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
xy−12P

For f(x, y) = −4xy + x² − 5y³, find the partial derivative ∂f/∂y at the point (−1, 2).

f(x, y) = −4xy + x² − 5y³P = (−1, 2)
fy(−1, 2) =
2.Easy
xyx = 1x = 2y = 3y = 21232

Evaluate I = ∫12 ∫23 (4x²y² − 2) dy dx over the rectangle 1 ≤ x ≤ 2, 2 ≤ y ≤ 3.

f = 4x²y² − 21 ≤ x ≤ 22 ≤ y ≤ 3
I =
3.Medium
xy−500246vP

For f(x, y) = −2xy² − 2x² + 4xy, find the directional derivative Duf at (−2, 3) in the direction of v = (4, −3).

f(x, y) = −2xy² − 2x² + 4xyP = (−2, 3)v = (4, −3)
Duf =
4.Medium
xy05−2024PQ

For f(x, y) = 4x²y² − 2x² − 4y² + 3xy, find the directional derivative Duf at P(3, 1) in the direction towards Q(5, 0).

f(x, y) = 4x²y² − 2x² − 4y² + 3xyP = (3, 1)Q = (5, 0)
Duf =
5.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 =
6.Hard
xyx = 1y = 3x13

Evaluate I = ∬T (4x + 2y) dA over the triangle T bounded by y = 0, x = 1 and y = 3x.

f = 4x + 2y0 ≤ x ≤ 10 ≤ y ≤ 3x
I =

Answer key

  1. 1. fy(−1, 2) = −56
  2. 2. I = 57.11
  3. 3. Duf = −8
  4. 4. Duf = −19.23
  5. 5. D = 2048
  6. 6. I = 7

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