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 = 1x = 3y = 2132

Evaluate I = ∫13 ∫02 (6x²y − 1) dy dx over the rectangle 1 ≤ x ≤ 3, 0 ≤ y ≤ 2.

f = 6x²y − 11 ≤ x ≤ 30 ≤ y ≤ 2
I =
2.Easy
xyx = 2x = 4y = 3y = 22432

Evaluate I = ∫24 ∫23 (−3x²y² + 3) dy dx over the rectangle 2 ≤ x ≤ 4, 2 ≤ y ≤ 3.

f = −3x²y² + 32 ≤ x ≤ 42 ≤ y ≤ 3
I =
3.Medium
xy−1−1∇fP

For f(x, y) = 3 sin(−2x − 2y), find the maximum rate of change of f at the point (−1, −1), that is |∇f|.

f(x, y) = 3 sin(−2x − 2y)P = (−1, −1)
|∇f| =
4.Medium
xy−23P

The tangent plane to z = 2xy + 3x² + 4y³ at the point where (x, y) = (−2, 3) can be written as z = px + qy + c. Find c.

f(x, y) = 2xy + 3x² + 4y³(a, b) = (−2, 3)
c =
5.Hard
xy−505−4−202AB

The marked points are the critical points of f(x, y) = −x³ + 27x − y² − 2y. Click the local maximum.

f(x, y) = −x³ + 27x − y² − 2y
  1. 1A (3, −1)
  2. 2B (−3, −1)
6.Hard
xy−202−202(0, 0)(2, 2)(−2, −2)

Find the critical points of f(x, y) = x⁴ + y⁴ − 16xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x⁴ + y⁴ − 16xy
fmin =

Answer key

  1. 1. I = 100
  2. 2. I = −348.7
  3. 3. |∇f| = 5.546
  4. 4. c = −216
  5. 5. A (3, −1)
  6. 6. fmin = −32

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