Domain of Combinations of Functions Video 4
Finding the domain of a function with a root and a logarithm in the denominator
Problem: Find the domain of the functions:
Finding the domain of a function with a root and a logarithm in the denominator
Explaining domain restrictions for denominators, even roots, and logarithms
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
Explaining how to solve equations with power and exponential terms
Explaining the basic properties of logarithmic functions
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
Determining where functions are continuous
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 exponential functions by factoring
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Determining the domain of a three-dimensional vector function
Finding and sketching the domain of a function of two variables
Finding the domain of even and odd roots
Finding the domain of functions with even and odd roots
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
Discussing properties of linear functions
Discussing the types of linear graphs including horizontal and vertical lines and their properties
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 the general properties of even and odd root functions
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
Solving equations with exponential terms
Solving equations with base e exponential terms
Properties of exponential growth and decay functions
Properties of the exponential function with base e
Explaining the terminology and rules of logarithms
Discussing parent functions of rational functions and their properties
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
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 domain of a piecewise function and evaluating it
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
Using properties of logarithms to determine whether a statement is true or false
Using properties of logarithms to determine whether a statement is true or false
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
Examples of finding derivatives and calculating approximate profit
Derivative examples with logarithms
Determining where functions are continuous
Examples of differentiating exponential and logarithmic functions
Explaining the formulas for the derivatives of exponential functions
Calculating a derivative of a quotient with an exponential function
Explaining the definition of logarithmic scale and double-log plots
Explaining the logarithmic scale and a semilog plot
Using logarithms to transform a power function into a linear function
Writing a non-linear function for the graph containing two points on a double-log plot
Using logarithms to transform an exponential function to a linear function
Writing a non-linear function to represent the line graphed on a semilog plot
Using the second derivative to find the maximum rate of change in a word problem
Solving a rational equation and checking the solutions
Finding the domains of functions
Finding the domains of expressions
Determining the properties of a rational function and graphing it
Determining the properties of a rational function and graphing it
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
Determining properties of an exponential function
Simplifying logarithmic expressions using the properties of logarithms
Using the properties of logarithms to expand a logarithmic expression
Using the properties of logarithms to expand a logarithmic expression
Using the properties of logarithms to condense the expression as a single logarithm
Using the properties of logarithms to condense the expression as a single logarithm
Using the formula to change the base of a logarithm
Using the formula to change the base of a logarithm
Solving an equation with an exponential function
Solving an equation with an exponential function
Modeling a population size with an exponential function
Using a graph to evaluate a function
Cartesian equations and parametric equations of curves
Review of limits and derivatives of inverse trigonometric functions
Derivatives of exponential functions and the exponential model
Review of limits and derivatives of inverse trigonometric functions
Mean Value Theorem and using derivatives to find the shape of curves
Properties and derivatives of inverse trigonometric functions
Continuity of Functions and the Intermediate Value Theorem
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 L'Hospital's Rule to solve limits
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Finding the derivative of a function containing a logarithmic function
Finding the derivatives of functions containing logarithms
Finding the derivative of a function containing a logarithms
Finding the equation of a tangent line to a function containing a logarithm
Using logarithmic differentiation to find the derivative of a function
Using logarithmic differentiation to find the derivative of a function
Using logarithmic differentiation to find the derivative of a function
Using the Product and Chain Rules to differentiate functions with logarithms
Differentiating functions with logarithmic and inverse trigonometric functions
Using the chain rule to find a derivative
Using logarithmic differentiation to find the derivative of a function
Using logarithmic differentiation to find the derivative of a function
Finding a number that satisfies the conclusion of the Mean Value Theorem on an interval
Evaluating a limit using L'Hospital's Rule
Evaluating a limit with an indeterminate power using L'Hospital's Rule
Evaluating a limit with an indeterminant power using L'Hospital's Rule
Antidifferentiating twice to find a function from its second derivative
Antidifferentiating to find a function from its derivative
Antidifferentiating twice to find a function from its second derivative
Antidifferentiating twice to find a function from its second derivative
Using antidifferentiation to find the height of a cliff from the impact speed of a dropped stone
Evaluating a limit using logarithm properties
Evaluating a limit using L'Hospital's Rule
Evaluating a limit with an indeterminant power using L'Hospital's Rule
Evaluating a limit at infinity by combining logarithms
Finding a derivative using the Chain Rule and Product Rule
Finding a derivative using the Quotient Rule
Using logarithmic differentiation to find the derivative of a function
Using L'Hospital's Rule to solve a limit with an indeterminant product
Using L'Hospital's Rule to solve a limit with an indeterminant difference
Evaluating an indefinite integral by simplifying a fraction to powers of x
Evaluating a definite integral by first simplifying the integrand
Evaluating a definite integral with an exponential and sine function
Using u-substitution to evaluate an indefinite integral
Examples reviewing integration techniques from Calculus I
Review of estimating the sum of a series and the remainder
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Review of power series properties and writing functions as power series
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Determining properties of sequences
Finding power series representations for functions and the radius and interval of convergence of the power series
Examples of integration by substitution
Examples finding the area bounded by curves
Using the disk, washer, and shell method to find a volume of revolution
Examples using integration by parts
Using the method of partial fractions to integrate rational functions
Integrating a rational function using u-substitution
Integrating a rational function using u-substitution
Finding the volume of a solid of revolution using the washer method
Finding the volume of a solid of revolution using cylindrical shells
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Determining if a sequence converges or diverges
Determining if a sequence is increasing, decreasing, or not monotonic and if it is bounded
Using the comparison test to determine if a series converges or diverges
Using the comparison test to determine if a series converges or diverges
Finding a power series representation for a function
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
Finding the interval and radius of convergence for a power series
Solving an indefinite integral with a natural logarithm using integration by parts
Solving a definite integral with a natural logarithm using integration by parts
Using u-substitution to evaluate a definite integral with a natural logarithm
Solving an improper integral where the upper limit of integration is infinity
Solving an improper integral where the upper limit of integration is infinity
Solving an improper integral where the upper limit of integration is infinity
Cartesian equations and parametric equations of curves
Review of limits and derivatives of inverse trigonometric functions
Derivatives of exponential functions and the exponential model
Review of limits and derivatives of inverse trigonometric functions
Mean Value Theorem and using derivatives to find the shape of curves
Properties of inverse trig functions and the derivative of arctangent
Properties and derivatives of inverse trigonometric functions
Continuity of Functions and the Intermediate Value Theorem
Properties of inverse trig functions and the derivative of arctangent
Properties and derivatives of inverse trigonometric functions
Proving facts about logarithms and exponentials including the derivative of an exponential with an arbitrary base
Proving the derivative of ln(x) and the power and product rules using logarithmic differentiation
Derivatives of exponential and logarithmic functions and the exponential model
Proving L'Hospital's Rule and using it to evaluate limits
Using L'Hospital's Rule to solve limits
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Using u substitution to evaluate integrals and prove facts about logarithms and integrals and
Examples of integration by substitution
Examples reviewing integration techniques from Calculus I
Review of estimating the sum of a series and the remainder
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Review of power series properties and writing functions as power series
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Determining properties of sequences
Finding power series representations for functions and the radius and interval of convergence of the power series
Examples of integration by substitution
Examples finding the area bounded by curves
Using the disk, washer, and shell method to find a volume of revolution
Examples using integration by parts
Using the method of partial fractions to integrate rational functions
Finding a partial fraction decomposition and integrating using partial fractions
Finding the position vector function given the velocity and an initial position
Finding and sketching the domain of a function of two variables
Finding the partial derivatives of a function of two variables
Performing basic numerical calculations in Python
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 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 when a quadratic equation does not have a real solution
How to rewrite radical terms as power 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
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
Adding and subtracting rational expressions by finding a common denominator
Finding the limit of a sequence using logarithm properties
Finding the limit of a sequence
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