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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
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
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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 4: Tangent Planes and Linear Approximations
Section 4: Tangent Planes and Linear Approximations
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
Section 5
or return to the
main several variable calculus page
.
Concepts
The equation of the tangent plane
Differentials
Applications of differentials
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.
Tangent Planes and Linear Approximations Conceptual V1.1
Exercises
Directions:
You should attempt to solve the problems first and then watch the video to see the solution.
Find the equation of the the tangent plane to the surface \(z=x^2+3y^2\) at the point \((1, -1, 4)\). What is the equation of the normal line to the surface at the point \((1, -1, 4)\)?
Reveal Answer
Tangent Plane: \(2x+6y+z=-4\)
Normal Line: \(\vec{r}(t)=\langle 1,-1,4\rangle + t\langle-2,6,1\rangle\) or \(x=1-2t, y=1+6t, z=4+t\)
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.
Tangent Planes and Linear Approximations Exercise V3.1
Find the differential, \(dz\), if \(z=f(x,y)=x^2+y^2\). If \(x\) changes from 2 to 2.5 and \(y\) changes from 3 to 2.96, compare the values of \(\Delta z\) and \(dz\).
Reveal Answer
\(\Delta z=1.866\)
\(dz=1.76\)
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.
Tangent Planes and Linear Approximations Exercise V4
Use differentials to approximate \(\sqrt{9(1.95)^2+(8.1)^2}.\)
Reveal Answer
\(\sqrt{9(1.95)^2+(8.1)^2}\approx 9.99\)
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.
Tangent Planes and Linear Approximations Exercise V1
The length and width of a rectagle are measured to be 25 cm and 35 cm, respectively, with an error in measurement of at most 0.1 in the length and 0.2 in the width. Use differentials to estimate the maximum error in the calculated area of the rectangle.
Reveal Answer
The maximum error is approximately 8.5 cm\(^2.\)
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.
Tangent Planes and Linear Approximations Exercise V2
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.
How is the tangent plane useful in approximating a surface \(z=f(x,y)\) near the point of tangency?
What is the difference between \(\Delta z\) and \(dz\), and under what conditions does \(dz\) approximate \(\Delta z\)?
How can we use differentials to estimate the amount of metal in a closed cylindrical can if we know the radius and height of the can?
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