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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
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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
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Section 1: Probabilistic Models and Probability Laws
Section 2: Conditional Probability, Bayes’ Rule, and Independence
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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
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Section 6: Directional Derivatives and the Gradient Vector
Section 6: Directional Derivatives and the Gradient Vector
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 directional derivative
The gradient vector
Links & Resources
Download Notes
Return to Main Calculus Page
Return to Mini-Course Main Page
Watch Concepts Video 1
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.
Directional Derivatives and The Gradient Vector Conceptual V1
Watch Concepts Video 2
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.
Directional Derivatives and The Gradient Vector Conceptual V2
Exercises
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.
Find \(D_{{\bf{u}}}f(x,y)\) at the point \((1,2)\) in the direction of \(\left<1,-3\right>\) to the surface \(f(x,y)=x^3+2x^2y^2.\)
Reveal Answer
\(D_{{\bf {u}}}f(1,2)=-\dfrac{5}{\sqrt{10}}\)
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.
Directional Derivatives and The Gradient Vector Exercise V1
Suppose \(f(x,y)=x^3-2xy+y^2\). Find \(D_{{\bf{u}}}f(x,y)\) at the point \((1,2)\) where \({\bf{u}}\) is the unit vector corresponding to \(\dfrac{\pi}{3}.\)
Reveal Answer
\(D_{{\bf {u}}}f(1,2)=-\dfrac{1}{2}+\sqrt{3}\)
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.
Directional Derivatives and The Gradient Vector Exercise V5
If \(f(x,y,z)=z^3-x^2y\), find \(D_{{\bf{u}}}f(1,6,2)\) if \({\bf{u}}=\left<\dfrac{3}{13},\dfrac{4}{13},\dfrac{12}{13}\right>.\)
Reveal Answer
\(D_{{\bf {u}}}f(1,6,2)=\dfrac{104}{13}\)
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.
Directional Derivatives and The Gradient Vector Exercise V2
Let \(f(x,y)=xe^y\)
Find the rate of change of \(f\) at the point \((2,0)\) in the direction of the point \(P(2,0)\) to the point \(Q\left(\dfrac{1}{2},2\right).\)
At the point \((2,0)\), in what direction does \(f\) have the maximum rate of change? What is the maximum rate of change?
Reveal Answer
\(D_{{\bf{u}}}f(2,0)=1\)
The direction of the maximum rate of change at the point \((2,0)\) is \(\nabla f(2,0)= \langle 1, 2\rangle\), and the maximum rate of change is \( \sqrt{5}.\)
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.
Directional Derivatives and The Gradient Vector Exercise V3
Find the maximum rate of change of \(f(x,y)=\tan(3x+2y)\) at the point \(\left(\displaystyle{{\pi}\over{6}}, -\displaystyle{{\pi}\over{8}}\right)\) and the direction in which it occurs.
Reveal Answer
The direction of the maximum rate of change at \(\left(\dfrac{\pi}{6},-\dfrac{\pi}{8}\right)\) is \(\nabla f\left(\dfrac{\pi}{6},-\dfrac{\pi}{8}\right)=\langle6,4\rangle\), and the maximum rate of change is \(|\nabla f|=\sqrt{52}.\)
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.
Directional Derivatives and The Gradient Vector Exercise V4
Self-Assessment Questions
Directions:
The following self-assessment question are a measure of how well you understood the material in this section.
How is the geometric interpretation of the directional derivative of \(z=f(x,y)\) at a point \(P(x_0, y_0)\) in the direction of the unit vector \({\bf{u}}\) different from \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\)?
How can we use the directional derivative to find the steepest ascent or descent of a surface \(z=f(x,y)\)?
What is the difference between the direction of the maximum rate of change of \(z=f(x,y)\) at a point \(P\) and the maximum rate of the change of \(z=f(x,y)\) at a point \(P\)?
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