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
xy20P

For f(x, y) = 3x²e−y, find the partial derivative ∂f/∂y at the point (2, 0).

f(x, y) = 3x²e−yP = (2, 0)
fy(2, 0) =
2.Easy
xyx = 1x = 3y = 2132

Evaluate I = ∫13 ∫02 (4x²y + 1) dy dx over the rectangle 1 ≤ x ≤ 3, 0 ≤ y ≤ 2.

f = 4x²y + 11 ≤ x ≤ 30 ≤ y ≤ 2
I =
3.Medium
xy02402vP

For f(x, y) = 2 ln(x² + 4y²), find the directional derivative Duf at (1, 0.5) in the direction of v = (4, 2).

f(x, y) = 2 ln(x² + 4y²)P = (1, 0.5)v = (4, 2)
Duf =
4.Medium
xy2−1∇fP

For f(x, y) = x³y³ + xy, find the maximum rate of change of f at the point (2, −1), that is |∇f|.

f(x, y) = x³y³ + xyP = (2, −1)
|∇f| =
5.Hard
xy−505−4−202(3, −2)(−3, −2)

Find the critical points of f(x, y) = −x³ + 27x − y² − 4y, classify them, and give the value fmax of f at its local maximum.

f(x, y) = −x³ + 27x − y² − 4y
fmax =
6.Hard
xy−202−202(0, 0)(2, 2)(−2, −2)

f(x, y) = x⁴ + y⁴ − 16xy has a critical point at (2, 2). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x⁴ + y⁴ − 16xyP = (2, 2)
D =

Answer key

  1. 1. fy(2, 0) = −12
  2. 2. I = 73.33
  3. 3. Duf = 3.578
  4. 4. |∇f| = 29.07
  5. 5. fmax = 58
  6. 6. D = 2048

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