Domain of Rational Functions Video 3
Explaining how to find any holes and vertical asymptotes of a rational function
Problem: Find the domain of the function and any holes or vertical asymptotes: \(f(x)=\dfrac{x^2+x}{x^3+2x^2+x}\)
Explaining how to find any holes and vertical asymptotes of a rational function
Discussing parent functions of rational functions and their properties
Finding the domain, intercepts, asymptotes, and holes of a rational function
Finding the domain, intercepts, asymptotes, and holes of a rational function
Finding the domain, intercepts, asymptotes, and holes of a rational function
Determining the properties of a rational function and graphing it
Determining the properties of a rational function and graphing it
Solving a rational equation and checking the solutions
Finding the domain of a function with a denominator, square root, and logarithm
Finding the domain of a function with a natural logarithm and denominator
Finding the domain of several rational functions
Finding the domain of a rational function
Explaining the basic properties of logarithmic functions
How to simplify rational expressions
Explaining how to multiply and divide rational expressions
Performing algebraic operations with rational expressions and simplifying the answer
Solving equations that contain rational expressions
Solving an equation that contains rational expressions
Solving an equation that contains rational expressions
Finding the domain of a piecewise function and evaluating it
Subtracting rational functions and then simplifying the result
Multiplying two rational functions and simplifying the result
Dividing rational functions and simplifying the result
Dividing and subtraction rational functions and simplifying the result
Determining where functions are continuous
Using Information From the Derivative to Sketch a Graph
Dividing rational expressions and simplifying
Subtracting rational expressions and simplifying
Performing operations with rational expressions and simplifying
Performing operations with rational expressions and simplifying
Finding the domains of expressions
Determining properties of an exponential function
Determining the properties of a logarithmic function
Determining the graph of a logarithmic function
Determining the properties of a logarithmic function
Determining the graph of a logarithmic function
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving an equation with exponential functions by factoring
Solving an equation with logarithmic functions
Review of limits and derivatives of inverse trigonometric functions
Review of limits, continuity, and the Intermediate Value Theorem
Review of limits and derivatives of inverse trigonometric functions
Properties and derivatives of inverse trigonometric functions
Evaluating limits of functions
Evaluating Limits of Functions
Continuity of Functions and the Intermediate Value Theorem
Limits at infinity and asymptotes, along with physics applications
Limits at infinity, asymptotes, and tangent lines
Properties and derivatives of inverse trigonometric functions
Using derivatives to find properties of graphs
Mean Value Theorem and properties of a graph
Mean Value Theorem and properties of a graph
Using derivatives to find properties of graphs
Finding the vertical and horizontal asymptotes for a rational function
Review of limits, continuity, and the Intermediate Value Theorem
Review of limits and derivatives of inverse trigonometric functions
Review of limits and derivatives of inverse trigonometric functions
Properties and derivatives of inverse trigonometric functions
Evaluating limits of functions
Evaluating limits of functions and using the Squeeze Theorem
Continuity of Functions and the Intermediate Value Theorem
Limits at infinity and asymptotes, along with physics applications
Limits at infinity, asymptotes, and tangent lines
Properties and derivatives of inverse trigonometric functions
Using derivatives to find properties of graphs
Mean Value Theorem and properties of a graph
Mean Value Theorem and properties of a graph
Using derivatives to find properties of graphs
Determining the domain of a three-dimensional vector function
Finding the limit of a three-dimensional vector function
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Finding the domain of a function with a root and a logarithm in the denominator
Finding the domain of a logarithmic function
Finding the domain of even and odd roots
Finding the domain of functions with even and odd roots
Explaining domain restrictions for denominators, even roots, and logarithms
Discussing properties of linear functions
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Explaining the standard form for a quadratic equation and the possible number of solutions
Solving quadratic equations with the difference of two squares formula
Solving quadratic equations by factoring
Solving a quadratic equation by factoring
Solving a quadratic equation by factoring
Solving quadratic equations with the quadratic formula and discussing the number of possible solutions
Explaining the graph and properties for the parent function of quadratics
Discussing properties of quadratic functions including the vertex, domain, range, and end behavior
Explaining how to solve radical equations and then solving example problems
Solving equations with even roots and checking the solutions
Solving equations with odd radicals
Solving equations with multiple radicals
Explaining the general properties of even and odd root functions
Properties of exponential growth and decay functions
Properties of the exponential function with base e
Explaining how to solve equations with power and exponential terms
Expanding logarithmic expressions using logarithm rules
Condensing a logarithmic expression into a single logarithm
Explaining how to solve logarithmic equations
Solving logarithmic equations with a single logarithm term
Solving logarithmic equations with multiple logarithm terms
Solving logarithmic equations with multiple logarithm terms on both sides
Finding horizontal asymptotes for rational functions
Explaining how to do polynomial long division
Finding horizontal asymptotes for rational functions
Determining the end behavior and horizontal asymptotes for rational functions
Adding and subtracting rational expressions by finding a common denominator
Determining the domain and range of a relation and if the relation is a function
Determining the domain and range of a relation and if the relation is a function
Determining the domain and range of a relation and if the relation is a function
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Finding the piecewise-defined function for a given graph along with its domain and range
Finding the properties of an exponential function including domain, range, end behavior, and intercepts
Finding the properties of an exponential function including domain, range, end behavior, and intercepts
Finding the properties of an exponential function including domain, range, end behavior, and intercepts
Finding the domain of a function that is a fraction containing an exponential and root
Finding the domain of a function that is a fraction containing an exponential and root
Finding the domain of a function that is a fraction containing exponentials and a square root
Solving an equation with exponential functions
Computing and simplifying the difference quotient for a rational function
Finding the difference quotient for a quadratic function
Adding and simplifying rational functions
Review of limits, continuity, and differentiability
Review of limits at infinity and continuity of functions
Examples of calculating infinite limits with exponentials in a rational function
Review of derivative rules such as product, quotient, and chain rule
Absolute extrema on closed intervals
Factoring rational functions to solve limits
Limits at infinity of rational functions
Determining where functions are continuous
Examples of Product, Quotient, and Chain Rule
Examples of Product, Quotient, and Chain Rule
Determining concavity and inflection points and sketching the graph
Using the Derivative to Sketch the Graph of a Function
Using Critical Values to Find Absolute Extrema on Closed Intervals
Finding Absolute Extrema on Open Intervals
Solving examples finding the area between curves
Finding the area between two curves
Factoring an algebraic expression
Factoring a quadratic
Factoring a polynomial
Factoring a quadratic
Solving a polynomial equation using factoring by grouping
Finding the domains of functions
Finding the x-intercepts of quadratic functions
Finding properties of a polynomial including zeros and end behavior
Finding properties of a polynomial including zeros and end behavior
Finding properties of a polynomial including zeros and end behavior
Using properties of a polynomial to find its graph
Solving a nonlinear inequality
Solving a nonlinear inequality
Solving a nonlinear inequality
Solving a nonlinear inequality
Restricting the domain of a function so it is one-to-one and has an inverse
Determining algebraically the inverse of a function
Determining algebraically the inverse of a function
Determining algebraically the inverse of a function
Determining properties of an exponential function
Determining properties of an exponential function
Determining properties of an exponential function
Identifying the graph of an exponential function
Solving an equation with logarithmic functions
Solving an equation with an exponential function
Solving an equation with logarithmic functions
Solving a trigonometric equation by factoring
Solving a trigonometric equation
Calculating the difference quotient for a rational function
Calculating the difference quotient for a rational function
Using a graph to evaluate a function
Cartesian equations and parametric equations of curves
Review of the limit definition of a derivative and calculating the derivative
Related rates problems, differentials, linear and quadratic approximations
Review of the limit definition of a derivative and calculating the derivative
Mean Value Theorem and using derivatives to find the shape of curves
Review of the limit definition of a derivative and calculating the derivative
Differentials, linear approximations and quadratic approximations
Review of the limit definition of a derivative and calculating the derivative
Derivatives of exponential and logarithmic functions and the exponential model
Derivatives of exponential and logarithmic functions and the exponential model
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Using the Quotient Rule to differentiate a rational function
Evaluating a limit using L'Hospital's Rule
Using the first and second derivatives to sketch the graph of a function
Evaluating a limit using logarithm properties
Evaluating a limit with an indeterminate difference using a common denominator
Finding the antiderivative of a rational function
Evaluating a a limit from the left for a rational function
Evaluating the limit of a rational function by factoring and cancelling
Using L'Hospital's Rule to evaluate a limit representing a derivative at a point
Evaluating the limit at infinity of a rational function
Evaluating a definite integral by first simplifying the integrand
Reviewing u-Substitution
Review of partial fractions
Review of partial fractions
Examples of integration by substitution
Using the method of partial fractions to integrate rational functions
Using the method of partial fractions to integrate rational functions
Determining divergence or convergence of improper integrals
Integrating a rational function using u-substitution
Integrating a rational function using u-substitution
Finding the length of a curve given by parametric equations
Using u-substitution to integrate a rational function
Explaining all four cases of partial fraction decomposition
Using partial fraction decomposition to rewrite and then integrate a rational function
Using partial fraction decomposition to rewrite and then integrate a rational function
Using partial fraction decomposition to rewrite and then integrate a rational function
Using partial fraction decomposition to rewrite and then integrate a rational function
Using partial fraction decomposition to rewrite and then integrate a rational function
Proving a certain parameterization gives the unit circle and showing vector equations of lines are parallel
Cartesian equations and parametric equations of curves
Using the limit definition to find derivatives of functions and vector functions
Review of the limit definition of a derivative and calculating the derivative
Related rates problems, differentials, linear and quadratic approximations
Review of the limit definition of a derivative and calculating the derivative
Mean Value Theorem and using derivatives to find the shape of curves
Properties of inverse trig functions and the derivative of arctangent
Proving a piecewise function and a polynomial are continuous
Using the limit definition to find derivatives of functions and vector functions
Using the limit definition to find derivatives of functions and vector functions
Review of the limit definition of a derivative and calculating the derivative
Differentials, linear approximations, and quadratic approximations
Finding a formula for a higher order derivative and a formula for the second derivative of a product
Review of the limit definition of a derivative and calculating the derivative
Using the Chain Rule to prove facts about derivatives
Finding a formula for a higher order derivative and a formula for the second derivative of a product
Properties of inverse trig functions and the derivative of arctangent
Derivatives of exponential and logarithmic functions and the exponential model
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Reviewing u-Substitution
Review of partial fractions
Review of partial fractions
Examples of integration by substitution
Using the method of partial fractions to integrate rational functions
Using the method of partial fractions to integrate rational functions
Finding a partial fraction decomposition and integrating using partial fractions
Determining divergence or convergence of improper integrals
Finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Evaluating a double integral over a given region by splitting it into two double integrals
Finding the domain of a function with a square root in the denominator
Explaining the main three forms for linear equations
Discussing how to write the equation of a line through two given points
Writing a linear equation of profit from a business word problem
Solving a linear equation algebraically and showing how this relates to the graph
Solving a linear equation algebraically
Solving a linear equation algebraically and showing how this relates to the graph
Explaining the format and possible solutions for a system of two linear equations
Solving a system of linear equations with two variables
Solving a system of two linear equations with two variables using the elimination method
Solving a system of two linear equations using a TI-84 calculator
Solving a system of two linear equations using a TI-84 calculator
Solving a system of two linear equations using a TI-84 calculator
Explaining when a quadratic equation does not have a real solution
How to rewrite radical terms as power terms
Solving equations with exponential terms
Solving equations with base e exponential terms
Simplifying power and exponential expressions using properties of exponents
Simplifying power and exponential expressions using properties of exponents
Simplifying power and exponential expressions using properties of exponents
Simplifying an expression with both power and exponential functions using properties of exponents
Properties of exponents applied to power and exponential expressions
Properties of multiplying and simplifying power and exponential expressions
Properties of exponents applied to power and exponential expressions
Explaining the terminology and rules of logarithms
Explaining basic algebraic operations for fractions
Evaluating and then finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Sketching the level curves for a function of two variables
Sketching the level curves for a function of two variables
Explaining all four cases of partial fraction decomposition
Explaining the format and possible solutions for a system of two linear equations