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
Tangents to Parametrized Curves at a Given Point in Python
Author: David Manuel
This video shows how to find a tangent line to a parametric curve at a point by finding the value of the parameter at that point and then finding the derivative. Then both the parametric curve and its tangent line are plotted using Python.
Transcript 17
Transcript 17
Problem: Find the first several derivatives of \(f(x)=\frac{1}{x}\) and generalize a pattern for the nth derivative.
Python Code:
from sympy import *
x,t=symbols('x t')
xt=cos(t)**3
yt=sin(t)**3
eq1=xt-1/27
# To get exact answers, use the Rational command: eq1=xt-Rational(1,27)
eq2=yt-16*sqrt(2)/27
# To get exact answers, use the Rational command: eq2=yt-Rational(16,27)*sqrt(2)
tval=solve([eq1,eq2],t)
print(tval)
dydx=diff(yt,t)/diff(xt,t)
print('dy/dx=',dydx)
m=dydx.subs(t,tval[0][0])
print(m)
tanline=16*sqrt(2)/27+m*(x-1/27)
print('The equation of the tangent line is y=',tanline.evalf())
matplotlib notebook
pcurve=plot_parametric(xt,yt,(t,0,pi/2))
pcurve.extend(plot(tanline,(x,0,1)))
pcurve.show()
Open Python Notebook File
from sympy import *
x,t=symbols('x t')
xt=cos(t)**3
yt=sin(t)**3
eq1=xt-1/27
# To get exact answers, use the Rational command: eq1=xt-Rational(1,27)
eq2=yt-16*sqrt(2)/27
# To get exact answers, use the Rational command: eq2=yt-Rational(16,27)*sqrt(2)
tval=solve([eq1,eq2],t)
print(tval)
dydx=diff(yt,t)/diff(xt,t)
print('dy/dx=',dydx)
m=dydx.subs(t,tval[0][0])
print(m)
tanline=16*sqrt(2)/27+m*(x-1/27)
print('The equation of the tangent line is y=',tanline.evalf())
matplotlib notebook
pcurve=plot_parametric(xt,yt,(t,0,pi/2))
pcurve.extend(plot(tanline,(x,0,1)))
pcurve.show()
Open Python Notebook File
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Finding a derivative using the Quotient Rule
MLC WIR 20B M151 week10 #10a
Finding the derivative of an implicit function
MLC WIR 20B M151 week10 #10b
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 #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
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
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 V89
Find the integral representing 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
Vector Functions: MATH 171 Problems 1 & 2
Proving a certain parameterization gives the unit circle and showing vector equations of lines are parallel
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 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 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 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
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 V6
Finding the tangent plane and the normal line for a surface
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 V13
Using the Fundamental Theorem of Line Integrals to evaluate a line integral of a vector field
WIR9 20B M251 V8
Using a double integral to find the surface area of a parametric surface
WIR10 20B M251 V3
Evaluating a surface integral of a vector field
WIR10 20B M251 V7
Using Stokes' Theorem to express a surface integral of the curl of a vector field as a single integral
Matrix Derivative and Integral
Explaining how to find the derivative and integral of matrix and vector functions
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
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
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
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
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
Solving Equations Symbolically in Python
Solving equations symbolically in Python and interpreting the results
Simplifying Difference Quotients in Python
Using Python to simplify a difference quotient and find a derivative using the limit definition
Approximating Definite Integrals with Riemann Sums in Python
Using Python to approximate a definite integral using left endpoint Riemann sums
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
Word Problems and the Algorithmic Process for Multistep Problem Solving in Python
Solving a multistep word problem in Python and graphing the resulting function
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
Volumes of Revolution in Python
Finding the volume of a solid of revolution in Python
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
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 10-13
Review of vector functions and parametric equations