with a Variable - Simplifying Required, Add Square Root Expressions with Variables, Add and Subtract Square Root Expressions with Variables (Adv), Solving Radical Equations with One Radical, Solve a Radical Equation with One Cube Root (Fraction Answer), Solve a Radical Equation with One Square Root (Fraction Answer), Solve a Basic Radical Equation Given Function Notation, Solve a Radical Equation Given Function Notation, Solving Radical Equations with Two Radicals, Ex 1: Solve a Basic Radical Equation - Square Roots, Ex 2: Solve Radical Equations - Square Roots, Ex 3: Solve Radical Equations - Square Roots, Ex 4: Solve Radical Equations - Square Roots, Ex 5: Solve Radical Equations - Two Square Roots, Ex 6: Solve Radical Equations - Two Square Roots, Ex 7: Solve Radical Equations - Two Square Roots, Ex: Solve a Radical Equation with One Radical: Factoring and Extraneous Solution, Ex: Solve a Radical Equation with Two Radicals: a*sqrt(b)=sqrt(d), Ex: Solve Radical Equations - Cube Roots / Fourth Roots, Solve Equations with Rational Exponents (One Solution), Solve Equations with Rational Exponents (Two Solutions), Solve an Equation with a Rational Exponent (x+b)^(c/d)=f (One Solution), Solve an Equation with a Rational Exponent (x+b)^(c/d)=f (Two Solutions), Ex: Solving an Equation with Rational Exponents Using Reciprocal Powers, Solve a Radical Equation Given Function Notation (Extraneous Sol), Radical Equation Application - Vehicle Speed from Skid Mark Length, Application: Evaluate a Square Function (Safe Speed), Application: Solve a Square Root Equation (Year of Obesity Percent), Ex: Radical Equation Application - Obesity Percentage, Ex: Radical Function Application Finding Inputs and Outputs (Speed and Length of Skids), Ex: Radical Function Outputs and Inputs Application - BMI, Ex: Radical Function Outputs and Inputs Application - Pendulum, Ex: Rational Function Outputs and Inputs Application - Average Cost, Kinetic Energy - Radical Equation Application, Volume of a Cone -Radical Equation Application, Volume of a Cone -Radical Equation Application (Special Formula), Ex: Determine a Real, Imaginary, and Complex Number, Simplify Square Roots to Imaginary Numbers, Write Number in the Form of Complex Numbers, Ex : Simplify Imaginary and Complex Numbers, Ex: Simplify, Add, and Subtract Imaginary and Complex Numbers, Ex 1: Adding and Subtracting Complex Numbers, Ex 2: Adding and Subtracting Complex Numbers, Ex 1: Simplify and Multiply Complex Numbers, Ex: Subtract and Multiply Complex Number, Simplifying Powers of i (Method of Dividing by 4), Ex: Raising the imaginary unit i to powers, Ex: Determine if a Triangle is a Right Triangle Given the Length of 3 Sides, Ex: Determine the Length of the Hypotenuse of a Right Triangle, Ex: Determine the Length of the Leg of a Right Triangle, Ex: Pythagorean Theorem App - Find the Width of a Laptop, Ex: Determine the Distance Between Two Points Using the Pythagorean Theorem, Ex: Find the Shortest Distance Between Two Locations North and West, Use The Pythagorean Theorem to Determine the Diagonal of a TV, Distance Formula App: Find the Closest Boat, Ex: Find the Endpoint of a Segment Given the Midpoint and One Endpoint, Ex: Determine the Distance Between Two Points (length of segment), Ex: Domain and Range of Square Root Functions, Ex: Domain and Range of Radical Functions, Ex: Domain of a Square Root Function with a Quadratic Radicand, Horizontal and Vertical Shifts of the Square Root Function, Horizontal and Vertical Stretches and Compressions of the Square Root Function, Ex: Match the Graph of a Horizontal and Vertical Shifted Graph to the Function, Ex: Match the Graph of a Reflected or Horizontally Compressed or Stretched Graph to a Function, Function Transformation Summary - The Square Root Function, Ex 1: Find the Equation of a Transformed Square Root Function from the Graph, Ex 2: Find the Equation of a Transformed Square Root Function From the Graph, Ex 3: Find the Equation of a Transformed Square Root Function From a Graph, Ex 4: Find the Equation of a Transformed Square Root Function From a Graph, Ex 1: Graphing a Transformation of the Square Root Function, Ex 2: Graphing a Transformation of the Square Root Function, Simplify and Give the Domain of Rational Expressions, Simplify Rational Expressions: (linear/quad) and (quad/quad), Ex 1: Simplifying Rational Expressions – Monomials, Ex: Find Values Where a Rational Expression is Undefined, Multiply Rational Expressions and Give the Domain, Divide Rational Expressions and Give the Domain, Ex 1: Multiply Rational Expressions – Monomials, Multiply Basic Rational Expressions: (ax/by)*(cy/dz) and (x/a)*(b/(x-c)), Multiply Rational Expressions: (quad/quad)*(quad/quad) a Not 1, Ex 1: Dividing Rational Expressions – Monomials, Divide Basic Rational Expressions: (Monomials and Linear Factors), Divide Rational Expressions: (quad/linear)/(quad/quad) a Not 1, Ex 1: Simplify a Complex Fraction (No Variables), Ex 2: Simplify a Complex Fraction (Variables), Ex 3: Simplify a Complex Fraction (Variables), Ex 4: Simplify a Complex Fraction (Variables), Ex 5: Simplify a Complex Fraction (Variables), Ex 6: Simplify a Complex Fraction (Variables), Ex 7: Simplify a Complex Fraction (Variables), Ex 1: Simplify a Complex Fraction with Variables (Basic), Ex 2: Simplify a Complex Fraction with Variables (Basic), Ex: Simplify a Complex Fraction with Variables (Factoring), Ex: Simplify a Complex Fraction with Addition and Subtraction and Constant Denominators, Ex: Simplify a Complex Fraction Subtraction and Variable Denominators, Ex: Simplify a Complex Fraction Subtraction and Variable Denominators with Factoring, Ex: Simplify a Complex Fraction with Addition and Constant and Variable Denominators, Ex: Simplify a Complex Fraction with Addition and Subtraction and Binomial Denominators, Complex Fraction Application: Simplify a Resistance Formula, Add and Subtract Rational Expressions with Like Denominators and Give the Domain, Add or Subtract Basic Rational Expressions with Like Denominators: (a and x+b), Add or Subtract Basic Rational Expressions with Like Denominators: (ax and bx^2), Add or Subtract Rational Expressions with Like Denominators: (x+a) and (x^2-bx+c), Ex 1: Add and Subtract Rational Expressions - Like Denominators, Ex 2: Add and Subtract Rational Expressions - Like Denominators, Ex 3: Add and Subtract Rational Expressions - Like Denominators, Add or Subtract Basic Rational Expressions with Unlike Denominators, Ex: Add and Subtract Rational Expressions - Opposite Denominators, Add Rational Expressions with Unlike Denominators and Give the Domain (Mono Denom), Subtract Rational Expressions with Unlike Denominators and Give the Domain, Subtract Rational Expressions with Unlike Denominators - 3 Expressions, Add and Subtract Rational Expressions with Unlike Denominators - 3 Expressions, Ex 1: Add and Subtract Rational Expressions - Unlike Denominators, Ex 2: Add and Subtract Rational Expressions - Unlike Denominators, Ex 3: Add and Subtract Rational Expressions - Unlike Denominators, Ex 4: Add and Subtract Rational Expressions - Unlike Denominators, Ex: Add and Subtract Rational Expressions with Unlike Monomial Denominators, Ex: Add Rational Expressions with Unlike Denominators, Ex: Subtract Rational Expressions with Unlike Denominators, Ex: Subtracting Rational Expressions (Factoring With A Not 1), Function Arithmetic (Add): f(x)+g(x) with Rational Functions, Function Arithmetic (Subtract): f(x)-g(x) with Rational Functions, Solve Rational Equations with Like Denominators, Solve a Rational Equation: (x-a)/x-b/c=d/x, Solve a Rational Equation: a/(x+b)=-c/(x+d) - No Cross Products, Ex 1: Solve Rational Equations Algebraically by Clearing Fractions and Solve Graphically, Ex 2: Solve Rational Equations Algebraically by Clearing Fractions and Solve Graphically, Ex: A Rational Equation with No Solution, Ex 1: Solve Equations with Fractions (Alternative Method), Ex 2: Solve Equations with Fractions (Alternative Method), Ex 1: Solve a Rational Equation (Alternative Method), Ex 2: Solve a Rational Equation (Alternative Method), Ex: Solve a Rational Equation - Alternative Method, Solve A Rational Equation (No Solution) x/(3x-9)-6=1/(x-3), Given f(x)=3/x-4/(3x), Solve f(x)=-7 (Rational Equation), Ex: Rational Function Outputs and Inputs Application - Time, Distance, Rate, Ex 1: Rational Equation Application - Painting Together, Ex 2: Rational Equation Application - Fill a Pool with Drain Open, Ex 3: Rational Equation Application - Plane and Car Travelling the Same Time, Ex 4: Rational Equation Application - Two Bikers Riding Different Distances, Ex: Rational Equation App - Find Individual Working Time Given Time Working Together, Ex: Rational Equation App - Find a Number Given the Sum of Reciprocals, Ex: Find How Faster Than the Speed Limit Is Needed Given a Travel Time, Rational Equation Application (Quadratic): Wind Speed, Rational Equation App (Quadratic): Find Individual Time Given Together Time, Rational Function Application: Function Value, Equation, End Behavior, Direct Variation Application: Currency Conversion, Ex: Direct Variation Application - Aluminum Can Usage, Inverse Variation Application: Water Temperature and Depth, Ex 2: Inverse Variation - Change of Variables, Ex 3: Inverse Variation - Fractional Variation Constant, Ex: Inverse Variation Application - Number of Workers and Job Time, Ex: Inverse Variation Application - Loudness and Distance, Joint Variation: Determine the Variation Constant (Volume of a Cone), Ex: Determine Rational Function Outputs and Inputs, Determining Vertical and Horizontal Asymptotes of Rational Functions, Determining Slant Asymptotes of Rational Functions, Ex 1: Determine Asymptotes and Graph a Rational Function, Ex 2: Determine Asymptotes and Graph a Rational Function, Ex 3: Determine Asymptotes and Graph a Rational Function, Ex 4: Determine Asymptotes and Graph a Rational Function (Slant), Ex: Determine Horizontal Asymptotes of Rational Functions, Ex: Find the Intercepts and Asymptotes of a Rational Function, Ex: Find the Intercepts, Asymptotes, and Hole of a Rational Function, Ex: Find a Rational Function Given the Vertical Asymptotes and Intercepts, Ex 1: Determine the Vertical and Slant Asymptotes of a Rational Function, Ex 2: Determine the Vertical and Slant Asymptotes of a Rational Function, Ex 1: Domain, Range, Asymptotes of a Basic Rational Function Using Translations, Ex 2: Domain, Range, Asymptotes of a Basic Rational Function Using Translations, Ex 1: Domain, Range, Asymptotes of a Basic Rational Function Using a Graph and Procedure, Ex 2: Domain, Range, Asymptotes of a Basic Rational Function Using a Graph and Procedure, Ex: Match Equations of Rational Functions to Graphs, Find the Intercepts and Asymptotes of a Rational Function (quad/quad with a not 1), Determine the Domain of Various Functions, Determine Vertical Intercepts of Various Functions, Determine Horizontal Intercepts of Various Functions (P1), Determine Horizontal Intercepts of Various Functions (P2), Determine Domain, Holes, and Asymptotes of a Rational Function (1), Determine Domain, Holes, and Asymptotes of a Rational Function (2), The Equation of a Rational Function from a Graph, Graph and Determine Key Components of a Rational Function (linear/linear) 1, Determine Key Components and Graph a Rational Function (linear/linear) 2, Match Rational Functions and Graphs: Translations, Graphing the Basic Rational Function f(x)=1/x, Graphing Reflections of the Basic Rational Function f(x)=1/x, Graphing Translations of the Basic Rational Function f(x)=1/x, Ex 1: Graph Two Translations of the Basic Rational Function f(x)=1/x, Ex 2: Graph Two Translations of the Basic Rational Function f(x)=1/x, Ex 1: Find the Equation of Rational Function From a Graph with a Hole, Ex 2: Find the Equation of Rational Function From a Graph with a Hole, Ex 3: Find the Equation of Rational Function From a Graph, Ex 4: Find the Equation of Rational Function From a Graph (Squared Intercept), Ex 5: Find the Equation of Rational Function From a Graph (Squared VA), Ex 6: Find the Equation of Rational Function From a Graph (Squared Intercept / VA), Rational Function Application - Concentration of a Mixture, Ex: Setting Up Partial Fraction Decomposition, Ex 1: Partial Fraction Decomposition (Linear Factors), Ex 2: Partial Fraction Decomposition (Linear Factors), Ex 3: Partial Fraction Decomposition (Repeated Linear Factors), Ex 4: Partial Fraction Decomposition (Repeated Linear Factors), Ex 5: Partial Fraction Decomposition (Linear and Quadratic Factors), Ex 6: Partial Fraction Decomposition (Repeating Quadratic Factors), Ex: Partial Fraction Decomposition - Degree 2 / Degree 3, Ex: Evaluate Functions and Composite Function in Context of a Story (Graphing Calculator), Ex: Intro Composite Function Notation Application Problem, Ex: Evaluate Composite Functions Using Tables of Values, Ex: Evaluate Composite Functions from Graphs, Ex 1: Determine Composite Function Values Using Table, Graph, and Function Rule, Ex 2: Determine Composite Function Values Using Table, Graph, and Function Rule, Determine f(g(x)) and g(f(x)) with Linear and Quadratic Functions, Determine f(f(x)) and g(g(x)) with Linear and Quadratic Functions, Ex: Find and Evaluate a Composition of Three Functions, Ex 2: Find Composite Function Values With Fractions, Ex: Find a Composition of Functions Involving Rational Functions, Ex: Domain of a Quotient and Composite Functions, Ex: Domain of Composite Function From Graphs, Ex: Inverse Function Notation and Reciprocal of a Function, Linear and Exponential Growth: Complete a Salary Table, Graphing by Plotting Points - Exponential, Graphing by Plotting Points - Exponential (6.3), Complete a Table of Values for an Exponential Equation in Two Variables, Evaluate Exponential Functions: Base 3 and 1/3, Write Exponential Equations Given Initial Values and Growth or Decay Rate, Write Exponential or Linear Equations to Model the Value of an Investment (Level 1), Write Exponential or Linear Equations to Model a Population (Level 2), Determine Exponential Function Values and Graph the Function, Graph a Basic Exponential Function Using a Table of Values, Graph an Exponential Function Using a Table of Values, Evaluate a Given Exponential Function to Predict a Future Population, Introduction to Exponential Equations in Two Variables, Determine the Initial Value and Percent Rate of Change from an Exponential Equation, Write an Exponential Equation that Models a Decreasing Population (Fox), Write an Exponential Equation to Model Wage Percent Increase over Years, Write an Exponential Equation to Model an Account Balance Over Years, Write an Exponential Equation to Model World Population Growth, Determine if Equations Are Linear or Exponential and Increasing or Decreasing, Interpret an Exponential Equation Modeling Depreciation, Interpret an Exponential Equation Modeling Rising Home Value, Compare Exponential Equations Modeling Account Values, Introduction to Exponential Functions in the Form f(x)=ab^x - Part 1, Introduction to Exponential Functions in the Form f(x)=ab^x - Part 2, Introduction to Exponential Functions in the Form f(x)=ae^(kx) - Part 1, Introduction to Exponential Functions in the Form f(x)=ae^(kx) - Part 2, Comparing Forms of Exponential Functions: y = ab^x and y = ae^(kx), Graphing Basic Exponential Functions: Growth and Decay, Interpret the Meaning of Ordered Pairs from a Graph (Exponential), Ex: Determine Exponential Graphs that Have Specific Characteristics - y = ab^x, Ex: Match Exponential Functions to Graphs, Match Exponential Growth and Decay Function with Graphs (Reflections), Describe an Exponential Function Transformation: y=e^(x)+3, Describe an Exponential Function Transformation: y=-2^x-1, Ex: Exponential Application Solved Using a Graphing Calculator, Determine if a Table Represents a Linear or Exponential Function, Ex: End (Long Run) Behavior of Exponential Functions, Ex: Find the Equation of a Transformed Exponential Function From a Graph, Ex: Match the Graphs of Translated Exponential Function to Equations, Ex: Match the Graphs of Reflected Exponential Functions to Equations, Ex 1: Determine if a Table of Value Represents a Linear or Exponential Function, Ex 2: Determine if a Table of Value Represents a Linear or Exponential Function, Ex 1: Determine if a Table of Value Represents a Linear or Exponential Function (Fractions/Decimals), Ex 2: Determine if a Table of Value Represents a Linear or Exponential Function (Fractions/Decimals), Ex 1: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Linear), Ex 2: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Linear), Ex 1: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Exponential), Ex 2: Determine if a Table Represents a Linear or Exponential Function and Find Equation (Exponential), Determine if Two Linear Functions Are Inverses (1), Determine if Two Linear Functions Are Inverses (2), Determine if a Relation Given as a Table is a One-to-One Function, Ex 1: Determine if the Graph of a Relation is a One-to-One Function, Ex 2: Determine if the Graph of a Relation is a One-to-One Function, Ex 1: Determine if Two Functions Are Inverses, Ex 2: Determine if Two Functions Are Inverses, Ex: Find an Inverse Function From a Table, Ex: Function and Inverse Function Values Using a Table, Ex: Function and Inverse Function Values Using a Graph, Ex: Restrict the Domain to Make a Function 1 to 1, Then Find the Inverse, Ex: Find Inverse Function Values Without Finding the Inverse Function, Ex: Function and Inverse Function Values, Ex: Find the Inverse of a Square Root Function with Domain and Range, Ex: Find the Inverse of a Rational Function, Ex: Find the Inverse of a Rational Function and an Inverse Function Value, Ex 1: Determine If Two Functions Are Inverses, Ex 2: Determine If Two Functions Are Inverses, Ex: Find the Inverse Function of an Exponential Function, Ex: Write Exponential Equations as Logarithmic Equations, Ex: Write Logarithmic Equations as Exponential Equations, Ex: Write Exponential Equations as Logarithmic Equations - Variables, Ex: Write Logarithmic Equations As Exponential Equations - Variables, Ex: Write Exponential Equations with base 10 as Common Logarithmic Equations, Ex: Write Exponential Equations as Logarithmic Equations - Natural Logarithms, Use the Definition of a Logarithm to Show the Zero Exponent and Identity Property, Ex 1: Evaluate Logarithms Without a Calculator - Whole Numbers, Ex: Evaluate Logarithmic Expressions without a Calculator - Different Bases, Ex: Evaluate Logarithmic Expressions without a Calculator - Common Log, Ex: Solve a Basic Exponential Equation with Base Ten Using Logarithm Definition, Ex: Solve a Exponential Equation with Base Ten Using Logarithm Definition (Multiple Steps), Ex: Solve a Basic Exponential Equation with Base e Using Logarithm Definition, Ex: Solve a Exponential Equation with Base e Using Logarithm Definition (Multiple Steps), Ex 2: Evaluate Logarithms Without a Calculator - Fractions, Ex: Evaluate Common Logarithms on a Calculator, Ex: Evaluate Common Logarithms Without a Calculator, Ex: Evaluate Common Logarithms on the Calculator, Ex: Evaluate Natural Logarithms on the Calculator, Ex: Determine the Value of a Number on a Logarithmic Scale (Log Form), Ex: Determine the Value of a Number on a Logarithmic Scale (Exponential Form), Ex: Determine the Difference in Order of Magnitude to Two Quantities, Ex: Determine the Difference in Order of Magnitude to Two Quantities (Application), Ex: Graph an Exponential Function and Logarithmic Function, Ex: Properties and Characteristics of a Logarithmic Function, Ex 1: Match Graphs with Exponential and Logarithmic Functions, Ex 2: Match Graphs with Exponential and Logarithmic Functions - Base 10 and e, Ex: Vertical Asymptotes and Domain of Logarithmic Functions, Ex: Find the Domain of Logarithmic Functions, Determine the Domain, Range, and Asymptote of a Log Function y=-ln(x-6), Determine the Domain, Range, and Asymptote of a Log Function y=-log_3(x)+4, Graph Exponential and Logarithmic Functions on the TI-84 (Inverses), Graphing Logarithmic Functions Using Desmos.com, Graphing Log Functions on the TI-84: y=log_(1/2)(x), Graphing Log Functions on the TI-84: y=-log_(3)(x), Graphing Log Functions on the TI-84: y=log_(2)(x-3)+2, Graphing Log Functions on the TI-84: y=2log_(3)(x+1)-3, Graphing Log Functions by Hand: y=log_(1/2)(x), Graphing Log Functions by Hand: y=-log_(3)(x), Graphing Log Functions by Hand: y=log_(2)(x-3)+2, Graphing Log Functions by Hand: y=2log_(3)(x+1)-3, Graphing Log Functions by Hand: y=log_4(x), Graphing Log Functions by Hand: y=log_3(x)+2, Graphing Basic Logarithmic Functions Using Desmos, Expand Logarithms Using the Product Rule for Logs, Expand Logarithms Using Properties of Logarithms Rule and Factoring, Expand Logarithms Using Properties of Logarithms (Expressions), Ex: Expand a Logarithm Containing a Radical, Combine Logarithms Using Properties of Logarithms, Ex: Combine a Sum and Difference of Two Logarithms, Ex 1: Combine a Logarithmic Expression Into One Logarithm, Ex 2: Combine a Logarithmic Expression Into One Logarithm, Ex 1: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 2: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 3: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 4: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Ex 5: Evaluate a Natural Logarithmic Expressing Using the Properties of Logarithms, Using the Inverse Property of Logarithms and Exponentials to Evaluate Expressions, Ex 1: Evaluate a Logarithmic Expression Using the Properties of Logarithms, Ex 2: Evaluate a Logarithmic Expression Using the Properties of Logarithms, Ex: Simplify Log Expression with the Base and Base of the Number are the Same, Ex: Evaluate Exponential Expressions with Logarithmic Exponents, Ex: Change of Base Formula to Evaluate Logarithmic Expressions, Ex: Change of Base Formula to Solve Basic Exponential Equations, Ex: Evaluate Logarithmic Functions Using the Change of Base Formula, Evaluate Logarithms of any Base on the TI-84 (Not using Change of Base), Determine the Human Age of a Dog And the Dog Age, Given the Human Age, Solve Logarithmic Equations for the Base (No Calculator), Solve a Log Equation with Two Logs Equal to Each Other, Ex: Solve a Logarithmic Equations - One Step by Dividing with Rational Solution, Ex: Solve a Logarithmic Equations - One Step by Dividing with Irrational Solution, Ex: Solve a Logarithmic Equations - One Step by Dividing with Log of a Quantity, Ex: Solving a Variety of Logarithmic Equations, Ex: Solve a Basic Logarithmic Equation - Linear and Quadratic, Ex: Solve Basic Logarithmic Equation - Radicals, Ex: Solve Logarithmic Equations Containing Only Logarithms, Ex: Solve a Logarithmic Equation With a Difference - Linear, Ex: Solve a Logarithmic Equation With a Sum - Quadratic Formula, Ex: Solve a Logarithmic Equation With a Difference - Quadratic Formula, Ex: Solve a Logarithmic Equation Requiring the Quadratic Formula, Ex 3: Solve Logarithmic Equations - Base and Number are the Same, Ex: Solve a Logarithmic Equation - Composite Log Expression, Ex: Solve a Logarithmic Equation in Terms of Other Variables, Ex: Solve a Logarithmic Equation with a Fractional Exponent, Solve a Log Equation by Factoring (Trinomial), Logarithm Application: Magnitude of an Earthquake, Logarithm Application: Intensity of Two Sounds (Decibels), Ex: Logarithmic Function Application - Test Scores, Ex: Exponential Function Application with Logarithms, Ex: Logarithmic Function Application - pH (Outputs and Inputs), Ex: Logarithmic Function Application - Preston Curve (Outputs and Inputs), Solving Exponential Equations by Obtaining a Common Base, Ex 1: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 2: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 3: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 4: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 5: Solve Exponential Equations Using Like Bases - No Logarithms, Ex 6: Solve Exponential Equations Using Like Bases - No Logarithms, Ex: Solve an Exponential Equation Graphically on the TI84, Solving Exponential Equations Using Logarithms, Ex 1: Solve a Basic Exponential Equation Using the Definition of a Logarithm, Ex 2: Solve a Basic Exponential Equation Using the Definition of a Logarithm, Ex 1: Solve Exponential Equations Using Logarithms, Ex 2: Solve Exponential Equations Using Logarithms, Ex 3: Solve Exponential Equations Using Logarithms, Ex 4: Solve Exponential Equations Using Logarithms, Ex 5: Solve Exponential Equations with Two Exponential Parts Using Logarithms, Ex 6: Solve an Exponential Equation Using Common Log, Ex 7: Solve an Exponential Equation Using Natural Log (Factoring), Ex: Solve an Exponential Equation Using Logarithms, Ex: Solve an Exponential Equation with Logarithms - Variable on Both Sides, Ex: Rewrite Exponential Functions: y = ab^t to y = ae^(kt), Ex: Rewrite Exponential Functions: y = ae^(kt) to y = ab^t, Determine a Continuous Exponential Decay Function and Make a Prediction, Ex: Write Linear and Exponential Functions Based Upon Given Information, Write an Exponential Function for Growth over Different Time Intervals, Ex: Find a Linear and Exponential Model for Population Growth, Ex: Use a Given Exponential Growth Function, Ex: Exponential Growth of Bacteria (Intro Question), Ex: Solve an Exponential Growth Equation Graphically Using the TI84 (Application), Ex: Compare Simple Interest and Annual Compounded Interest, Ex: Solve an Exponential Growth Equation Graphically Using the Desmos (Application), Ex: Solve an Exponential Decay Equation Graphically Using the TI84 (Application), Ex: Solve an Exponential Decay Equation Graphically Using the Desmos (Application), Ex: Determine the Doubling Time of an Investment Account Graphically (TI84), Exponential Growth App (y=ab^t) - Given Doubling Time, Ex: Determine the Half-Life of a Fish Population Graphically (TI84), Ex: Exponential Function Applications - Increasing Investment Value (Change of Base Used), Ex: Exponential Function Applications - Decreasing Water Level (Change of Base Used), Ex: Exponential Growth Function - Population, Ex: Exponential Function Application Using Logs - Export Values, Ex: Exponential Growth Function - Bacterial Growth, Ex: Exponential Decay Function with Logarithms, Ex: Exponential Growth Application - Predicting World Population, Ex: Basic Example of Exponential Decay Model, Ex: Exponential Decay Function - Half Life, Ex: Find the Initial Value and Exponential Growth or Decay Rate Given an Exponential Function, Ex: Exponential Functions: Growth Rate and Growth Factor, Ex: Exponential Functions: Decay Rate and Decay Factor, Ex: Determine Growth Exponential Functions Given Growth Rate and Initial Value (y=ab^x), Ex: Determine Exponential Decay Functions Given Decay Rate and Initial Value (y=ab^x), Exponential Decay App (y=ab^x) - Given Half Life, Exponential Decay App (y=ae^(kt)) - Given Half Life, Exponential Decay App (y=ab^t) - Find Initial Amount Given Half Life, Exponential Decay App with Logs (y=ae^(kt)) - Find Half Life, Ex: Exponential Model - Determine Age Using Carbon-14 Given Half Life, Find Initial Amount and Amount After Time From Half-Life (y=ae^(kt)), Exponential Growth App (y=ab^t) - Find Initial Amount Given Doubling Time, Ex: Practice Writing Exponential Equations - Doubling Equation and Halving Equation, Exponential Growth App with Logs (y=ae^(kt)) - Find Initial Amount Given Doubling Time, Ex: Find an Exponential Growth Function Given Two Points - Initial Value Given, Ex: Find an Exponential Decay Function Given Two Points - Initial Value Given, Ex: Find an Exponential Function Given Two Points - Initial Value Not Given, Exponential Function Application (y=ab^x) - Population Growth of India, Exponential Function Application (y=ab^x) - Population Decline of Chicago, Exponential Function Application (y=ab^x) - Depreciation of a Car, Annual Depreciation of a New Car: Find the Future Value, Exponential Function Application (y=ab^x) - Declining Computer Value, Determine an Exponential Decay Function P(t)=a(b)^t (No Logs), Exponential Function Application (y=ab^x) - Home Values, Exponential Function Application (y=ae^(kt)) - Bacteria Growth, Ex: Application - Cyclical Around Linear Growth, Ex: Application - Cyclical Around Exponential Growth, Ex: Find Annual Interest Rate Given f(t)=ae^(kt), Ex: Find Annual Depreciation Rate Given f(t)=ae^(kt), Ex: Find Continuous Interest Rate Given f(t)=ab^t, Ex: Find Continuous Depreciation Rate Given f(t)=ab^t, Determine an Exponential Function on the TI-84 Given Two Points (Growth), Determine an Exponential Function on the TI-84 Given Two Points (Decay), Determine the Total Return of an Investment as Percent, Compounded Interest: Solve for Principal (Present Value), Ex 2: Compounded Interest with Logarithms, Ex: Find Exponential Growth Rate and Make Prediction of a Future Home Value, Compounded Interest (Slightly Different Form of Equation): Find Future Value, Effective Interest Rate (Effective Yield), Ex 2: Continuous Interest with Logarithms, Ex 3: Continuous Interest with Logarithms and Doubling Time, Continuous Interest - Find the Initial Investment Needed, Ex: Perform Exponential Regression on a Graphing Calculator, Perform Exponential Regression and Make Predictions Using Desmos, Ex: Exponential Regression Application on the TI84 (Decreasing Polio Case), Ex: Exponential Decay Regression Model (Declining Population), Ex: Exponential Growth Regression Model (Investment Account), Ex: Comparing Linear and Exponential Regression, Ex: Find an Exponential Function for a Semi-Log Graph, Ex: Newton's Law of Cooling - Exponential Function App, Exponential Function App. Pyrex® Red Lid For 8" Square Glass Baking Dish,
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