Review for Common Exam: MATH 151 Exam 3 Review Problems 15-19
Mean Value Theorem and using derivatives to find the shape of curves
tags:
Concave Down Concave Up Concavity Critical Value Decreasing Derivative First Derivative Increasing Python Second DerivativeFinding Intervals of Increase/Decrease and Concavity in Python
Author: David Manuel
In this video, it is shown how to plot a function in Python to try to determine where the function is increasing and decreasing and where it is concave up or down. The presenter then shows how to take the first derivative in Python, and use it to determine the intervals where the function is increasing and decreasing. It is also mentioned how to alter this technique to find the intervals where the function is concave up and concave down.
Transcript 19
Transcript 19
Problem: Analyze the graph of \begin{align*}y=&2x^7-21x^6-36x^5+3x^4\\ &-38x^3+24x^2 \end{align*} and find where the function is increasing or decreasing and where the function is concave upward or downward.
Python Code:
Analyze the graph of y=2x^7-21x^6-36x^5+3x^4-38x^3+24x^2
from sympy import *
x=symbols('x')
y=2*x**7-21*x**6-36*x**5+3*x**4-38*x**3+24*x**2
matplotlib notebook
plot(y,(x,-10,10),ylim=[-10,10])
dy=diff(y,x)
# To find concavity, repeat this process but use dy=diff(y,x,2) to get the second derivative
cvals=solve(dy,x)
cvalsfloat=[i.evalf() for i in cvals]
print(cvalsfloat)
matplotlib notebook
plot(dy,(x,-2,11),ylim=[-10,10])
print('The function is increasing from(-∞,-1.625),(0,0.35),(10,253,∞)')
print('and decreasing from (-1.625,0) and (0.35,10.253)')
Open Python Notebook File
Analyze the graph of y=2x^7-21x^6-36x^5+3x^4-38x^3+24x^2
from sympy import *
x=symbols('x')
y=2*x**7-21*x**6-36*x**5+3*x**4-38*x**3+24*x**2
matplotlib notebook
plot(y,(x,-10,10),ylim=[-10,10])
dy=diff(y,x)
# To find concavity, repeat this process but use dy=diff(y,x,2) to get the second derivative
cvals=solve(dy,x)
cvalsfloat=[i.evalf() for i in cvals]
print(cvalsfloat)
matplotlib notebook
plot(dy,(x,-2,11),ylim=[-10,10])
print('The function is increasing from(-∞,-1.625),(0,0.35),(10,253,∞)')
print('and decreasing from (-1.625,0) and (0.35,10.253)')
Open Python Notebook File
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WIR 20B M152 V76
Finding a power series representation for a function
Vectors and Derivatives (Product Rule): MATH 171 Problems 1-3
Proving facts about the derivatives of vector functions including the product rule
Vectors and Derivatives: MATH 151 K.1 Problems 1-8
Derivatives and vectors with some physics applications
Derivatives and Applications: MATH 151 K.2 Problems 9-15
Tangent lines to parametric equations and related rates examples
Derivatives and Applications: MATH 171 Problems 7 & 8
Calculating slopes of tangent lines to parametric curves
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 2 Review Problems 21-29
Related rates problems, differentials, linear and quadratic approximations
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 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 10-15
Implicit differentiation and physics applications of derivatives
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
Derivatives and Rates of Change: MATH 171 Problems 1-3
Using the limit definition to find derivatives of functions and vector functions
Differentiation Formulas: MATH 171 Problems 4 & 5
Using the limit definition to find a derivative and proving the power rule
Differentiation Formulas: MATH 171 Problems 6 & 7
Proving the Product Rule and the rule for the derivative of a difference of 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
Differentials and Approximations: MATH 151 3.10 Problems 1-6
Differentials, linear approximations and quadratic approximations
Differentials: MATH 171 Problems 1 & 2
Proving facts about differentials including a product rule
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
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 171 Problems 4 & 5
Using the Chain Rule to prove facts about derivatives
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
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
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
Implicit Differentiation: MATH 171 Problems 6 & 7
Using implicit differentiation to solve a problem and to prove the power rule for rational exponents
Derivatives of Logarithmic Functions: MATH 171 Problems 1-3
Proving the derivative of ln(x) and the power and product rules using logarithmic differentiation
Differentiation and Applications: MATH 151 Problems 1-8
Derivatives of exponential and logarithmic functions and the exponential model
Derivatives and Applications: MATH 151 3.9 Problems 9-15
Tangent lines to parametric equations and related rates examples
Mean Value Theorem: MATH 171 Problems 6-8
Using the Mean Value Theorem to prove facts about functions
L'Hospital's Rule: MATH 171 Problems 7-9
Proving L'Hospital's Rule and using it to evaluate limits
Antiderivatives: MATH 171 Problems 4-6
Proving facts about antiderivatives and a physics application
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
Alternating Series and Ratio Tests: MATH 172 Problems 8-10
Proving the Alternating Series and Ratio Test and covers absolute convergence
Integration by Parts: MATH 172 Problems 1 & 2
Proving the formula for integration by parts and a reduction formula
WIR8 20B M251 V1
Evaluating a line integral along half a circle
WIR8 20B M251 V2
Evaluating a line integral along a curve in space
WIR8 20B M251 V3
Evaluating a line integral along a curve
WIR8 20B M251 V4
Evaluating a line integral along a curve
WIR8 20B M251 V5
Evaluating a line integral along a curve in space
WIR8 20B M251 V6
Evaluating a line integral of a vector field along an curve
WIR8 20B M251 V7
Evaluating a line integral of a vector field along a curve in space
WIR8 20B M251 V8
Using a line integral to find the work done by a force field moving a particle along a curve
Matrix Derivative and Integral
Explaining how to find the derivative and integral of matrix and vector functions
MATH 152: Sequences Exercise 4
Finding the limit of a sequence
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 V50
Determining the limit of a bounded and increasing sequence
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
WIR 20B M152 V45
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
MATH 152: Sequences Exercise 9
Determining if a sequence is increasing, decreasing, or non monotonic
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
The Chain Rule (multivariable) Exercise V3
Using the Chain Rule for multivariable functions to find the rate a volume is changing
Finding the nth Taylor Polynomial Exercise
Finding the nth Taylor polynomial of the natural log function centered at 1
Approximation Using Taylor Polynomials Exercise
Using a Taylor polynomial to approximate the value of a function and estimating the error
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
WIR 20B M152 V51
Determining 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 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
Numerical Calculations in Python
Performing basic numerical calculations in Python
Solving Equations Symbolically in Python
Solving equations symbolically in Python and interpreting the results
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
Plotting Multiple Expressions in One Graph in Python
Graphing a piecewise function using Python
Simplifying Difference Quotients in Python
Using Python to simplify a difference quotient and find a derivative using the limit definition
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
Definite and Indefinite Integrals in Python
Solving definite and indefinite integrals in Python
List Comprehension and Numerical, Graphical, and Symbolic Limits in Python
Using Python to numerically estimate a limit, graphically estimate a limit, and find the exact limit
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
For Loops and Recursive Sequences in Python
Using for loops in Python to generate a recursively defined sequence
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
Volumes of Revolution in Python
Finding the volume of a solid of revolution in Python
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
Plotting a Function and Partial Sums of its Power Series in Python
Video 29 in a series of Python instructional videos
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
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 3 Review Problems 8-15
Review of limits and derivatives of inverse trigonometric functions