Skip to lesson

Choose “Save as PDF” in the print dialog to keep a copy.

Cogito

Multivariable Calculus · Chapter 3 · Lesson 1

Practice Worksheet

Functions of Two Variables

Two inputs in, one height out.

12 problems · about 22 minutes · F-IF.A.1, F-IF.B.5

NameDate

Figures — use these to answer the problems

Follow the mesh along the x-axis, then along the y-axis. The surface rises one way and falls the other.
  • The surfacez = a(x² − y²)
  • Height at the origin0
a, the x-curvature1
b, the y-curvature1
Figure 1
The same controls on a bowl. Both directions now curve the same way.
  • The surfacez = a(x² + y²)
  • Height at the origin0
a, the x-curvature1
b, the y-curvature1
Figure 2

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f(x, y) = 2x² + y. What is f(3, −4)?

    Answer

  2. What does the graph of a function of two variables look like?

    • A surface in space, one height above each point of the plane.
    • A curve in the plane.

  3. f(x, y) = x + y². What is f(3, 2)?

    Answer

  4. f(x, y) = xy. What is f(−4, 5)?

    Answer

  5. f(x, y) = x² − y². What is f(5, 3)?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. f(x, y) = x² − y². What is f(3, 5)?

    Answer

  2. f(x, y) = 1/(x − y). Is (4, 4) in the domain? 1 yes, 0 no.

    Answer

  3. The trace of z = x² + y² at y = 0. Which curve is it? 1 line, 2 parabola, 3 circle.

    Answer

Stretch Yourself

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

4 problems

  1. Match each surface to its shape.

    Draw a line from each item on the left to its match on the right.

    • z = x² + y²
    • z = x² − y²
    • z = 2x + 3y
    • A bowl opening upward
    • A saddle
    • A flat tilted plane

  2. f(x, y) = 40 − x² − y². What is the largest value it can take?

    Answer

  3. The Plate

    Temperature is T(x, y) = 40 − x² − y² degrees. What is the temperature at (3, 4)?

    Answer

  4. The Job

    Cost is C(h, m) = 25h + 4m dollars for h hours and m materials. What is the cost at h = 6, m = 10?

    Answer