Where the Graph of a Function is Not Continuous
Determining where a piecewise function is continuous from its graph
List Comprehension and Numerical, Graphical, and Symbolic Limits in Python
Author: David Manuel
In this video, it is explained how to use list comprehension in Python to numerically estimate a limit. The presenter also explains how to use Python to graphically estimate a limit and find the exact limit.
Transcript 07
Transcript 07
Problem: Numerically estimate, graphically estimate, and evaluate exactly the following limit \[\displaystyle \lim_{x\rightarrow 1} \frac{\sqrt{x}-1}{x^3-1}\]
Python Code:
from sympy import *
x=symbols('x')
f=(sqrt(x)-1)/(x**3-1)
xval=[0.9,0.99,0.999,1.1,1.01,1.001,1.0001]
yval=[f.subs(x,i) for i in xval]
print(yval)
print('The limit appears to be about 1.6666')
matplotlib notebook
plot(f,(x,0,2))L=limit(f,x,1)print('The limit is',L)
L=limit(f,x,1,'-')
print('The left hand limit is',L)
L=limit(f,x,1,'+')
print('The right hand limit is',L)
Open Python Notebook File
from sympy import *
x=symbols('x')
f=(sqrt(x)-1)/(x**3-1)
xval=[0.9,0.99,0.999,1.1,1.01,1.001,1.0001]
yval=[f.subs(x,i) for i in xval]
print(yval)
print('The limit appears to be about 1.6666')
matplotlib notebook
plot(f,(x,0,2))L=limit(f,x,1)print('The limit is',L)
L=limit(f,x,1,'-')
print('The left hand limit is',L)
L=limit(f,x,1,'+')
print('The right hand limit is',L)
Open Python Notebook File
See more videos from this section
Related Videos
Evaluating One-Sided Limits from a Graph
Finding one-sided limits from the graph of a piecewise function
Vertical Asymptotes and Holes of a Rational Function
Finding the holes and vertical asymptotes of a rational function
Sandwich Theorem Example
Solving a limit at infinity using the Sandwich (or Squeeze) Theorem
MLC WIR 20B M151 week2 #06
Evaluating limits where the denominator approaches zero
MLC WIR 20B M151 week2 #08
Evaluating left and right limits of a fraction containing an absolute value
MLC WIR 20B M151 week2 #09
Evaluating one-sided and two-sided limits of a piecewise
MLC WIR 20B M151 week2 #10
Explaining why a piecewise function is discontinuous at a point
MLC WIR 20B M151 week2 #11
Solving for the values for constants to make a piecewise function is continuous
MLC WIR 20B M151 week3 #18
Solving for the value of a constant that makes a function is continuous everywhere
MLC WIR 20B M151 week10 #7
Solving for a constant so that a piecewise function is continuous
Derivatives and Applications: MATH 171 Problems 7 & 8
Calculating slopes of tangent lines to parametric curves
Evaluating Limits from a Graph
Finding limits from the graph of a piecewise function
Sandwich Theorem (or Squeeze Theorem)
Explaining the Sandwich (or Squeeze) Theorem with a graphical example
Review for the Common Exam: MATH 151 Exam 1 Review Problems 14-25
Review of limits, continuity, and the Intermediate Value Theorem
Review for the Common Exam: MATH 151 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Review for the Common Exam: MATH 151 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Limits: MATH 151 Problems 6-20
Evaluating limits of functions
Limits: MATH 151 Problems 6-20
Evaluating Limits of Functions
Continuity: MATH 151 2.5 Problems 1-6
Continuity of Functions and the Intermediate Value Theorem
Approximations and Exponentials: MATH 151 2.6 Problems 7-14
Approximation and Newton's Method, and limits and derivatives of exponential functions
Review for the Common Exam: MATH 151 3.1 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Review for the Common Exam: MATH 151 3.2 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Approximations and Exponentials: MATH 151 3.4 Problems 7-14
Approximation and Newton's Method, and limits and derivatives of exponential functions
MLC WIR 20B M151 week10 #6a
Evaluating a a limit from the left for a rational function
MLC WIR 20B M151 week10 #19b
Using L'Hospital's Rule to solve a limit with an indeterminant product
MLC WIR 20B M151 week10 #19c
Using L'Hospital's Rule to solve a limit with an indeterminant difference
MLC WIR 20B M151 week10 #19d
Using L'Hospital's Rule to solve a limit with an indeterminant power
MATH 152: Improper Integrals Exercise 11
Solving a type two improper integral using a limit
Vectors and Derivatives (Product Rule): MATH 171 Problems 1-3
Proving facts about the derivatives of vector functions including the product rule
Review for the Common Exam: MATH 151 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Limits: MATH 151 Problems 6-20
Evaluating limits of functions
Limits: MATH 151 Problems 6-20
Evaluating Limits of Functions
Precise Definition of a Limit: MATH 171 Problems 5 & 6
Proving limits using the epsilon-delta definition
Continuity: MATH 171 Problems 6 & 7
Proving a piecewise function and a polynomial are continuous
Continuity: MATH 151 2.5 Problems 1-6
Continuity of Functions and the Intermediate Value Theorem
Approximations and Exponentials: MATH 151 2.6 Problems 7-14
Approximation and Newton's Method, and limits and derivatives of exponential functions
Derivatives and Rates of Change: MATH 171 Problems 1-3
Using the limit definition to find derivatives of functions and vector functions
Review for the Common Exam: MATH 151 3.1 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Review for the Common Exam: MATH 151 3.2 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Approximations and Exponentials: MATH 151 3.4 Problems 7-14
Approximation and Newton's Method, and limits and derivatives of exponential functions
Absolute Extrema: MATH 171 Problems 4 & 5
Proving Fermat's Theorem and graphing local and absolute extrema
L'Hospital's Rule: MATH 171 Problems 7-9
Proving L'Hospital's Rule and using it to evaluate limits
Fundamental Theorem of Calculus: MATH 171 Problems 1-3
Proving and then applying the Fundamental Theorem of Calculus
Plotting Multiple Expressions in One Graph in Python
Graphing a piecewise function using Python
Parametric Plots and Tangents at a Given t-value in Python
Using Python to find the tangent line to a parametric equation and plot the two graphs
Simplifying Difference Quotients in Python
Using Python to simplify a difference quotient and find a derivative using the limit definition
Implicit Plots and Implicit Differentiation in Python
Using Python to plot an implicit curve and find a tangent line using implicit differentiation
Word Problems and the Algorithmic Process for Multistep Problem Solving in Python
Solving a multistep word problem in Python and graphing the resulting function
Finding and Graphing Tangent Lines to a Curve in Python
Using Python to find the equation of the tangent line to a curve and graphing the result
Generating and Plotting Partials Sums of a Series in Python
Using Python to calculate the partial sums of a series to estimate the value of the series
Generating and Plotting Sequences in Python
Using Python to numerically, graphically, and analytically find the limit of a sequence
Review for the Common Exam: MATH 151 Exam 1 Review Problems 14-25
Review of limits, continuity, and the Intermediate Value Theorem
Review for the Common Exam: MATH 151 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
MATH 140 2.1 Exercise 7
Graphing a line without using technology
MATH 140 2.1 Exercise 8
Graphing a line without using technology
MATH 140 2.1 Exercise 9
Graphing a line without using technology
MATH 140 2.2 Exercise 1
Writing a function for the value of an item depreciating linearly over time
MATH 140 Exam 1 Review #13
Finding the solution to a system of linear equations by graphing
MATH 140 3.2 Exercise 1
Graphing the solution set of an inequality with two variables
MATH 140 3.2 Exercise 2
Graphing the solution set of an inequality with two variables
MATH 140 3.2 Exercise 4
Graphing the solution set of system of linear inequalities
Math 140 3.2 Video 1
Explaining how to graph linear inequalities
MATH 140 3.2 Exercise 3
Graphing two linear inequalities
Math 140 3.2 Exercise 8
Sketching the solution set and finding the corner points for a system of linear inequalities
MATH 140 5.1 Exercise 5
Using a graph to find specific values of a function
MATH 140 5.7 Exercise 1
Identifying the parent function and transformations for a given graph
MATH 140 5.7 Exercise 2
Identifying the parent function and transformations for a given graph
MATH 140 5.7 Exercise 3
Identifying the parent function and transformations for a given graph
MATH 140 5.7 Exercise 4
Identifying the parent function and transformations for a given graph
MATH 140 5.7 Exercise 7
Drawing the graph of a transformed function given the original graph
MATH 140 5.7 Exercise 8
Evaluating compositions of two given functions and the graph of a third function
MATH 140 5.8 Exercise 3
Using the Horizontal Line Test to determine if graphs represent invertible functions
Evaluating a Limit Numerically
Finding the limit of a function numerically using a calculator
Combinations of Limits from Graphs
Finding limits of combinations of functions using their graphs
Evaluating Limits Algebraically
Using the properties of limits to find several limits algebraically
Limit of a Piecewise Function
Finding one and two-sided limits for a piecewise function algebraically
Limit of a Rational Function
Finding the limit of a rational function algebraically
Limit of a Rational Function 2
Finding the limit of a rational function algebraically
Limit with Multiple Fractions
Finding the limit algebraically of a function with multiple fractions
Limit of a Fraction with a Square Root
Finding the limit algebraically of a function with a square root in the numerator of a fraction
Limit with an Absolute Value
Finding the limit algebraically of a function with an absolute value
Horizontal Asymptote of a Fraction with Exponential Functions
Finding the horizontal asymptotes of a function that is a fraction with exponentials in the numerator and denominator
Limit at Negative Infinity of a Rational Function
Solving a limit approaching negative infinity for a rational function
Limits at Infinity of a Rational Function
Solving the limits approaching infinity and negative infinity of a rational function
Limits at Infinity of a Rational Function 2
Solving the limits approaching infinity and negative infinity of a rational function
Limits at Infinity of a Fraction with Exponential Functions
Solving the limits approaching infinity and negative infinity with exponential functions in the numerator and denominator
Determining Where the Derivative Exists From a Graph
Determining where the derivative of a function does not exist from a graph
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Sequences
An introduction to sequences and their limits along with the limit of recurrence equations
Sequences Example 1
Finding the first few terms and limit of a sequence
Sequences Example 2
Finding the terms and limit of a recursive sequence
Definition of a Double-Log Plot
Explaining the definition of logarithmic scale and double-log plots
Definition of a Semilog Plot
Explaining the logarithmic scale and a semilog plot
Double-Log Plot Example 1
Using logarithms to transform a power function into a linear function
Semilog Plot Example 1
Using logarithms to transform an exponential function to a linear function
Trigonometric Limits Example 1
Solving a limit example with a trigonometric functions
Trigonometric Limits Example 2
Solving a limit example with a trigonometric function
Trigonometric Limits Example 3
Evaluating a trigonometric limit using trigonometric identities
WIR2 20B M150 V13
Identifying the parent function and transformations of a function
WIR2 20B M150 V14
Identifying the parent function and transformations of a function
WIR3 20B M150 V13
Determining the properties of a rational function and graphing it
WIR8 20B M150 V09
Determining the properties of a rational function and graphing it
WIR3 20B M150 V19
Graphically verifying if two functions are inverses
WIR3 20B M150 V20
Restricting the domain of a function so it is one-to-one and has an inverse
WIR3 20B M150 V21
Determining algebraically the inverse of a function
WIR3 20B M150 V22
Determining algebraically the inverse of a function
WIR3 20B M150 V23
Determining algebraically the inverse of a function
WIR4 20B M150 V1
Determining properties of an exponential function
WIR4 20B M150 V2
Determining properties of an exponential function
WIR4 20B M150 V3
Determining properties of an exponential function
WIR4 20B M150 V4
Identifying the graph of an exponential function
WIR8 20B M150 V10
Determining properties of an exponential function
WIR4 20B M150 V8
Determining the graph of a logarithmic function
WIR4 20B M150 V10
Determining the graph of a logarithmic function
WIR5 20B M150 V13
Determining the properties of a sine function and graphing it
WIR5 20B M150 V15
Determining the properties of a cosine function and graphing it
WIR5 20B M150 V17
Writing the equation for a sine function to match a given graph
WIR5 20B M150 V18
Writing the equation for a cosine function to match a given graph
WIR7 20B M150 V06
Adding and subtracting vectors graphically
WIR6 20B M150 V01
Finding the properties of a transformed tangent function and graphing it
WIR2 20B M150 V11
Using a graph to evaluate a function
WIR2 20B M150 V12
Determining if a function is even or odd
Vectors and Derivatives: MATH 151 K.1 Problems 1-8
Derivatives and vectors with some physics applications
Review for the Common Exam: MATH 151 Exam 3 Review Problems 8-15
Review of limits and derivatives of inverse trigonometric functions
Review for the Common Exam: MATH 151 Exam 2 Review Problems 1-9
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Review for the Common Exam: MATH 151 Exam 1 Review Problems 36-40
Review of derivatives and tangent lines to functions and vector equations
Review for the Common Exam: MATH 151 Exam 3 Review Problems 8-15
Review of limits and derivatives of inverse trigonometric functions
Review for Common Exam: MATH 151 Exam 3 Review Problems 15-19
Mean Value Theorem and using derivatives to find the shape of curves
Inverse Trigonometric Functions: MATH 151 1.5 Problems 9-12
Properties and derivatives of inverse trigonometric functions
Limits and Asymptotes: MATH 151 2.6 Problems 7-11
Limits at infinity and asymptotes, along with physics applications
Limits and Asymptotes: MATH 151 2.7 Problems 7-11
Limits at infinity, asymptotes, and tangent lines
Review for the Common Exam: MATH 151 3.1 Exam 1 Review Problems 36-40
Review of derivatives and tangent lines to functions and vector equations
Differentials and Approximations: MATH 151 3.10 Problems 1-6
Differentials, linear approximations and quadratic approximations
Review for the Common Exam: MATH 151 3.2 Exam 1 Review Problems 36-40
Derivatives and tangents to curves
Trigonometry Derivatives and the Chain Rule: MATH 151 3.3 Problems 1-9
Derivatives of trigonometric functions and using the Chain Rule
Trigonometry Derivatives and the Chain Rule: MATH 151 3.4 Problems 1-9
Derivatives of trigonometric functions and using the Chain Rule
Vectors and Derivatives: MATH 151 3.4 Problems 1-8
Derivatives and vectors with some physics applications
Inverse Trigonometric Functions: MATH 151 Problems 9-12
Properties and derivatives of inverse trigonometric functions
Vectors and Derivatives: MATH 151 3.7 Problems 1-8
Derivatives and vectors with some physics applications
Studying the Shape of Curves: MATH 151 4.1 Problems 2-7
Using derivatives to find properties of graphs
Studying the Shape of Curves: MATH 151 Problems 2-7
Using derivatives to find properties of graphs
L'Hospital's Rule: MATH 151 4.4 Problem 1(a-h)
Using L'Hospital's Rule to solve limits
Riemann Sums and Definite Integrals: MATH 151 5.2 Problems 6-12
Using Reimann sums and the Fundamental Theorem of Calculus
Riemann Sums and Definite Integrals: MATH 151 5.3 Problems 6-12
Using Riemann sums and the Fundamental Theorem of Calculus
MLC WIR 20B M151 week2 #01
Converting parametric equations into a Cartesian equation and graphing
MLC WIR 20B M151 week3 #16
Finding the vertical and horizontal asymptotes for a rational function
MLC WIR 20B M151 week3 #15
Using the Squeeze Theorem to evaluate the limit of a function
MLC WIR 20B M151 week3 #17
Using the graph of a function to sketch a graph of its derivative
MLC WIR 20B M151 week3 #04
Finding a Cartesian equation for a parametric equation and graphing it
MLC WIR 20B M151 week3 #12
Find the Cartesian form and sketch the graph for the parametric equation of a line
MLC WIR 20B M151 week4 #7
Finding the derivatives of compositions of functions from their graphs
MLC WIR 20B M151 week7 #3
Sketching a curve given information about the function and its derivatives
MLC WIR 20B M151 week7 #4a
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4b
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4c
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4d
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4e
Evaluating a limit with an indeterminate product using L'Hospital's Rule
MLC WIR 20B M151 week7 #4f
Evaluating a limit with an indeterminant difference using L'Hospital's Rule
MLC WIR 20B M151 week7 #4g
Evaluating a limit with an indeterminate power using L'Hospital's Rule
MLC WIR 20B M151 week7 #4h
Evaluating a limit with an indeterminant power using L'Hospital's Rule
MLC WIR 20B M151 week9 #1
Determining the properties of a function from the graph of its derivative
MLC WIR 20B M151 week9 #8a
Evaluating a limit using logarithm properties
MLC WIR 20B M151 week9 #8b
Evaluating a limit with an indeterminate difference using a common denominator
MLC WIR 20B M151 week9 #8c
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week9 #8d
Evaluating a limit with an indeterminant power using L'Hospital's Rule
MLC WIR 20B M151 week9 #8e
Evaluating a limit with an indeterminant product using L'Hospital's Rule
MLC WIR 20B M151 week9 #8f
Evaluating a limit with an indeterminant product using L'Hospital's Rule
MLC WIR 20B M151 week9 #13
Using a limit of Riemann sums to write an expression for the area under a graph
MLC WIR 20B M151 week10 #8a
Evaluating the limit at infinity of a rational function
MLC WIR 20B M151 week10 #8b
Evaluating the limit of a function at negative infinity
MLC WIR 20B M151 week10 #8c
Evaluating the limit at infinity of a fraction with exponential functions
MLC WIR 20B M151 week10 #8d
Evaluating the limit at negative infinity of a fraction with exponential functions
MLC WIR 20B M151 week10 #8e
Evaluating a limit at infinity by combining logarithms
MLC WIR 20B M151 week10 #19a
Using L'Hospital's Rule to solve a limit
MLC WIR 20B M151 week10 #6b
Evaluating the limit of a rational function by factoring and cancelling
MLC WIR 20B M151 week10 #6c
Using L'Hospital's Rule to evaluate a limit representing a derivative at a point
Review for the Common Exam: MATH 152 Exam 3 b Review Problems 1-3
Review of sequences and finding the sum of a series
WIR 20B M152 V84
Converting parametric equations into a Cartesian equation and graphing
WIR 20B M152 V85
Converting parametric equations into a Cartesian equation and graphing
WIR 20B M152 V86
Converting parametric equations into a Cartesian equation and graphing
WIR 20B M152 V87
Converting parametric equations into a Cartesian equation and graphing
WIR 20B M152 V68
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V69
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V70
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V71
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V72
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V75
Finding the interval and radius of convergence for a power series
MATH 152: Area Exercise 1
Finding the area of a region bounded by two curves
WIR 20B M152 V36
Determining if an improper integral converges or diverges
WIR 20B M152 V37
Determining if an improper integral converges or diverges
WIR 20B M152 V38
Determining if an improper integral converges or diverges
WIR 20B M152 V39
Determining if an improper integral converges or diverges
MATH 152: Improper Integrals Exercise 14
Using the Comparison Theorem to solve an improper integral
MATH 152: Improper Integrals Exercise 9
Solving a type two improper integral with a limit
MATH 152: Improper Integrals Exercise 10
Solving a type two improper integral using a limit
Vectors and Derivatives: MATH 151 K.1 Problems 1-8
Derivatives and vectors with some physics applications
Review for the Common Exam: MATH 151 Exam 1 Review Problems 36-40
Review of derivatives and tangent lines to functions and vector equations
Review for the Common Exam: MATH 151 Exam 3 Review Problems 8-15
Review of limits and derivatives of inverse trigonometric functions
Review for the Common Exam: MATH 151 Exam 2 Review Problems 1-9
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Review for Common Exam: MATH 151 Exam 3 Review Problems 15-19
Mean Value Theorem and using derivatives to find the shape of curves
Inverse Trigonometric Functions: MATH 151 1.5 Problems 9-12
Properties and derivatives of inverse trigonometric functions
Inverse Trigonometric Functions: MATH 171 Problems 4-6
Properties of inverse trig functions and the derivative of arctangent
Limits: MATH 171 Problem 7
Proving a property for limits of sums using the epsilon-delta definition
Limits: MATH 171 Problems 8 & 9
Proving a property of scalar multiplication for limits using the epsilon-delta definition and using the Squeeze Theorem for Limits.
Precise Definition of a Limit: MATH 171 Problems 3 & 4
Proving limits using the epsilon-delta definition
Continuity: MATH 171 Problems 8 & 9
Proving a product of continuous functions is continuous and using the Intermediate Value Theorem
Limits and Asymptotes: MATH 171 Problem 5
Proving a property for the limit of the difference of functions as x approaches infinity
Limits and Asymptotes: MATH 171 Problems 1 & 2
Proving the values of limits as x approaches infinity using the epsilon-delta definition
Limits and Asymptotes: MATH 171 Problems 3 & 4
Proving the values of limits as x approaches infinity using the epsilon-delta definition
Limits and Asymptotes: MATH 151 2.6 Problems 7-11
Limits at infinity and asymptotes, along with physics applications
Limits and Asymptotes: MATH 151 2.7 Problems 7-11
Limits at infinity, asymptotes, and tangent lines
Review for the Common Exam: MATH 151 3.1 Exam 1 Review Problems 36-40
Review of derivatives and tangent lines to functions and vector equations
Differentiation Formulas: MATH 171 Problems 6 & 7
Proving the Product Rule and the rule for the derivative of a difference of functions
Differentials and Approximations: MATH 151 3.10 Problems 1-6
Differentials, linear approximations and quadratic approximations
Review for the Common Exam: MATH 151 3.2 Exam 1 Review Problems 36-40
Derivatives and tangents to curves
Trigonometry Derivatives and the Chain Rule: MATH 151 3.3 Problems 1-9
Derivatives of trigonometric functions and using the Chain Rule
Trigonometry Derivatives and the Chain Rule: MATH 171 Problems 1-3
Proving the derivatives of trigonometric functions and that sine is continuous
Trigonometry Derivatives and the Chain Rule: MATH 151 3.4 Problems 1-9
Derivatives of trigonometric functions and using the Chain Rule
Vectors and Derivatives: MATH 151 3.4 Problems 1-8
Derivatives and vectors with some physics applications
Inverse Trigonometric Functions: MATH 151 Problems 9-12
Properties and derivatives of inverse trigonometric functions
Derivatives of Logarithmic Functions: MATH 171 Problems 7-9
Proving facts about logarithms and exponentials including the derivative of an exponential with an arbitrary base
Studying the Shape of Curves: MATH 151 4.1 Problems 2-7
Using derivatives to find properties of graphs
Studying the Shape of Curves: MATH 151 Problems 2-7
Using derivatives to find properties of graphs
Studying the Shape of Curves: MATH 171 Problems 1-3
What the derivatives of a function tell us about the shape of its graph
L'Hospital's Rule: MATH 151 4.4 Problem 1(a-h)
Using L'Hospital's Rule to solve limits
Optimization: MATH 171 Problems 1-3
Optimizing values in word problems
Definite Integrals: MATH 171 Problems 5-9
The definition and properties of definite integrals
Riemann Sums and Definite Integrals: MATH 151 5.2 Problems 6-12
Using Reimann sums and the Fundamental Theorem of Calculus
Riemann Sums and Definite Integrals: MATH 151 5.3 Problems 6-12
Using Riemann sums and the Fundamental Theorem of Calculus
Review for the Common Exam: MATH 152 Exam 3 b Review Problems 1-3
Review of sequences and finding the sum of a series
Limit of a Sequence: MATH 172 Problems 1-5
Using the precise definition for the limit of a sequence and proving the Squeeze Theorem
Monotone Sequence Theorem: MATH 172 Problems 6-8
Using the Monotone Sequence Theorem and induction to show a sequence converges
Taylor and Maclaurin Series: MATH 172 Problems 1-5
Reviewing Taylor and Maclaurin Series and Taylor's Inequality
Infinite Series: MATH 172 Problems 1-3
The definition of convergent series and the Divergence Test
The Sum of a Series
Defining the sum of a series as the limit of the sequence of partial sums
Test for Divergence
Explaining how to use the Test for Divergence for an infinite series
Finding the Sum of a Series
Finding the sum of a series when you know the sequence of partial sums
Alternating Series and Ratio Tests: MATH 172 Problems 8-10
Proving the Alternating Series and Ratio Test and covers absolute convergence
Power Series: MATH 172 Problems 1-3
Covering Radius and Interval of Convergence of Power Series
Improper Integrals: MATH 172 Problems 3-5
Definition of Improper Integrals and using the Comparison Theorem to determine convergence or divergence
Hydrostatic Force: MATH 172 Problems 6 & 7
Using Riemann Sums and Integrals to calculate hydrostatic force
WIR1 20B M251 V1
Showing how an equation in three dimensional space represents a plane
WIR2 20B M251 V4
Identifying the equation of a hyperboloid and sketching its graph
WIR2 20B M251 V5
Identifying the equation of a cone in three dimensions and sketching its graph
WIR2 20B M251 V7
Finding the limit of a three-dimensional vector function
WIR3 20B M251 V1
Finding and sketching the domain of a function of two variables
WIR3 20B M251 V2
Finding and sketching the domain of a function of two variables
WIR3 20B M251 V3
Sketching the graph of a function of two variables
WIR3 20B M251 V4
Sketching the graph of a function of two variables
WIR3 20B M251 V5
Sketching the level curves of a function of two variables
WIR3 20B M251 V6
Sketching the level curves of a function of two variables
WIR3 20B M251 V7
Describing the level surfaces of a function of three variables
WIR5 20B M251 V6
Evaluating a double integral over a given region
WIR5 20B M251 V7
Evaluating a double integral over a given region
WIR5 20B M251 V8
Evaluating a double integral and sketching the region of integration
WIR5 20B M251 V9
Changing the order of integration for a double integral
WIR5 20B M251 V10
Evaluating a double integral by reversing the order of integration
WIR5 20B M251 V11
Setting up double integrals of Type I and II to give the volume of a solid under a surface
WIR6 20B M251 V1
Evaluating a double integral over a circular region by changing to polar coordinates
WIR6 20B M251 V2
Evaluating a double integral by changing to polar coordinates
WIR6 20B M251 V3
Evaluating a double integral by changing to polar coordinates
WIR6 20B M251 V4
Converting a double integral to a double integral in polar coordinates
WIR6 20B M251 V5
Writing an iterated integral in polar coordinates that gives the volume of the solid that lies below a paraboloid
WIR6 20B M251 V6
Writing an iterated integral in polar coordinates that gives the volume of the solid bounded by a cone
WIR6 20B M251 V7
Writing an a triple integral over a given solid as an iterated integral
WIR6 20B M251 V8
Evaluating a triple integral over a solid bounded by given surfaces
WIR6 20B M251 V9
Writing an iterated integral that gives the volume of a solid
WIR7 20B M251 V1
Evaluating a triple integral over a solid bounded by elliptic paraboloids
WIR7 20B M251 V2
Writing a triple integral as an iterated integral in cylindrical coordinates
WIR7 20B M251 V3
Converting an iterated triple integral into cylindrical and spherical coordinates
WIR7 20B M251 V4
Using an iterated integral in spherical coordinates to find the volume of a solid
WIR7 20B M251 V5
Evaluating a triple integral for a given solid by writing an iterated integral in spherical coordinates
MATH 152: Sequences Exercise 4
Finding the limit of a sequence
MATH 152: Sequences Exercise 5
Finding the limit of a sequence with an arcsin
MATH 152: Sequences Exercise 6
Finding the limit of a sequence using logarithm properties
MATH 152: Sequences Exercise 7
Finding the limit of a sequence
WIR 20B M152 V45
Determining if a sequence converges or diverges
WIR 20B M152 V46
Determining if a sequence converges or diverges
WIR 20B M152 V47
Determining if a sequence converges or diverges
WIR 20B M152 V48
Determining if a sequence is increasing, decreasing, or not monotonic and if it is bounded
WIR 20B M152 V49
Determining if a sequence is increasing, decreasing, or not monotonic and if it is bounded
WIR 20B M152 V50
Determining the limit of a bounded and increasing sequence
Functions of Several Variables Conceptual V1
Introducing functions of several variables and level curves
Functions of Several Variables Exercise V1
Evaluating and then finding and sketching the domain of a function of two variables
Functions of Several Variables Exercise V2
Finding and sketching the domain of a function of two variables
Functions of Several Variables Exercise V3
Sketching the level curves for a function of two variables
Functions of Several Variables Exercise V4
Sketching the level curves for a function of two variables
Functions of Several Variables Exercise V5
Sketching the level curves for a function of two variables
Multivariable Limits Conceptual V1
Explaining how to calculate the limit of a surface for two dimensional functions
Multivariable Limits Conceptual V2
Explaining the definition of the limit of a function of two variables
Multivariable Limits Conceptual V3
Explaining limits of functions of two variables that approach infinity
Multivariable Limits Exercise 1
Solving several limits of functions of two variables
Multivariable Limits Exercise 2
Finding the limit of a function of two variables
Multivariable Limits Exercise 3
Solving a limit of a function of two variables
Multivariable Limits Exercise 4
Solving a limit of a function of two variables
What is a Graph?
Explaining that a graph is a collection of vertices and edges
Euler Circuits Example 1
Drawing a graph to depict a bus route
Euler Circuits Example 2
Counting the vertices and edges from a graph
Euler Circuits Example 3
Identifying the adjacent vertices in a graph
Euler Circuits Example 4
Finding two different paths between vertices in a graph
Euler Circuits Example 5
Determining if a list of vertices in a graph is a path, circuit, Euler path, Euler circuit, or none of the above
Euler Circuits Example 6
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Euler Circuits Example 7
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Euler Circuits Example 8
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Euler Circuits Example 9
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Euler Circuits Example 10
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Finding Euler Circuits
Explaining Euler's Theorem on when a graph has an Euler path or Euler circuit
Finding Euler Circuits Example 1
Finding the valence of each vertex in a graph
Hamiltonian Circuits
Explaining Hamiltonian Circuits and paths, the Method of Trees to find them, and the number of Hamiltonian circuits for complete graphs
Hamiltonian Circuits Example 1
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Hamiltonian Circuits Example 2
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Hamiltonian Circuits Example 3
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Hamiltonian Circuits Example 4
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Hamiltonian Circuits Example 5
Using the method of trees to find all Hamiltonian circuits of a graph starting at a given vertex
Hamiltonian Circuits Example 8
Determining if a graph is a complete graph
Hamiltonian Circuits Example 9
Determining if a graph is a complete graph
Hamiltonian Circuits Example 10
Determining if a graph is a complete graph
Hamiltonian Circuits Example 11
Determining if a graph is a complete graph
Minimum-Cost Spanning Trees Example 1
Finding two spanning trees for a graph
Cumulative Distribution Function Video 2
Explaining the properties of a cumulative distribution function for a random variable
Functions of Several Variables Conceptual V1
Introducing functions of several variables and level curves
Multivariable Limits Conceptual V1
Explaining how to calculate the limit of a surface for two dimensional functions
Multivariable Limits Conceptual V2
Explaining the definition of the limit of a function of two variables
Multivariable Limits Conceptual V3
Explaining limits of functions of two variables that approach infinity
Systems of Two Linear Equations in Two Variables Video 1
Explaining the format and possible solutions for a system of two linear equations
Graphs for Data
Explanation of graphs for data and which are appropriate for categorical or numerical data
Math 304 Section 1.1 Exercise 1
Solving a system of linear equations and giving a geometric interpretation of the solution
Math 140 2.3 Exercise 4
Solving a system of two linear equations in two unknowns by graphing
Math 140 Exam 3 Review Exercise 1
Determining if a graph represents a function and finding the domain and range
Convergence in Probability
Defining convergence for random variables in probability
Convergence in Probability Exercise 1
Proving that a sequence of random variables converges in probability
The Sum of a Series
Defining the sum of a series as the limit of the sequence of partial sums
Test for Divergence
Explaining how to use the Test for Divergence for an infinite series
Finding the Sum of a Series
Finding the sum of a series when you know the sequence of partial sums
Math 152: Series Exercise 2
Finding the first four terms of several given series
Math 152: Series Exercise 3
Determining if a series diverges by the Test for Divergence
Math 152: Series Exercise 5
Finding the sum of a series given a formula for the partial sums
WIR 20B M152 V51
Determining if a series converges or diverges
WIR 20B M152 V59
Using the limit comparison test to determine if a series converges or diverges
WIR 20B M152 V60
Using the limit comparison test to determine if a series converges or diverges
WIR 20B M152 V61
Using the limit comparison test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 8
Using the Limit Comparison Test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 9
Using the Limit Comparison Test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 10
Using the Limit Comparison Test to determine if a series converges or diverges
WIR 20B M152 V62
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V63
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V64
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V65
Determining if an Alternating Series converges or diverges
Introduction and Installation of Python
An introduction to installing and getting started with Python
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Using Variables in Python
Defining variables in Python and using the variables in equations
Plotting Expressions and Solving Equations Numerically in Python
Plotting an expression in Python and finding a numerical approximation for the solution
Numerical Calculations in Python
Performing basic numerical calculations in Python
Solving Equations Symbolically in Python
Solving equations symbolically in Python and interpreting the results
Tangents to Parametrized Curves at a Given Point in Python
Using Python to plot a parametrized curve and its tangent line at a point
List Comprehension and Patterns in Higher Derivatives in Python
Using Python's list comprehension tool to find several higher order derivatives of a function at once
Definite and Indefinite Integrals in Python
Solving definite and indefinite integrals in Python
Finding Absolute (Global) Extrema in Python
Using Python to find the absolute maximum and minimum of a function on a closed interval
Approximating Definite Integrals with Riemann Sums in Python
Using Python to approximate a definite integral using left endpoint Riemann sums
Finding Derivatives in Python
Finding derivatives in Python and solving for the rate of change of a force
Finding Horizontal Tangent Lines in Python
Finding when the tangent line to a function is horizontal in Python
Finding Intervals of Increase/Decrease and Concavity in Python
Using Python to find the intervals where a function is increasing or decreasing
For Loops and Recursive Sequences in Python
Using for loops in Python to generate a recursively defined sequence
Areas Between Curves in Python
Using Python to identify the region between curves and then evaluate the area of the region
Calculating Work in Python: Hooke’s Law
Using Python to calculate the amount of work done in moving a spring using Hooke's Law
Calculating Work in Python: Pumping Liquid from a Tank
Using Python to calculate the work done in pumping the liquid from a tank whose ends are semicircles
Radius and Interval of Convergence in Python
Using Python to find the radius and interval of convergence of a power series with the Ratio Test
Volumes of Revolution in Python
Finding the volume of a solid of revolution in Python
Plotting a Function and Partial Sums of its Power Series in Python
Video 29 in a series of Python instructional videos
Making Linear Functions Video 1
Discussing how to write the equation of a line through two given points
Solving Linear Equations Algebraically Video 1
Solving a linear equation algebraically and showing how this relates to the graph
Solving Linear Equations Algebraically Video 2
Solving a linear equation algebraically
Solving Linear Equations Algebraically Video 3
Solving a linear equation algebraically and showing how this relates to the graph
Systems of Two Linear Equations in Two Variables Video 1
Explaining the format and possible solutions for a system of two linear equations
Systems of Two Linear Equations in Two Variables Video 5
Solving a system of two linear equations using a TI-84 calculator
Systems of Two Linear Equations in Two Variables Video 6
Solving a system of two linear equations using a TI-84 calculator
Quadratic Form Video 1
Explaining the graph and properties for the parent function of quadratics
Quadratic Equations Video 7
Explaining when a quadratic equation does not have a real solution
Properties of Rational Functions and Expressions Video 1
Discussing parent functions of rational functions and their properties
Properties of Radical Functions and Expressions Video 1
Explaining the general properties of even and odd root functions
Solving Radical Equations Video 2
Solving equations with even roots and checking the solutions
Properties of Power and Exponential Expressions Video 4
Properties of exponential growth and decay functions
Properties of Power and Exponential Expressions Video 5
Properties of the exponential function with base e
Solving Power and Exponential Equations Video 1
Explaining how to solve equations with power and exponential terms
Properties of Logarithmic Functions and Expressions Video 2
Explaining the basic properties of logarithmic functions
Using Double Integrals to Compute Mass and Weight
Using double integrals to calculate the weight of a plate given its density and size
Changing Order of Integration for Iterated Integrals
Evaluating iterated integrals over the same region but with different orders of integration
Determining the Iterated Integral for a Given Region
Writing and then solving an iterated integral for a given region
Systems of Two Linear Equations in Two Variables Video 1
Explaining the format and possible solutions for a system of two linear equations
Properties of Power and Exponential Expressions Video 4
Properties of exponential growth and decay functions
Properties of Power and Exponential Expressions Video 5
Properties of the exponential function with base e
Properties of Logarithmic Functions and Expressions Video 2
Explaining the basic properties of logarithmic functions
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Functions of Several Variables Conceptual V1
Introducing functions of several variables and level curves
Cumulative Distribution Function Video 2
Explaining the properties of a cumulative distribution function for a random variable
Review for the Common Exam: MATH 151 Exam 3 Review Problems 8-15
Review of limits and derivatives of inverse trigonometric functions