Functions of Several Variables and Level Curves · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Reading surfaces through level curves

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. For f(x, y) = x^2 + y^2, what level value passes through (3, 4)?

    Answer: ______________

  2. Ben says where contour lines bunch together, the surface is steeper. Is Ben right?

    Circle one:   True   False

  3. Ben says where contour lines bunch together, the surface is steeper. Is Ben right?

    Circle one:   True   False

  4. On a contour map, what does wide spacing between lines indicate?

    • a flat, gently changing region
    • a steep cliff
    • a sharp peak
    • a deep hole

  5. What shape are the level curves of f(x, y) = x + y?

    • parallel straight lines
    • circles
    • parabolas

  6. On a contour map, what does wide spacing between lines indicate?

    • a steep cliff
    • a flat, gently changing region
    • a sharp peak

  7. What is the domain of f(x, y) = sqrt(9 - x^2 - y^2)?

    • disk x^2 + y^2 <= 9
    • the whole plane
    • the line y = x
    • the upper half plane

  8. For f(x, y) = sqrt(9 minus x squared minus y squared), what is the radius of the domain disk?

    Answer: ______________

  9. Alex says the pair (3, 1) is in the domain of sqrt(9 minus x squared minus y squared). Is Alex right?

    Circle one:   True   False

  10. A contour map shows closed loops with values rising toward the center. What sits there?

    • a flat plain
    • a valley floor
    • a hill
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Reading surfaces through level curves W1-mt_u04fZ_XiJ8-s1

  1. 25 · f(3, 4) = 9 + 16 = 25.
  2. True · Ben is right: tight spacing means height changes fast over short distance.
  3. True · Ben is right: tight spacing means height changes fast over short distance.
  4. a flat, gently changing region · Wide spacing means height changes slowly, so the ground is gentle.
  5. parallel straight lines · x + y = c rearranges to y = minus x + c, lines of slope minus 1.
  6. a flat, gently changing region · Wide spacing means height changes slowly, so the ground is gentle.
  7. disk x^2 + y^2 <= 9 · The square root needs 9 - x^2 - y^2 >= 0, which is the disk of radius 3.
  8. 3 · The root needs 9 minus x squared minus y squared >= 0, a disk of radius 3.
  9. False · 9 + 1 = 10 exceeds 9, so (3, 1) fails the domain test.
  10. a hill · Closed loops are hills or dips, and rising values toward the center mean a hill.
Worksheet · LightMySky