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
xyx = 1y = 4y = 2142

Evaluate I = ∫01 ∫24 (−3x²y² + 2) dy dx over the rectangle 0 ≤ x ≤ 1, 2 ≤ y ≤ 4.

f = −3x²y² + 20 ≤ x ≤ 12 ≤ y ≤ 4
I =
2.Easy
xy−31P

For f(x, y) = 4x³y³ + 4y³ − 3xy, find the partial derivative ∂f/∂x at the point (−3, 1).

f(x, y) = 4x³y³ + 4y³ − 3xyP = (−3, 1)
fx(−3, 1) =
3.Medium
xy1−1∇fP

For f(x, y) = xy + 4x² − 4y², find the maximum rate of change of f at the point (1, −1), that is |∇f|.

f(x, y) = xy + 4x² − 4y²P = (1, −1)
|∇f| =
4.Medium
xy3−2∇fP

For f(x, y) = −3x³y² − 5x³ − 2y³ + xy, find the maximum rate of change of f at the point (3, −2), that is |∇f|.

f(x, y) = −3x³y² − 5x³ − 2y³ + xyP = (3, −2)
|∇f| =
5.Hard
xyy = 4xy = x²416

Find the area I = ∬R 1 dA of the region R between the parabola y = x² and the line y = 4x.

y = x²y = 4xf = 1
I =
6.Hard
xyy = 2y = x22

The integral I = ∫02 ∫x2 1/(1 + y²)² dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + y²)²0 ≤ x ≤ y ≤ 2
I =

Answer key

  1. 1. I = −14.67
  2. 2. fx(−3, 1) = 105
  3. 3. |∇f| = 11.4
  4. 4. |∇f| = 551.7
  5. 5. I = 10.67
  6. 6. I = 0.4

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