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−2−3P

For f(x, y) = −3x³y³ + 5x³ + 2xy, find the partial derivative ∂f/∂x at the point (−2, −3).

f(x, y) = −3x³y³ + 5x³ + 2xyP = (−2, −3)
fx(−2, −3) =
2.Easy
xy21P

For f(x, y) = xe2y, find the partial derivative ∂f/∂y at the point (2, 1).

f(x, y) = xe2yP = (2, 1)
fy(2, 1) =
3.Medium
xy050246PQ

For f(x, y) = −x²y + 5x² − 4y² − 4xy, find the directional derivative Duf at P(2, 3) in the direction towards Q(−1, 2).

f(x, y) = −x²y + 5x² − 4y² − 4xyP = (2, 3)Q = (−1, 2)
Duf =
4.Medium
xy−2024−4−20PQ

For f(x, y) = −3x³y + 4x² − y³ + 3xy, find the directional derivative Duf at P(1, −2) in the direction towards Q(−2, 1).

f(x, y) = −3x³y + 4x² − y³ + 3xyP = (1, −2)Q = (−2, 1)
Duf =
5.Hard
xyy = 1.2y = x1.21.2

The integral I = ∫01.2 ∫x1.2 cos(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = cos(y²)0 ≤ x ≤ y ≤ 1.2
I =
6.Hard
xyy = 0.8y = x0.80.8

The integral I = ∫00.8 ∫x0.8 cos(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = cos(y²)0 ≤ x ≤ y ≤ 0.8
I =

Answer key

  1. 1. fx(−2, −3) = 1026
  2. 2. fy(2, 1) = 29.56
  3. 3. Duf = 15.18
  4. 4. Duf = −22.63
  5. 5. I = 0.4957
  6. 6. I = 0.2986

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