MLC WIR 20B M151 week7 #2a
Finding a number that satisfies the conclusion of the Mean Value Theorem on an interval
Finding Derivatives in Python
Author: David Manuel
In this video, it is demonstrated how to find derivatives in Python. The presenter solves an example the rate of change of force as a box is dragged along the ground by a rope attached to a pulley.
Transcript 08
Transcript 08
Problem: Suppose a mass \(m\) is dragged across the ground by a rope attached to a pulley. Suppose that \(x\) is the horizontal distance from the mass to the pulley, \(h\) is the height of the pulley, and \(F\) is the force applied to the rope. Let \(g\) be the acceleration due to gravity, and \(\mu\) be the friction coefficient of the box along the ground. It can be shown that the force is given by this equation
\[F=\frac{\mu m g\sqrt{x^2+h^2}}{x+\mu h}\]
How fast is the force changing with respect to \(x\) in general? How fast is it changing when \(m=18,\) \(h=10,\) \(\mu=0.55\), \(g=9.8,\) \(x=10\) and \(x=40.\)
Python Code:
from sympy import *
mu,m,g,x,h=symbols('mu m g x h')
F=mu*m*g*sqrt(x**2+h**2)/(x+mu*h)
dF=F.diff(x)print('The general derivative is',dF)
dFat10=F.subs({m:18,h:10,mu:0.55,g:9.8,x:10})
print('When x=10, the rate of change in t is',dFat10.evalf() ,'Newtons/meter')
dFat40=F.subs({m:18,h:10,mu:0.55,g:9.8,x:40})
print('When x=40, the rate of change in t is', dFat40.evalf(), 'Newtons/meter')
Open Python Notebook File
from sympy import *
mu,m,g,x,h=symbols('mu m g x h')
F=mu*m*g*sqrt(x**2+h**2)/(x+mu*h)
dF=F.diff(x)print('The general derivative is',dF)
dFat10=F.subs({m:18,h:10,mu:0.55,g:9.8,x:10})
print('When x=10, the rate of change in t is',dFat10.evalf() ,'Newtons/meter')
dFat40=F.subs({m:18,h:10,mu:0.55,g:9.8,x:40})
print('When x=40, the rate of change in t is', dFat40.evalf(), 'Newtons/meter')
Open Python Notebook File
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Proving facts about differentials including a product rule
Higher Order Derivatives: MATH 171 Problems 4-6
Finding a formula for a higher order derivative and a formula for the second derivative of a product
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
Approximations and Exponentials: MATH 151 3.4 Problems 7-14
Approximation and Newton's Method, and limits and derivatives of exponential functions
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 171 Problems 6 & 7
Using implicit differentiation to solve a problem and to prove the power rule for rational exponents
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 1-3
Proving the derivative of ln(x) and the power and product rules using logarithmic differentiation
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
Studying the Shape of Curves: MATH 151 4.1 Problems 2-7
Using derivatives to find properties of graphs
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: MATH 171 Problems 6-8
Using the Mean Value Theorem to prove facts about functions
Studying the Shape of Curves: MATH 171 Problems 1-3
What the derivatives of a function tell us about the shape of its 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
Studying the Shape of Curves: MATH 151 Problems 2-7
Using derivatives to find properties of graphs
L'Hospital's Rule: MATH 171 Problems 7-9
Proving L'Hospital's Rule and using it to evaluate limits
Optimization: MATH 171 Problems 1-3
Optimizing values in word problems
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
Fundamental Theorem of Calculus: MATH 171 Problems 1-3
Proving and then applying the Fundamental Theorem of Calculus
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
Definition of Work
Explaining the definition of work for constant and non-constant forces
Work Done Stretching a Spring Using Hooke's Law
How to use Hooke's Law to calculate the work done in stretching a spring
Work Done Pulling Up a Rope
How to calculate the work done in lifting up a rope possibly with a mass
Integration by Parts: MATH 172 Problems 1 & 2
Proving the formula for integration by parts and a reduction formula
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 V11
Finding the curvature of a three-dimensional vector function
WIR4 20B M251 V9
Finding the absolute maximum and minimum values of a function of two variables on a rectangle
WIR4 20B M251 V10
Finding the absolute maximum and minimum values of a function of two variables on a triangular region
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
WIR9 20B M251 V4
Using Green's Theorem to calculate the work done as a particle moves through a force field
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
Increasing, Decreasing, and Non Monotonic Sequences
Explaining increasing, decreasing, and non monotonic for sequences
MATH 152: Sequences Exercise 10
Determining if a sequence is increasing, decreasing, or non monotonic
WIR 20B M152 V45
Determining if a sequence converges or diverges
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 V63
Using the Alternating Series Test to determine if a series converges or diverges
Using Variables in Python
Defining variables in Python and using the variables in equations
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
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Introduction and Installation of Python
An introduction to installing and getting started with 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
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
Finding Absolute (Global) Extrema in Python
Using Python to find the absolute maximum and minimum of a function on a closed interval
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
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
Volumes of Revolution in Python
Finding the volume of a solid of revolution in Python
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
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