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 Horizontal Tangent Lines in Python
Author: David Manuel
In this video, it is shown how to use Python to find any horizontal tangent lines for a function. The presenter finds the derivative of the function in Python and then has Python solve when the derivative is equal to zero. This equation is solved numerically by giving Python a starting guess for the solution, which was found from a graph of the function.
Transcript 13
Transcript 13
Problem: Find the horizontal tangent lines to
\[y=\frac{6x^2-4x^3}{2\sqrt{x^3(2-x)}}\]
Python Code:
from sympy import *
x=symbols('x')
y=(6*x**2-4*x**3)/(2*sqrt(x**3*(2-x)))
Python Code:
matplotlib notebook
plot(y,(x,0,2),ylim=[-10,10])
dy=diff(y,x).simplify()
print(dy)
horiz=nsolve(dy,x,0.75)
print('The graph has a horizontal tangent line at x=',horiz)
Open Python Notebook File
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Finding the derivative of an implicit function
MLC WIR 20B M151 week10 #11
Using logarithmic differentiation to find the derivative of a function
MLC WIR 20B M151 week10 #15
Determining the rate that the area of a triangle is increasing based on the rate an angle is increasing
MLC WIR 20B M151 week10 #17
Finding the value for a function that satisfies the Mean Value Theorem on an interval
MLC WIR 20B M151 week10 #18
Determining the properties of a function from the graph of its derivative
MLC WIR 20B M151 week10 #19a
Using L'Hospital's Rule to solve a limit
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
WIR 20B M152 V88
Finding the length of a curve given by parametric equations
WIR 20B M152 V90
Finding the surface area of a rotated parametric curve
WIR 20B M152 V81
Finding the Taylor series of a function
WIR 20B M152 V82
Finding the Taylor series of a function
WIR 20B M152 V70
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V76
Finding a power series representation for a function
Parametric Curves: MATH 151 J.3 Problems 1-5
Cartesian equations and parametric equations of curves
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 3 Review Problems 1-7
Derivatives of exponential functions and the exponential model
Review for the Common Exam: MATH 151 Exam 2 Review Problems 21-29
Related rates problems, differentials, linear and quadratic approximations
Review for Common Exam: MATH 151 Exam 3 Review Problems 15-19
Mean Value Theorem and using derivatives to find the shape of curves
Review for Common Exam: MATH 151 Exam 3 Review Problems 22-24
Optimization problems
Review for Common Exam: MATH 151 Exam 3 Review Problems 25-30
Antiderivatives and physics applications
Inverse Trigonometric Functions: MATH 171 Problems 4-6
Properties of inverse trig functions and the derivative of arctangent
Inverse Trigonometric Functions: MATH 151 1.5 Problems 9-12
Properties and derivatives of inverse trigonometric 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
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
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
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
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
Integration by Parts: MATH 172 Problems 1 & 2
Proving the formula for integration by parts and a reduction formula
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
MATH 308 WIR22A V1
Finding the slope of solutions and the critical points of a differential equation
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
Partial Derivatives Conceptual V1.2
Explanation of partial derivatives for functions with more than one variable
Partial Derivatives Exercise V1
Finding the partial derivatives of a function
Partial Derivatives Exercise V2
Finding the partial derivatives of a function with two variables
Partial Derivatives Exercise V3
Example of finding a partial derivative of a function with two variables
Tangent Planes and Linear Approximations Conceptual V1.1
Explaining tangent planes for functions of two variables and how to calculate them
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
Partial Derivatives Conceptual V1.2
Explanation of partial derivatives for functions with more than one variable
Tangent Planes and Linear Approximations Conceptual V1.1
Explaining tangent planes for functions of two variables and how to calculate them
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
Notation
Explaining the notation used for statistics and parameter for mean, standard deviation, variance, proportion, and slope
Math 140 2.3 Exercise 2
Determining how many solutions systems of equations have without solving or graphing
Math 140 2.3 Exercise 3
Solving for a coefficient so that a linear system has exactly one solution
Math 140 2.3 Exercise 4
Solving a system of two linear equations in two unknowns by graphing
Math 140 2.3 Exercise 5
Solving a system of two linear equations in two unknowns by graphing
Math 140 2.3 Exercise 6
Solving a system of two linear equations in two unknowns by graphing
Math 140 2.3 Exercise 8
Solving a system of two linear equations in two unknowns by substitution
Equation of a Line Given Two Points
Writing the equation of a line passing through two points
Slope-Intercept Form of an Equation for a Line
Writing the equation of a line through two points in slope-intercept form
Standard Form of an Equation for a Line
Writing the equation of a graphed line in standard form
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
Numerical Calculations in Python
Performing basic numerical calculations in Python
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
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
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
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
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
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
Linear Forms Video 1
Explaining the main three forms for linear equations
Linear Forms Video 2
Discussing properties of linear functions
Linear Graphs Video 1
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Making Linear Functions Video 1
Discussing how to write the equation of a line through two given points
Making Linear Functions Video 2
Writing a linear equation of profit from a business word problem
Linear Forms Video 1
Explaining the main three forms for linear equations
Linear Forms Video 2
Discussing properties of linear functions
Linear Graphs Video 1
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Partial Derivatives Conceptual V1.2
Explanation of partial derivatives for functions with more than one variable