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 = 2y = 4y = 11241

Evaluate I = ∫12 ∫14 (−3y² − 2) dy dx over the rectangle 1 ≤ x ≤ 2, 1 ≤ y ≤ 4.

f = −3y² − 21 ≤ x ≤ 21 ≤ y ≤ 4
I =
2.Easy
xyx = 1x = 2y = 3123

Evaluate I = ∫12 ∫03 (xy + 1) dy dx over the rectangle 1 ≤ x ≤ 2, 0 ≤ y ≤ 3.

f = xy + 11 ≤ x ≤ 20 ≤ y ≤ 3
I =
3.Medium
xy1−1∇fP

For f(x, y) = x²e2y, find the maximum rate of change of f at the point (1, −1), that is |∇f|.

f(x, y) = x²e2yP = (1, −1)
|∇f| =
4.Medium
xy33P

The tangent plane to z = −2x³y² − 3x² + 2y² − xy at the point where (x, y) = (3, 3) can be written as z = px + qy + c. Find c.

f(x, y) = −2x³y² − 3x² + 2y² − xy(a, b) = (3, 3)
c =
5.Hard
xyy = 4xy = x²416

Evaluate I = ∬R x dA, where R is the region between the parabola y = x² and the line y = 4x.

y = x²y = 4xf = x
I =
6.Hard
xy02−1012AB

The marked points are the critical points of f(x, y) = x³ + y³ − 3xy. Click the saddle point.

f(x, y) = x³ + y³ − 3xy
  1. 1A (1, 1)
  2. 2B (0, 0)

Answer key

  1. 1. I = −69
  2. 2. I = 9.75
  3. 3. |∇f| = 0.3828
  4. 4. c = 1962
  5. 5. I = 21.33
  6. 6. B (0, 0)

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