Multivariable calculus worksheet with answers

Free printable Multivariable calculus worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
xy−32P

For f(x, y) = 4x²y² − 3x² − 3y² − xy, find the partial derivative ∂f/∂x at the point (−3, 2).

f(x, y) = 4x²y² − 3x² − 3y² − xyP = (−3, 2)
fx(−3, 2) =
2.Easy
xy1−1P

For f(x, y) = 3x²y − 2x³ + y² + xy, find the partial derivative ∂f/∂y at the point (1, −1).

f(x, y) = 3x²y − 2x³ + y² + xyP = (1, −1)
fy(1, −1) =
3.Medium
xy−20−20vP

For f(x, y) = 3x³ey, find the directional derivative Duf at (−1, −1) in the direction of v = (2, −2).

f(x, y) = 3x³eyP = (−1, −1)v = (2, −2)
Duf =
4.Medium
xy2−1∇fP

For f(x, y) = −2xy³ − 2x² − 4y³ + 4xy, find the maximum rate of change of f at the point (2, −1), that is |∇f|.

f(x, y) = −2xy³ − 2x² − 4y³ + 4xyP = (2, −1)
|∇f| =
5.Hard
xy−505−2024(3, 2)(−3, 2)

Find the critical points of f(x, y) = −x³ + 27x − y² + 4y, classify them, and give the value fmax of f at its local maximum.

f(x, y) = −x³ + 27x − y² + 4y
fmax =
6.Hard
xyx = 1.2y = x1.2

The integral I = ∫01.2 ∫y1.2 sin(x²) dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = sin(x²)0 ≤ y ≤ x ≤ 1.2
I =

Answer key

  1. 1. fx(−3, 2) = −80
  2. 2. fy(1, −1) = 2
  3. 3. Duf = 3.122
  4. 4. |∇f| = 18.87
  5. 5. fmax = 58
  6. 6. I = 0.4348

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