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 = 2x = 4y = 5y = 22452

Evaluate I = ∫24 ∫25 (2y + 6) dy dx over the rectangle 2 ≤ x ≤ 4, 2 ≤ y ≤ 5.

f = 2y + 62 ≤ x ≤ 42 ≤ y ≤ 5
I =
2.Easy
xyx = 1y = 3y = 1131

Evaluate I = ∫01 ∫13 (6y + 1) dy dx over the rectangle 0 ≤ x ≤ 1, 1 ≤ y ≤ 3.

f = 6y + 10 ≤ x ≤ 11 ≤ y ≤ 3
I =
3.Medium
xy0.50.5P

The tangent plane to z = ln(x² + 4y²) at the point where (x, y) = (0.5, 0.5) can be written as z = px + qy + c. Find c.

f(x, y) = ln(x² + 4y²)(a, b) = (0.5, 0.5)
c =
4.Medium

z = 2 ln(x² + 4y²) with x = −3t − 1 and y = 2t². Use the chain rule to find dz/dt at t = 2.

z = 2 ln(x² + 4y²)x = −3t − 1y = 2t²t = 2
dz/dt =
5.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 (−2, −2)
  3. 3C (0, 0)
6.Hard
xy02−2−101(2, −1)

Find the critical points of f(x, y) = 2x² − 2xy + 3y² − 10x + 10y − 3, classify them, and give the value fmin of f at its local minimum.

f(x, y) = 2x² − 2xy + 3y² − 10x + 10y − 3
fmin =

Answer key

  1. 1. I = 78
  2. 2. I = 26
  3. 3. c = −1.777
  4. 4. dz/dt = 3.633
  5. 5. C (0, 0)
  6. 6. fmin = −18

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