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
xy31P

For f(x, y) = x²y² + 4x³ + y³ + 4xy, find the partial derivative ∂f/∂x at the point (3, 1).

f(x, y) = x²y² + 4x³ + y³ + 4xyP = (3, 1)
fx(3, 1) =
2.Easy
xyx = 1x = 3y = 3y = 21332

Evaluate I = ∫13 ∫23 (2x²y² − 3) dy dx over the rectangle 1 ≤ x ≤ 3, 2 ≤ y ≤ 3.

f = 2x²y² − 31 ≤ x ≤ 32 ≤ y ≤ 3
I =
3.Medium

z = −4x²y² − x² − 4y³ + 4xy with x = t − 1 and y = −2t² − 1. Use the chain rule to find dz/dt at t = 0.

z = −4x²y² − x² − 4y³ + 4xyx = t − 1y = −2t² − 1t = 0
dz/dt =
4.Medium
xy−500246vP

For f(x, y) = xy − 5x² − 5y², find the directional derivative Duf at (−2, 3) in the direction of v = (4, 3).

f(x, y) = xy − 5x² − 5y²P = (−2, 3)v = (4, 3)
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
xy−20−20AB

The marked points are the critical points of f(x, y) = x³ + y³ + 6xy. Click the local maximum.

f(x, y) = x³ + y³ + 6xy
  1. 1A (−2, −2)
  2. 2B (0, 0)

Answer key

  1. 1. fx(3, 1) = 118
  2. 2. I = 103.8
  3. 3. dz/dt = 6
  4. 4. Duf = −0.8
  5. 5. D = 2048
  6. 6. A (−2, −2)

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