Vectors and Derivatives: MATH 151 K.1 Problems 1-8
Derivatives and vectors with some physics applications
Parametric Plots and Tangents at a Given t-value in Python
Author: David Manuel
In this video, the presenter shows how to plot a parametric curve in Python. The presenter then shows how to find the tangent line and plot it with the original parametric curve.
Transcript 16
Transcript 16
Problem: Plot the parametrized curve \(x=t^3-3t^2,\) \(y=t^3-3t\) on \(t=[-2,3]\) and the tangent line at \(t=-2.\)
Python Code:
from sympy import *
x,t=symbols('x t')
xt=t**3-3*t**2
yt=t**3-3*t
matplotlib notebook
pcurve=plot_parametric(xt,yt,(t,-2,3))
dydx=diff(yt,t)/diff(xt,t)
m=dydx.subs(t,-2)
x0=xt.subs(t,-2)
y0=yt.subs(t,-2)
tanline=y0+m*(x-x0)
print('The equation of the tangent line is y=',tanline)
pcurve.extend(plot(tanline,(x,-20,0)))
pcurve.show()
Open Python Notebook File
from sympy import *
x,t=symbols('x t')
xt=t**3-3*t**2
yt=t**3-3*t
matplotlib notebook
pcurve=plot_parametric(xt,yt,(t,-2,3))
dydx=diff(yt,t)/diff(xt,t)
m=dydx.subs(t,-2)
x0=xt.subs(t,-2)
y0=yt.subs(t,-2)
tanline=y0+m*(x-x0)
print('The equation of the tangent line is y=',tanline)
pcurve.extend(plot(tanline,(x,-20,0)))
pcurve.show()
Open Python Notebook File
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MLC WIR 20B M151 week6 #1ij
Differentiating functions with logarithmic and inverse trigonometric functions
MLC WIR 20B M151 week6 #1kl
Differentiating functions with inverse trigonometric functions
MLC WIR 20B M151 week6 #2a
Finding the derivative of an implicit function
MLC WIR 20B M151 week6 #2b
Finding the derivative of an implicit function
MLC WIR 20B M151 week6 #3
Using the chain rule to find a derivative
MLC WIR 20B M151 week6 #4a
Using logarithmic differentiation to find the derivative of a function
MLC WIR 20B M151 week6 #4b
Using logarithmic differentiation to find the derivative of a function
MLC WIR 20B M151 week6 #5a
Using the Product Rule to find the derivative of a product of functions
MLC WIR 20B M151 week6 #5b
Using the Quotient Rule to find a derivative
MLC WIR 20B M151 week6 #5c
Using the Chain Rule to find the derivative of a composition of functions
MLC WIR 20B M151 week6 #6
Finding the second derivative of an exponential functino
MLC WIR 20B M151 week6 #7
Finding the speed of an object given a vector function for its position
MLC WIR 20B M151 week6 #8
Finding the point when the tangent line to a function has a particular slope
MLC WIR 20B M151 week6 #12
Using the derivative to find the total distance traveled by a particle
MLC WIR 20B M151 week6 #10
Finding a higher order derivative of a function
MLC WIR 20B M151 week7 #1a
Finding the absolute maximum and absolute minimum values of a function on a closed interval
MLC WIR 20B M151 week7 #1b
Finding the absolute maximum and absolute minimum values of a function on a closed interval
MLC WIR 20B M151 week7 #1c
Finding the absolute maximum and absolute minimum values of a function on a closed interval
MLC WIR 20B M151 week6 #17
Finding the linear approximation for a function
MLC WIR 20B M151 week7 #3
Sketching a curve given information about the function and its derivatives
MLC WIR 20B M151 week7 #4a
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4b
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4c
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4d
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week7 #4e
Evaluating a limit with an indeterminate product using L'Hospital's Rule
MLC WIR 20B M151 week7 #4f
Evaluating a limit with an indeterminant difference using L'Hospital's Rule
MLC WIR 20B M151 week7 #4g
Evaluating a limit with an indeterminate power using L'Hospital's Rule
MLC WIR 20B M151 week7 #4h
Evaluating a limit with an indeterminant power using L'Hospital's Rule
MLC WIR 20B M151 week7 #5
Using the first and second derivatives to sketch the graph of a function
MLC WIR 20B M151 week8 #1
Solving a word problem minimizing area
MLC WIR 20B M151 week8 #2
Solving a word problem maximizing area
MLC WIR 20B M151 week8 #3
Finding the point on a line closest to a point
MLC WIR 20B M151 week9 #2
Using the derivative to determine where a function is increasing and decreasing
MLC WIR 20B M151 week9 #3
Finding the absolute maximum and minimum values of a function on a closed interval
MLC WIR 20B M151 week9 #4
Finding the intervals where a function is increasing and decreasing and its local maximums and minimums
MLC WIR 20B M151 week9 #5
Finding where a function is increasing, decreasing, concave up and concave down and its local extrema and inflection points
MLC WIR 20B M151 week9 #6
Finding where a function is increasing, decreasing, concave up and concave down and its local extrema and inflection points
MLC WIR 20B M151 week9 #7
Finding where a function is increasing, decreasing, concave up and concave down and its local extrema and inflection points
MLC WIR 20B M151 week9 #8c
Evaluating a limit using L'Hospital's Rule
MLC WIR 20B M151 week9 #8d
Evaluating a limit with an indeterminant power using L'Hospital's Rule
MLC WIR 20B M151 week9 #8e
Evaluating a limit with an indeterminant product using L'Hospital's Rule
MLC WIR 20B M151 week9 #9
Finding a number that satisfies the conclusion of the Mean Value Theorem for a function on an interval
MLC WIR 20B M151 week9 #8f
Evaluating a limit with an indeterminant product using L'Hospital's Rule
MLC WIR 20B M151 week9 #10a
Finding the antiderivative of a polynomial
MLC WIR 20B M151 week9 #10b
Finding the antiderivative of a rational function
MLC WIR 20B M151 week9 #10c
Finding the antiderivative of a function with secant and tangent
MLC WIR 20B M151 week9 #10d
Finding the antiderivative of a function and using a function value to find the constant
MLC WIR 20B M151 week9 #10e
Antidifferentiating twice to find a function from its second d
MLC WIR 20B M151 week9 #10f
Antidifferentiating twice to find a function from its second d
MLC WIR 20B M151 week10 #2
Converting parametric equations into a Cartesian equation
MLC WIR 20B M151 week10 #3
Converting parametric equations into a Cartesian equation
MLC WIR 20B M151 week10 #6c
Using L'Hospital's Rule to evaluate a limit representing a derivative at a point
MLC WIR 20B M151 week10 #9a
Finding a derivative using the Product Rule
MLC WIR 20B M151 week10 #9c
Finding a derivative using the Chain Rule and Product Rule
MLC WIR 20B M151 week10 #9d
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 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
MATH 152: Area Exercise 1
Finding the area of a region bounded by two curves
Vector Functions: MATH 171 Problems 1 & 2
Proving a certain parameterization gives the unit circle and showing vector equations of lines are parallel
Parametric Curves: MATH 151 J.3 Problems 1-5
Cartesian equations and parametric equations of curves
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 22-24
Optimization problems
Review for Common Exam: MATH 151 Exam 3 Review Problems 25-30
Antiderivatives and physics applications
Limits: MATH 151 Problems 6-20
Evaluating limits of functions
Limits: MATH 151 Problems 6-20
Evaluating Limits of Functions
Continuity: MATH 151 2.5 Problems 1-6
Continuity of Functions and the Intermediate Value Theorem
Continuity: MATH 171 Problems 6 & 7
Proving a piecewise function and a polynomial are continuous
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
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
Mean Value Theorem: MATH 171 Problems 6-8
Using the Mean Value Theorem to prove facts about functions
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
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
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
WIR1 20B M251 V1
Showing how an equation in three dimensional space represents a plane
WIR2 20B M251 V4
Identifying the equation of a hyperboloid and sketching its graph
WIR2 20B M251 V5
Identifying the equation of a cone in three dimensions and sketching its graph
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
WIR3 20B M251 V1
Finding and sketching the domain of a function of two variables
WIR3 20B M251 V2
Finding and sketching the domain of a function of two variables
WIR3 20B M251 V3
Sketching the graph of a function of two variables
WIR3 20B M251 V4
Sketching the graph of a function of two variables
WIR3 20B M251 V5
Sketching the level curves of a function of two variables
WIR3 20B M251 V6
Sketching the level curves of a function of two variables
WIR3 20B M251 V7
Describing the level surfaces of a function of three variables
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
WIR5 20B M251 V6
Evaluating a double integral over a given region
WIR5 20B M251 V7
Evaluating a double integral over a given region
WIR5 20B M251 V8
Evaluating a double integral and sketching the region of integration
WIR5 20B M251 V9
Changing the order of integration for a double integral
WIR5 20B M251 V10
Evaluating a double integral by reversing the order of integration
WIR5 20B M251 V11
Setting up double integrals of Type I and II to give the volume of a solid under a surface
WIR6 20B M251 V1
Evaluating a double integral over a circular region by changing to polar coordinates
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
WIR6 20B M251 V4
Converting a double integral to a double integral in polar coordinates
WIR6 20B M251 V5
Writing an iterated integral in polar coordinates that gives the volume of the solid that lies below a paraboloid
WIR6 20B M251 V6
Writing an iterated integral in polar coordinates that gives the volume of the solid bounded by a cone
WIR6 20B M251 V7
Writing an a triple integral over a given solid as an iterated integral
WIR6 20B M251 V8
Evaluating a triple integral over a solid bounded by given surfaces
WIR6 20B M251 V9
Writing an iterated integral that gives the volume of a solid
WIR7 20B M251 V1
Evaluating a triple integral over a solid bounded by elliptic paraboloids
WIR7 20B M251 V2
Writing a triple integral as an iterated integral in cylindrical coordinates
WIR7 20B M251 V3
Converting an iterated triple integral into cylindrical and spherical 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 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
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
Functions of Several Variables Conceptual V1
Introducing functions of several variables and level curves
Functions of Several Variables Exercise V1
Evaluating and then finding and sketching the domain of a function of two variables
Functions of Several Variables Exercise V2
Finding and sketching the domain of a function of two variables
Functions of Several Variables Exercise V3
Sketching the level curves for a function of two variables
Functions of Several Variables Exercise V4
Sketching the level curves for a function of two variables
Functions of Several Variables Exercise V5
Sketching the level curves for a function of two variables
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
Euler Circuits Example 10
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
What is a Graph?
Explaining that a graph is a collection of vertices and edges
Euler Circuits Example 1
Drawing a graph to depict a bus route
Euler Circuits Example 2
Counting the vertices and edges from a graph
Euler Circuits Example 3
Identifying the adjacent vertices in a graph
Euler Circuits Example 4
Finding two different paths between vertices in a graph
Euler Circuits Example 5
Determining if a list of vertices in a graph is a path, circuit, Euler path, Euler circuit, or none of the above
Euler Circuits Example 6
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Euler Circuits Example 7
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Euler Circuits Example 8
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Euler Circuits Example 9
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Finding Euler Circuits
Explaining Euler's Theorem on when a graph has an Euler path or Euler circuit
Finding Euler Circuits Example 1
Finding the valence of each vertex in a graph
Hamiltonian Circuits
Explaining Hamiltonian Circuits and paths, the Method of Trees to find them, and the number of Hamiltonian circuits for complete graphs
Hamiltonian Circuits Example 1
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Hamiltonian Circuits Example 2
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Hamiltonian Circuits Example 3
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Hamiltonian Circuits Example 4
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Hamiltonian Circuits Example 5
Using the method of trees to find all Hamiltonian circuits of a graph starting at a given vertex
Hamiltonian Circuits Example 8
Determining if a graph is a complete graph
Hamiltonian Circuits Example 9
Determining if a graph is a complete graph
Hamiltonian Circuits Example 10
Determining if a graph is a complete graph
Hamiltonian Circuits Example 11
Determining if a graph is a complete graph
Minimum-Cost Spanning Trees Example 1
Finding two spanning trees for a graph
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
Functions of Several Variables Conceptual V1
Introducing functions of several variables and level curves
The Chain Rule (multivariable) Conceptual V1
Explaining the Chain Rule for multivariable functions
Systems of Two Linear Equations in Two Variables Video 1
Explaining the format and possible solutions for a system of two linear equations
Graphs for Data
Explanation of graphs for data and which are appropriate for categorical or numerical data
Math 304 Section 1.1 Exercise 1
Solving a system of linear equations and giving a geometric interpretation of the solution
Math 140 2.3 Exercise 4
Solving a system of two linear equations in two unknowns by graphing
Math 140 Exam 3 Review Exercise 1
Determining if a graph represents a function and finding the domain and range
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
Plotting Expressions and Solving Equations Numerically in Python
Plotting an expression in Python and finding a numerical approximation for the solution
Introduction and Installation of Python
An introduction to installing and getting started with Python
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
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
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
Simplifying Difference Quotients in Python
Using Python to simplify a difference quotient and find a derivative using the limit definition
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
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
Making Linear Functions Video 1
Discussing how to write the equation of a line through two given points
Solving Linear Equations Algebraically Video 1
Solving a linear equation algebraically and showing how this relates to the graph
Solving Linear Equations Algebraically Video 2
Solving a linear equation algebraically
Solving Linear Equations Algebraically Video 3
Solving a linear equation algebraically and showing how this relates to the graph
Systems of Two Linear Equations in Two Variables Video 1
Explaining the format and possible solutions for a system of two linear equations
Systems of Two Linear Equations in Two Variables Video 5
Solving a system of two linear equations using a TI-84 calculator
Systems of Two Linear Equations in Two Variables Video 6
Solving a system of two linear equations using a TI-84 calculator
Quadratic Form Video 1
Explaining the graph and properties for the parent function of quadratics
Quadratic Equations Video 7
Explaining when a quadratic equation does not have a real solution
Properties of Rational Functions and Expressions Video 1
Discussing parent functions of rational functions and their properties
Properties of Radical Functions and Expressions Video 1
Explaining the general properties of even and odd root functions
Solving Radical Equations Video 2
Solving equations with even roots and checking the solutions
Properties of Power and Exponential Expressions Video 4
Properties of exponential growth and decay functions
Properties of Power and Exponential Expressions Video 5
Properties of the exponential function with base e
Solving Power and Exponential Equations Video 1
Explaining how to solve equations with power and exponential terms
Properties of Logarithmic Functions and Expressions Video 2
Explaining the basic properties of logarithmic functions
Using Double Integrals to Compute Mass and Weight
Using double integrals to calculate the weight of a plate given its density and size
Changing Order of Integration for Iterated Integrals
Evaluating iterated integrals over the same region but with different orders of integration
Determining the Iterated Integral for a Given Region
Writing and then solving an iterated integral for a given region
Systems of Two Linear Equations in Two Variables Video 1
Explaining the format and possible solutions for a system of two linear equations
Properties of Power and Exponential Expressions Video 4
Properties of exponential growth and decay functions
Properties of Power and Exponential Expressions Video 5
Properties of the exponential function with base e
Properties of Logarithmic Functions and Expressions Video 2
Explaining the basic properties of logarithmic functions
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Functions of Several Variables Conceptual V1
Introducing functions of several variables and level curves
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