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 = 4y = 3y = 11431

Evaluate I = ∫14 ∫13 (−3y + 1) dy dx over the rectangle 1 ≤ x ≤ 4, 1 ≤ y ≤ 3.

f = −3y + 11 ≤ x ≤ 41 ≤ y ≤ 3
I =
2.Easy
xy0.50.5P

For f(x, y) = 5 sin(3x + 3y), find the partial derivative ∂f/∂y at the point (0.5, 0.5).

f(x, y) = 5 sin(3x + 3y)P = (0.5, 0.5)
fy(0.5, 0.5) =
3.Medium
xy−4−202024vP

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

f(x, y) = 3y·exyP = (−1, 2)v = (−3, 3)
Duf =
4.Medium
xy−4−20−20PQ

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

f(x, y) = x³y + x² − y³ + xyP = (−2, −1)Q = (−4, −3)
Duf =
5.Hard
xyx = 0.8y = x0.8

The integral I = ∫00.8 ∫y0.8 1/(1 + x²)² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + x²)²0 ≤ y ≤ x ≤ 0.8
I =
6.Hard
xy−202−202ABC

The marked points are the critical points of f(x, y) = x⁴ + y⁴ − 16xy. Click the saddle point.

f(x, y) = x⁴ + y⁴ − 16xy
  1. 1A (2, 2)
  2. 2B (0, 0)
  3. 3C (−2, −2)

Answer key

  1. 1. I = −30
  2. 2. fy(0.5, 0.5) = −14.85
  3. 3. Duf = −1.435
  4. 4. Duf = 21.21
  5. 5. I = 0.1951
  6. 6. B (0, 0)

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