Functions of Two Variables

Two inputs in, one height out.

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

What this lesson teaches

The student evaluates functions of two variables and identifies their domain and range.

A function of two variables sends a point in the plane to a height, so its graph is a surface.
The domain is now a region of the plane rather than intervals on a line.
Fixing one variable slices the surface into an ordinary one-variable trace.

Straightforward practice. Get the method working first.

5 problems

f(x, y) = 2x² + y. What is f(3, −4)?
Answer 14
Why 14.
What does the graph of a function of two variables look like?
Answer A surface in space, one height above each point of the plane.
Why A surface.
f(x, y) = x + y². What is f(3, 2)?
Answer 7
Why 3 + 4.
f(x, y) = xy. What is f(−4, 5)?
Answer -20
Why Multiply, keeping the sign.
f(x, y) = x² − y². What is f(5, 3)?
Answer 16
Why 25 − 9.

The same ideas with more to keep track of.

3 problems

f(x, y) = x² − y². What is f(3, 5)?
Answer -16
Why 9 − 25. Order matters on a saddle.
f(x, y) = 1/(x − y). Is (4, 4) in the domain? 1 yes, 0 no.
Answer 0
Why The denominator would be zero.
The trace of z = x² + y² at y = 0. Which curve is it? 1 line, 2 parabola, 3 circle.
Answer 2
Why Setting y = 0 leaves z = x².

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

4 problems

Match each surface to its shape.
Answer z = x² + y² → A bowl opening upward; z = x² − y² → A saddle; z = 2x + 3y → A flat tilted plane
Why The sign between the two squares decides the first two.
f(x, y) = 40 − x² − y². What is the largest value it can take?
Answer 40
Why Both squares are smallest at zero.
The Plate: Temperature is T(x, y) = 40 − x² − y² degrees. What is the temperature at (3, 4)?
Answer 15
Why 15 degrees.
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 190
Why 190 dollars.