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Linear Algebra for MATH 308
Lecture 1: Vectors, Linear Independence, and Spanning Sets
Lecture 2: Operations with Matrices and Vectors
Lecture 3: Systems of Equations
Lecture 4: Determinant
Lecture 5: Eigenvectors and Eigenvalues
Lecture 6: Matrix Inverses and Diagonalization
Lecture 7: Systems of Differential Equations
Lecture 8: Systems of Differential Equations
Quantitative Finance
Several Variables Calculus
Section 1: Functions of Several Variables
Section 2: Limits and Continuity
Section 3: Partial Derivatives
Section 4: Tangent Planes and Linear Approximations
Section 5: The Chain Rule
Section 6: Directional Derivatives and the Gradient Vector
Section 7: Maximum and Minimum Values
Section 8: Lagrange Multipliers
Differential Equations
Section 1: Integrating Factor
Section 2: Separable Equations
Section 3: Compound Interest
Section 4: Variation of Parameters
Section 5: Systems of Ordinary Differential Equations
Section 6: Matrices
Section 7: Systems of Equations, Linear Independence, and Eigenvalues & Eigenvectors
Section 8: Homogeneous Linear Systems with Constant Coefficients
Section 9: Complex Eigenvalues
Section 10: Fundamental Matrices
Section 11: Repeated Eigenvalues
Section 12: Nonhomogeneous Linear Systems
Mathematical Probability
Section 1: Probabilistic Models and Probability Laws
Section 2: Conditional Probability, Bayes’ Rule, and Independence
Section 3: Discrete Random Variable, Probability Mass Function, and Cumulative Distribution Function
Section 4: Expectation, Variance, and Continuous Random Variables
Section 5: Discrete Distributions
Section 6: Continuous Distributions
Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution
Section 8: Independence of Two Random Variables, Covariance, and Correlation
Section 9: Conditional Distribution and Conditional Expectation
Section 10: Moment Generating Function
Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers
Section 12: Convergence and the Central Limit Theorem
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Section 5: The Chain Rule
Section 5: The Chain Rule
Instructions
First, you should watch the concepts videos below explaining the topics in the section.
Second, you should attempt to solve the exercises and then watch the videos explaining the exercises.
Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
When you have finished the material below, you can start on the
next section
or return to the
main several variable calculus page
.
Concepts
The Chain Rule for functions of more than one variable
Related rates
Links & Resources
Download Notes
Return to Main Calculus Page
Return to Mini-Course Main Page
Watch Concepts Video
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
The Chain Rule (multivariable) Conceptual V1
Exercises
Directions:
You should attempt to solve the problems first and then watch the video to see the solution.
If \(z=\ln(x^2+y^2)\), \(x=1+e^{6t}\), \(y=\sec^2(5t)\), find \(\dfrac{dz}{dt}.\)
Reveal Answer
\(\dfrac{dz}{dt}=\dfrac{2x}{x^2+y^2}\cdot6e^{6t}+\dfrac{2y}{x^2+y^2}\cdot 2\sec(5t)\cdot \sec(5t)\tan(5t)\)
Watch Video Solution
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
The Chain Rule (multivariable) Exercise V1
If \(z=x^2-3x^2y^3\), \(x=se^t\), \(y=se^{-t}\), find \(\dfrac{\partial z}{\partial s}\) and \(\dfrac{\partial z}{\partial t}.\)
Reveal Answer
\(\dfrac{dz}{ds}=\left(2x-6xy^3\right)e^t-9x^2y^2e^{-t}\)
\(\dfrac{dz}{dt}=\left(2x-6xy^3\right)\left(5e^t\right)+45x^2y^2e^{-t}\)
Watch Video Solution
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
The Chain Rule (multivariable) Exercise V2
If \(u=x^4y+y^2z^3\), \(x=rse^t\), \(y=rs^2e^{-t}\), \(z=r^2s\sin t\), find \(\displaystyle{\frac{\partial u}{\partial t}}\) when \(r=2\), \(s=3\) and \(t=0.\)
Reveal Answer
\(\left.\dfrac{\partial u}{\partial t}\right|_{r=2,s=3,t=0}=69,984\)
Watch Video Solution
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
The Chain Rule (multivariable) Exercise V4
The radius of a circular cone is increasing at a rate of 1.8 inches per second, while the height is decreasing at a rate of 2.5 inches per second. At what rate is the volume of the cone changing when the radius is 120 inches and the height is 140 inches?
Reveal Answer
\(8160\pi \frac{\text{in}^3}{\text{sec}}\)
Watch Video Solution
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
The Chain Rule (multivariable) Exercise V3
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
What are the requirements in order to use the chain rule? Write your own question that uses the chain rule where \(w=f(x,y,z)\), where \(x\), \(y\) and \(z\) are all functions of \(u\) and \(v\). Draw the tree diagram and solve the problem.
Write your own related rates word problem where two or more quantities are changing with respect to time and find the rate at which a third quantity is changing by using a formula that relates these quantities.
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