Review for the Common Exam: MATH 152 Exam 3 b Review Problems 1-3
Review of sequences and finding the sum of a series
Generating and Plotting Sequences in Python
Author: David Manuel
In this video, we use Python to numerically, graphically, and analytically find the limit of a sequence. We will use Python's list comprehension tool to easily generate the first 20 terms and plot the first 50 terms. Lastly, we use Python's built-in limit command to find the exact limit of the sequence.
Transcript 25
Transcript 25
Problem: Suppose that
\[\lim_{n\rightarrow\infty} a_n = n\sin\left(\frac{\pi}{n}\right)\]
- List the first 20 terms (numerical estimate).
- Plot the first 50 terms (graphical estimate).
- Find the exact limit.
Python Code:
Limit of a_n = n * sin(pi/n)
a) List the first 20 terms (numerical estimate)
b) Plot the first 50 terms (graphical estimate)
c) Find the exact limit
from sympy import *
n=symbols('n',positive=True,integer=True)
a=n*sin(pi/n)
a1to20=[a.subs(n,i).evalf() for i in range(1,21)]
print(a1to20)
matplotlib notebook
import matplotlib.pyplot as plt
n1to50=range(1,51)
a1to50=[a.subs(n,i) for i in n1to50]
plt.plot(n1to50,a1to50,'o')
L=limit(a,n,oo)
print('The limit of the sequence is',L)
Open Python Notebook File
Limit of a_n = n * sin(pi/n)
a) List the first 20 terms (numerical estimate)
b) Plot the first 50 terms (graphical estimate)
c) Find the exact limit
from sympy import *
n=symbols('n',positive=True,integer=True)
a=n*sin(pi/n)
a1to20=[a.subs(n,i).evalf() for i in range(1,21)]
print(a1to20)
matplotlib notebook
import matplotlib.pyplot as plt
n1to50=range(1,51)
a1to50=[a.subs(n,i) for i in n1to50]
plt.plot(n1to50,a1to50,'o')
L=limit(a,n,oo)
print('The limit of the sequence is',L)
Open Python Notebook File
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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
Vectors and Derivatives: MATH 151 3.4 Problems 1-8
Derivatives and vectors with some physics applications
Exponentials and Hyperbolic Trigonometric Functions: MATH 171 Problems 3 & 4
Examples involving the tangent line to an exponential function and finding the derivative of hyperbolic cosine
Implicit Differentiation: MATH 151 3.5 Problems 10-13
Implicit differentiation and finding tangent lines
Inverse Trigonometric Functions: MATH 151 Problems 9-12
Properties and derivatives of inverse trigonometric functions
Differentiation and Applications: MATH 151 Problems 1-8
Derivatives of exponential and logarithmic functions and the exponential model
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
Derivatives and Applications: MATH 151 3.9 Problems 9-15
Tangent lines to parametric equations and related rates examples
Absolute Extrema: MATH 171 Problems 4 & 5
Proving Fermat's Theorem and graphing local and absolute extrema
Mean Value Theorem and Studying the Shape of Curves: MATH 151 4.2 Problems 1-6
Mean Value Theorem and properties of a graph
Mean Value Theorem and Studying the Shape of Curves: MATH 151 4.3 Problems 1-6
Mean Value Theorem and properties of a graph
L'Hospital's Rule: MATH 151 4.4 Problem 1(a-h)
Using L'Hospital's Rule to solve limits
L'Hospital's Rule: MATH 171 Problems 7-9
Proving L'Hospital's Rule and using it to evaluate limits
Riemann Sums: MATH 151 5.1 Problems 1-5
Riemann sums, including approximating areas under curves.
Definite Integrals: MATH 171 Problems 5-9
The definition and properties of definite integrals
Fundamental Theorem of Calculus: MATH 171 Problems 1-3
Proving and then applying the Fundamental Theorem of Calculus
Integration by Substitution: MATH 171 Problems 4-6
Using u substitution to evaluate integrals and prove facts about logarithms and integrals and
Review for the Common Exam: MATH 152 Exam 1 b Review Problems 29-33
Review of trigonometric substitution
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 13-18
Review of Taylor and Maclaurin Series and their properties
Introduction to Sequences
An introduction to sequences including the definition and notation
The Limit of a Sequence
Explaining the limit of a sequence
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
Alternating Sequences
Properties of alternating sequences and finding the limit of alternating sequences
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 152 11.10 Problems 4-10
Finding Taylor and Maclaurin Series for functions
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
Sequence of Partial Sums
Defining the sequence of partial sums for an infinite series
Infinite Series: MATH 172 Problems 1-3
The definition of convergent series and the Divergence Test
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
Area Between Curves: MATH 172 Problems 1-3
Interpreting integrals to represent areas between curves
Proving Trigonometric Identities: MATH 172 Problems 3 & 4
Proving trigonometric identities useful for integration
Integrating Powers of Sine and Cosine
Using trigonometric identities to integrate powers of sine and cosine
Cases for Trigonometric Substitution
Explaining the cases for using trigonometric substitution
Trigonometric Substitution: MATH 172 Problems 6 & 7
Integrating using a trigonometric substitution
Integration Using Partial Fractions: MATH 172 Problems 8 & 9
Finding a partial fraction decomposition and integrating using partial fractions
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 V15
Calculating the length of the cross product of two vectors given information about the vectors
WIR2 20B M251 V8
Finding a vector equation for the tangent line to a three-dimensional vector function
WIR2 20B M251 V9
Finding the angle of intersection for two three-dimensional vector functions
WIR2 20B M251 V10
Finding the length of a three-dimensional curve
WIR2 20B M251 V12
Finding the position vector function given the velocity and an initial position
WIR3 20B M251 V9
Finding the partial derivatives of a function of two variables
WIR3 20B M251 V10
Finding the partial derivatives of a function of three variables
WIR3 20B M251 V11
Finding all second-order partial derivatives of a function of two variables
WIR4 20B M251 V5
Finding the maximum rate of change and the direction it occurs for a function of two variables
WIR5 20B M251 V4
Evaluating a double integral over a rectangle
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
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
WIR8 20B M251 V1
Evaluating a line integral along half a circle
WIR8 20B M251 V6
Evaluating a line integral of a vector field along an curve
WIR9 20B M251 V2
Using Green's Theorem to evaluate a line integral on a closed curve
WIR9 20B M251 V6
Finding the curl and divergence of a three dimensional vector field
WIR10 20B M251 V1
Evaluating a surface integral over a cylinder
WIR10 20B M251 V9
Using the Divergence Theorem to evaluate the surface integral of a vector field
WIR 20B M152 V44
Finding a formula for the terms of a sequence
MATH 152: Sequences Exercise 7
Finding the limit of a sequence
Introduction to Sequences
An introduction to sequences including the definition and notation
The Limit of a Sequence
Explaining the limit of a sequence
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
Alternating Sequences
Properties of alternating sequences and finding the limit of alternating sequences
MATH 152: Sequences Exercise 1
Finding the first ten terms of a sequence
MATH 152: Sequences Exercise 2
Finding the first ten terms of a sequence
MATH 152: Sequences Exercise 8
Using L'Hopital's rule to find the limit of a sequence
MATH 152: Sequences Exercise 10
Determining if a sequence is increasing, decreasing, or non monotonic
MATH 152: Sequences Exercise 11
Determining if a sequence is increasing, decreasing, or non monotonic
MATH 152: Sequences Exercise 12
Determining if the limit of an alternating sequence exists
MATH 152: Sequences Exercise 13
Finding the limit of an alternating sequence
MATH 152: Sequences Exercise 14
Determining if a sequence is bounded
MATH 152: Sequences Exercise 15
Determining if a sequence is bounded
MATH 152: Sequences Exercise 16
Determining if every bounded, decreasing sequence has a limit
MATH 152: Sequences Exercise 17
Finding the limit of a recursive sequence
MATH 152: Sequences Exercise 3
Finding the general formula for a given sequence
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
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
Using a Rotation Map
Using a rotation map matrix to rotate a 2-dimensional vector by the angle \(\pi\)
Sine, Cosine, and Tangent
Explaining the trigonometric ratios of right triangles for sine, cosine, and tangent
Trig Functions with the Unit Circle
Finding the values of trig functions with the unit circle
Trig Functions with the Unit Circle Exercise 4
Finding the exact values of sine and secant for several given angles using the unit circle
Trig Functions with the Unit Circle Exercise 5
Finding the angles where sine has a given value using the unit circle
The Graph of Sine
Using the unit circle to sketch the graph of the sine function
The Graph of Tangent
Using the unit circle to sketch the graph of the tangent function
The Graphs of the Reciprocal Trig Functions
Graphing the reciprocal trig functions cosecant, secant, and cotangent
Transformations of Trig Graphs Exercise 1
Graphing a period of a transformed sine function
Transformations of Trig Graphs Exercise 3
Writing the sine and cosine functions for a given graph
Deriving the Double Angle Trig Identities
Using the sum identities of sine and cosine to derive the double angle identities
Deriving the Cofunction Trig Identities
Using the difference identities of sine and cosine to derive the cofunction identities
Deriving the Half-Angle Trig Identities
Using the double angle identities for cosein to derive the half-angle identities for sine and cosine
Using Identities to Find Exact Trig Values Exercise 2
Using the Sum Identity for Trig to find the exact value of sine
Using Identities to Find Exact Trig Values Exercise 3
Using a Double Angle Identity to find the exact value of an expression with sine and cosine
Using Identities to Find Exact Trig Values Exercise 4
Using the Half-Angle Identities to find the exact values of sine, cosine, and tangent
Solving Trigonometric Equations Exercise 1
Finding all solutions to a trigonometric equation with sine
Solving Trigonometric Equations Exercise 2
Solving a trigonometric equation with sine and cosine by factoring
Solving Trigonometric Equations Exercise 4
Solving a trigonometric equation with tangent and sine by factoring
Solving Trig Equations Using Identities Exercise 2
Solving a trig equation with sine and cosine using aPythagorean Identity
Solving Trig Equations Using Identities Exercise 4
Solving a trig equation with sine and tangent using a trig identity
Solving Trig Equations Using Identities Exercise 5
Solving a trig equation with sine and cosine using trig identities
WIR5 20B M150 V8
Using trigonometry to determine the edge lengths of a right triangle
WIR5 20B M150 V9
Evaluating trigonometric functions given a point on the terminal side of an angle
WIR5 20B M150 V10
Evaluating trigonometric functions given a point on the terminal side of an angle
WIR5 20B M150 V11
Finding trigonometric functions given information about the angle
WIR5 20B M150 V4
Evaluating all six trigonometric functions for angles in degrees and radians
WIR5 20B M150 V5
Evaluating all six trigonometric functions for a right triangle
WIR5 20B M150 V6
Evaluating all six trigonometric functions for a right triangle
Evaluating All Six Trigonometric Functions
Finding the values of all six trigonometric functions of an angle given a right triangle
Evaluating the Other Trigonometric Functions Given Sine
Evaluating the other five trigonometric functions for an acute angle given the value of sine
Evaluating the Other Trigonometric Functions Given Sine (2)
Evaluating the other five trigonometric functions given sine and that tangent is negative
Solving an Equation with Sine and Cosine
Solving a trigonometric equation with sine and cosine on a given interval
Factoring a Quadratic-Like Equation with Sine
Solving an quadratic-like equation with sine by factoring
Solving an Equation with Sine
Solving a trigonometric equation with sine on a given interval
Sketching One Period of a Sine Function
Sketching one period of a transformed sine graph
Properties of a Sine Function
Finding the amplitude, period, phase shift, and vertical shift for a given sine function
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
Sequence of Partial Sums
Defining the sequence of partial sums for an infinite series
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 1
Explaining the difference between the terms and partial sums of a series
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
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 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
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
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Numerical Calculations in Python
Performing basic numerical calculations in Python
Plotting Expressions and Solving Equations Numerically in Python
Plotting an expression in Python and finding a numerical approximation for the solution
Plotting Multiple Expressions in One Graph in Python
Graphing a piecewise function using Python
Solving Equations Symbolically in Python
Solving equations symbolically in Python and interpreting the results
Introduction and Installation of Python
An introduction to installing and getting started with Python
Using Variables in Python
Defining variables in Python and using the variables in equations
Tangents to Parametrized Curves at a Given Point in Python
Using Python to plot a parametrized curve and its tangent line at a point
Word Problems and the Algorithmic Process for Multistep Problem Solving in Python
Solving a multistep word problem in Python and graphing the resulting function
Implicit Plots and Implicit Differentiation in Python
Using Python to plot an implicit curve and find a tangent line using implicit differentiation
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
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
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
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
Approximating Definite Integrals with Riemann Sums in Python
Using Python to approximate a definite integral using left endpoint Riemann sums
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
Plotting a Function and Partial Sums of its Power Series in Python
Video 29 in a series of Python instructional videos
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
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Review for the Common Exam: MATH 151 Exam 1 Review Problems 10-13
Review of vector functions and parametric equations
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