Solving Rational Equations Video 3
Solving an equation that contains rational expressions
Problem: Solve the equation \(\quad\dfrac{2}{x+3}+\dfrac{x}{x-1}=3.\)
Solving an equation that contains rational expressions
Solving equations that contain rational expressions
Finding the domain of a function with a denominator, square root, and logarithm
Explaining the standard form for a quadratic equation and the possible number of solutions
Solving quadratic equations with the quadratic formula and discussing the number of possible solutions
Explaining when a quadratic equation does not have a real solution
Explaining how to solve radical equations and then solving example problems
Solving equations with even roots and checking the solutions
Solving equations with multiple radicals
Performing algebraic operations with rational expressions and simplifying the answer
Finding the domain of a piecewise function and evaluating it
Subtracting rational expressions and simplifying
Performing operations with rational expressions and simplifying
Performing operations with rational expressions and simplifying
Solving an equation using the quadratic formula
Solving a rational equation and checking the solutions
Solving a quadratic equation using the quadratic formula
Finding the domains of expressions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Review of limits and derivatives of inverse trigonometric functions
Review of limits and derivatives of inverse trigonometric functions
Continuity of Functions and the Intermediate Value Theorem
Explaining all four cases of partial fraction decomposition
Review of limits and derivatives of inverse trigonometric functions
Review of limits and derivatives of inverse trigonometric functions
Continuity of Functions and the Intermediate Value Theorem
Determining the domain of a three-dimensional vector function
Finding the domain of several rational functions
Finding the domain of a rational function
Explaining how to find any holes and vertical asymptotes of a rational function
Finding the domain of a rational function using factoring
Explaining domain restrictions for denominators, even roots, and logarithms
Finding the domain of a function with a natural logarithm and denominator
Finding the domain of even and odd roots
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
Explaining the graph and properties for the parent function of quadratics
Discussing properties of quadratic functions including the vertex, domain, range, and end behavior
Solving equations with odd radicals
Explaining the general properties of even and odd root functions
Explaining how to solve equations with power and exponential terms
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
Discussing parent functions of rational functions and their properties
Adding and subtracting rational expressions by finding a common denominator
Explaining how to multiply and divide rational expressions
Explaining all four cases of partial fraction decomposition
Finding the maximum revenue, maximum profit, and break even quantity for given revenue and cost functions
Finding the piecewise-defined function for a given graph along with its domain and range
Solving an equation with exponential functions
Finding the domain of a function that is a fraction containing exponentials and a square root
Dividing and subtraction rational functions and simplifying the result
Computing and simplifying the difference quotient for 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
Finding the domain, intercepts, asymptotes, and holes of a rational function
Adding and simplifying rational functions
Subtracting rational functions and then simplifying the result
Determining where functions are continuous
Dividing rational expressions and simplifying
Solving a quadratic equation by completing the square
Finding the domains of functions
Determining the properties of a rational function and graphing it
Determining the properties of a rational function and graphing it
Solving an equation with an exponential function
Solving an equation with exponential functions by factoring
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving a trigonometric equation by factoring
Solving a trigonometric equation
Review of limits, continuity, and the Intermediate Value Theorem
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
Properties and derivatives of inverse trigonometric functions
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
Properties and derivatives of inverse trigonometric functions
Derivatives of exponential and logarithmic functions and the exponential model
Derivatives of exponential and logarithmic functions and the exponential model
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Evaluating a limit with an indeterminate difference using a common denominator
Review of work and average value
Reviewing u-Substitution
Examples of integration by substitution
Examples finding a volume using disks, washers, and by slicing
Using integrals to solve work problems and find the average value
Using the limit definition to find derivatives of functions and vector functions
Review of limits, continuity, and the Intermediate Value Theorem
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
Properties and derivatives of inverse trigonometric functions
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
Review of the limit definition of a derivative and calculating the derivative
Properties and derivatives of inverse trigonometric functions
Derivatives of exponential and logarithmic functions and the exponential model
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Review of work and average value
Reviewing u-Substitution
Examples of integration by substitution
Examples finding a volume using disks, washers, and by slicing
Using integrals to solve work problems and find the average value
Using integrals to solve work problems and find the average value
Finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Finding the domain of a function with a square root in the denominator
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 functions with even and odd roots
Solving a linear equation algebraically
Discussing properties of linear functions
Discussing the types of linear graphs including horizontal and vertical lines and their properties
How to rewrite radical terms as power terms
Properties of exponential growth and decay functions
Properties of the exponential function with base e
Solving equations with exponential terms
Solving equations with base e exponential terms
Explaining the basic properties of logarithmic functions
Expanding logarithmic expressions using logarithm rules
Condensing a logarithmic expression into a single logarithm
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
Explaining basic algebraic operations for fractions
How to simplify rational expressions
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
Identifying quadratic functions and their properties
Graphing a piecewise function
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
Solving an equation with exponential functions
Identifying the parent function and transformations for a given graph
Finding the difference quotient for a quadratic function
Multiplying two rational functions and simplifying the result
Dividing rational functions and simplifying the result
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
Using the power rule to find derivatives
Examples of Product, Quotient, and Chain Rule
Examples of Product, Quotient, and Chain Rule
Determining concavity and inflection points and sketching the graph
Using Information From the Derivative to Sketch a Graph
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
Explaining the Intermediate Value Theorem and the Bisection Method
Using the Intermediate Value Theorem and Bisection Method to approximate the solution of an equation
Using derivatives to find the velocity and acceleration functions from the position function
Simplifying an expression with radicals
Rationalizing the denominator of a fraction with square roots
Rationalizing the denominator of a fraction with square roots
Factoring a quadratic
Factoring a polynomial
Factoring a quadratic
Solving a polynomial equation using factoring by grouping
Determining if an equation represents a function
Putting a quadratic function in standard form and determining its properties
Putting a quadratic function in standard form and determining its properties
Finding the x-intercepts of quadratic functions
Putting a quadratic function in standard form and determining its properties
Finding the end behavior of a polynomial
Finding properties of a polynomial including zeros and end behavior
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 an exponential function
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 an exponential function
Solving an equation with an exponential function
Modeling a population size with an exponential function
Solving a trigonometric equation
Solving a trigonometric equation
Solving a trigonometric equation
Solving a trigonometric equation
Solving a trigonometric equation
Calculating the difference quotient for a rational function
Calculating the difference quotient for function with a square root
Calculating the difference quotient for a rational function
Using a graph to evaluate a function
Using a double angle formula to solve a trigonometric equation
Using a double angle formula to solve a trigonometric equation
Using a double angle formula to solve a trigonometric equation
Using double angle formulas to evaluate trigonometric functions
Cartesian equations and parametric equations of curves
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Mean Value Theorem and using derivatives to find the shape of curves
Evaluating limits of functions
Evaluating Limits of Functions
Approximation and Newton's Method, and limits and derivatives of exponential functions
Limits at infinity and asymptotes, along with physics applications
Limits at infinity, asymptotes, and tangent lines
Approximation and Newton's Method, and limits and derivatives of exponential functions
Mean Value Theorem and properties of a graph
Mean Value Theorem and properties of a graph
Riemann sums, including approximating areas under curves.
Finding the vertical and horizontal asymptotes for a rational function
Finding the derivative of a function using the limit definition
Using the Quotient Rule to differentiate a rational function
Using a linear approximation to approximate the value of a function
Evaluating a limit using L'Hospital's Rule
Finding the antiderivative of a rational function
Evaluating a limit using logarithm properties
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
Examples reviewing integration techniques from Calculus I
Review of partial fractions
Review of partial fractions
Review of finding the radius and interval of convergence for a power series
Review of Taylor and Maclaurin Series and their properties
Review of Taylor Polynomials and Taylor's Inequality
Finding a Taylor Polynomial for a function
Examples finding the area bounded by curves
Using the disk, washer, and shell method to find a volume of revolution
Finding volume using using disks, washers and the shell method
Examples using integration by parts
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
Integrating a function with a square root using u-substitution
Finding the area of a region enclosed by two curves
Finding the area of a region bounded by curves
Finding the volume of a solid of revolution using the washer method
Finding the volume of a solid of revolution using cylindrical shells
Using the comparison test to determine if an improper integral converges or diverges
Using the comparison test to determine if an improper integral converges or diverges
Converting parametric equations into a Cartesian equation and graphing
Using u-substitution to 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
Using partial fraction decomposition to rewrite and then integrate a rational function
Proving associative and scalar multiplication properties for vectors
Proving a certain parameterization gives the unit circle and showing vector equations of lines are parallel
Cartesian equations and parametric equations of curves
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Mean Value Theorem and using derivatives to find the shape of curves
Properties of inverse trig functions and the derivative of arctangent
Evaluating limits of functions
Proving a property of scalar multiplication for limits using the epsilon-delta definition and using the Squeeze Theorem for Limits.
Evaluating limits of functions and using the Squeeze Theorem
Proving a piecewise function and a polynomial are continuous
Approximation and Newton's Method, and limits and derivatives of exponential functions
Limits at infinity and asymptotes, along with physics applications
Limits at infinity, asymptotes, and tangent lines
Finding a formula for a higher order derivative and a formula for the second derivative of a product
Finding a formula for a higher order derivative and a formula for the second derivative of a product
Approximation and Newton's Method, and limits and derivatives of exponential functions
Properties of inverse trig functions and the derivative of arctangent
Mean Value Theorem and properties of a graph
Mean Value Theorem and properties of a graph
Riemann sums, including approximating areas under curves.
The definition and properties of definite integrals
Examples reviewing integration techniques from Calculus I
Review of partial fractions
Review of partial fractions
Review of finding the radius and interval of convergence for a power series
Review of Taylor and Maclaurin Series and their properties
Review of Taylor Polynomials and Taylor's Inequality
Finding a Taylor Polynomial for a function
Examples finding the area bounded by curves
Using the disk, washer, and shell method to find a volume of revolution
Finding volume using using disks, washers and the shell method
Examples using integration by parts
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 the limit of a three-dimensional vector function
Finding a vector equation for the tangent line to a three-dimensional vector function
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Defining variables in Python and using the variables in equations
Evaluating a double integral over a given region by splitting it into two double integrals
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
Explaining the main three forms for linear equations
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
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
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 the format and possible solutions for a system of two linear equations