Partial Derivatives · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
به انگلیسی

Slopes in the x and y directions

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

  1. To find fx(2, 3) for f(x, y) = x²y + y², which substitution may you make before differentiating?

    • Set x = 2, then differentiate in x.
    • Set y = 3, then differentiate in x.
    • Set both inputs, then differentiate in x.

  2. For f(x, y) = y³ + 4y, what is fx(2, -1)?

    Answer: ______________

  3. At (a, b), fy(a, b) = -3. Which statement describes the tangent to the trace with x fixed at a?

    • The y trace has tangent slope +3.
    • The x trace has tangent slope -3.
    • The y trace has tangent slope -3.
    • The y trace has tangent slope 0.

  4. For f(x, y) = x squared + 3xy + y squared, what is fy(2, 1)?

    Answer: ______________

  5. For f(x, y) = x² + y², Noor proposes fx = 3x and fy = 3y. Their mixed derivatives both equal 0, so Noor says both first partials are correct. Is this conclusion valid?

    Circle one:   True   False

  6. Eli says that a smooth function f has fxy = fyx at every point of its open domain. Is Eli right?

    Circle one:   True   False

  7. For f(x, y) = x²y + sin y, using radians, what is fy(2, 0)?

    Answer: ______________

  8. What is fx for f(x, y) = x cubed + 2x squared y + 5y squared?

    • 3x squared + 2xy
    • 3x squared + 4xy
    • 3x squared + 4xy + 10y

  9. A smooth function is claimed to have fx = y² + 3xy and fy = 2xy + 1. Using these proposed formulas, find fxy(2, 1) - fyx(2, 1). Read fxy as x then y, and fyx as y then x.

    Answer: ______________

  10. For f(x, y) = x squared y cubed, what is fxy(1, 2)? Here fxy means differentiate in x, then y.

    Answer: ______________

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Answer key

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

Slopes in the x and y directions W1-mt_tu11fd9Xk9-s1

  1. Set y = 3, then differentiate in x. · y is held fixed, so setting y = 3 gives 3x² + 9. Its x derivative is 6x, which can then be evaluated at x = 2.
  2. 0 · With y fixed, the whole expression is constant in x. Its x derivative is 0 at every point, including (2, -1).
  3. The y trace has tangent slope -3. · fy is the tangent slope when x stays fixed and y varies. Its value here is -3, so the tangent slopes downward in the increasing-y direction.
  4. 8 · fy = 3x + 2y, so fy(2, 1) = 6 + 2 = 8.
  5. False · Both proposed mixed derivatives are 0, but the actual first partials are 2x and 2y. Matching mixed derivatives do not prove the first partials are correct.
  6. True · Yes. Smoothness supplies continuous mixed second partials near every point of the open domain, so the two orders agree there.
  7. 5 · Hold x fixed: fy = x² + cos y. At (2, 0), this is 4 + 1 = 5.
  8. 3x squared + 4xy · Hold y fixed. The first term gives 3x squared, the second gives 4xy, and the third gives 0.
  9. 6 · The proposals give fxy = 2y + 3x and fyx = 2y. At (2, 1), their difference is 8 - 2 = 6. These cannot both be first partials of a smooth function near that point.
  10. 24 · fx = 2x y cubed, then fxy = 6x y squared, so fxy(1, 2) = 6 times 1 times 4 = 24.
Worksheet · LightMySky