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
xy11P

For f(x, y) = 5 sin(−3x + y), find the partial derivative ∂f/∂x at the point (1, 1).

f(x, y) = 5 sin(−3x + y)P = (1, 1)
fx(1, 1) =
2.Easy
xyx = 1x = 2y = 4y = 21242

Evaluate I = ∫12 ∫24 (−xy² + 2) dy dx over the rectangle 1 ≤ x ≤ 2, 2 ≤ y ≤ 4.

f = −xy² + 21 ≤ x ≤ 22 ≤ y ≤ 4
I =
3.Medium
xy−4−202024vP

For f(x, y) = 2 ln(x² + 2y²), find the directional derivative Duf at (−1, 2) in the direction of v = (3, 0).

f(x, y) = 2 ln(x² + 2y²)P = (−1, 2)v = (3, 0)
Duf =
4.Medium
xy−500246PQ

For f(x, y) = 4x²y² − 4x² + 2y³ + 4xy, find the directional derivative Duf at P(−2, 3) in the direction towards Q(0, 2).

f(x, y) = 4x²y² − 4x² + 2y³ + 4xyP = (−2, 3)Q = (0, 2)
Duf =
5.Hard
xyy = 2xy = x²24

Evaluate I = ∬R xy dA, where R is the region between the parabola y = x² and the line y = 2x.

y = x²y = 2xf = xy
I =
6.Hard
xy0−2024(2, 3)(−2, 3)

Find the critical points of f(x, y) = x³ − 12x + y² − 6y, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x³ − 12x + y² − 6y
fmin =

Answer key

  1. 1. fx(1, 1) = 6.242
  2. 2. I = −24
  3. 3. Duf = −0.4444
  4. 4. Duf = −167.3
  5. 5. I = 2.667
  6. 6. fmin = −25

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