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Cogito

Multivariable Calculus · Chapter 4 · Lesson 2

Practice Worksheet

Tangent Planes and Linear Approximation

Zoom in far enough and every surface is flat.

12 problems · about 24 minutes · F-IF.B.6, G-GPE.A.1

NameDate

Figure — use these to answer the problems

A plane has one constant slope in each direction. Set a and b and read those two slopes off the tilt.
  • The surfacez = a·x + b·y
  • Height at the origin0
a, the slope in x1
b, the slope in y2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f(2, 3) = 20, fₓ = 5, fᵧ = 1. Estimate f(2.2, 3.0).

    Answer

  2. What is a tangent plane?

    • The best flat approximation to the surface at a point.
    • Any plane that cuts through the surface.

  3. f(1, 1) = 10, fₓ = 4, fᵧ = 0. Estimate f(1.5, 1).

    Answer

  4. f(1, 1) = 10, fₓ = 4, fᵧ = 6. Estimate f(1, 1.5).

    Answer

  5. fₓ = 2 and fᵧ = 3, with dx = 1 and dy = 1. What is dz?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A tangent plane at a point where fₓ = 0 and fᵧ = 0. Is it horizontal? 1 yes, 0 no.

    Answer

  2. f(0, 0) = 7, fₓ = −2, fᵧ = 5. Estimate f(1, 1).

    Answer

  3. Does linear approximation get better or worse further from the point? Enter 1 for better, 2 for worse.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Order the steps for building a tangent plane.

    Write 1 to 4 in the boxes to put these in order.

    • Evaluate f at the point
    • Compute both partial derivatives
    • Evaluate both partials at the point
    • Assemble the plane equation

  2. f(4, 2) = 30 and both partials are 0. Estimate f(4.1, 2.1).

    Answer

  3. The Small Step

    f(2, 5) = 40, fₓ = 3 and fᵧ = −1. Estimate f(2.2, 5.4).

    Answer

  4. The Differential

    fₓ = 5 and fᵧ = 2. Both inputs rise by 0.1. What is dz?

    Answer