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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
 



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. 
  1. 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 \((1, -1, 4)\)?

    Tangent Plane: \(2x+6y+z=-4\)
    Normal Line: \(\vec{r}(t)=\langle 1,-1,4\rangle + t\langle-2,6,1\rangle\) or \(x=1-2t, y=1+6t, z=4+t\)


    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


  2. 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 \(dz\).

    \(\Delta z=1.866\)
    \(dz=1.76\)


    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


  3. Use differentials to approximate \(\sqrt{9(1.95)^2+(8.1)^2}.\)

    \(\sqrt{9(1.95)^2+(8.1)^2}\approx 9.99\)


    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


  4. 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.

    The maximum error is approximately 8.5 cm\(^2.\)


    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.
    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.
  1. How is the tangent plane useful in approximating a surface \(z=f(x,y)\) near the point of tangency?
  2.  What is the difference between \(\Delta z\) and \(dz\), and under what conditions does \(dz\) approximate \(\Delta z\)?
  3. 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?