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 = 2y = 4y = 1241

Evaluate I = ∫02 ∫14 (5y + 2) dy dx over the rectangle 0 ≤ x ≤ 2, 1 ≤ y ≤ 4.

f = 5y + 20 ≤ x ≤ 21 ≤ y ≤ 4
I =
2.Easy
xyx = 1x = 3y = 5y = 21352

Evaluate I = ∫13 ∫25 (3y + 1) dy dx over the rectangle 1 ≤ x ≤ 3, 2 ≤ y ≤ 5.

f = 3y + 11 ≤ x ≤ 32 ≤ y ≤ 5
I =
3.Medium
xy024024vP

For f(x, y) = xy − 3x³ + y², find the directional derivative Duf at (2, 1) in the direction of v = (−2, 3).

f(x, y) = xy − 3x³ + y²P = (2, 1)v = (−2, 3)
Duf =
4.Medium
xy1−1∇fP

For f(x, y) = 2 sin(3x − 3y), find the maximum rate of change of f at the point (1, −1), that is |∇f|.

f(x, y) = 2 sin(3x − 3y)P = (1, −1)
|∇f| =
5.Hard
xy02−1012AB

The marked points are the critical points of f(x, y) = x³ + y³ − 3xy. Click the local minimum.

f(x, y) = x³ + y³ − 3xy
  1. 1A (1, 1)
  2. 2B (0, 0)
6.Hard
xy1

Use polar coordinates to evaluate I = ∬R e−(x² + y²) dA, where R is the upper half of the disc x² + y² ≤ 1 (y ≥ 0).

f = e−(x² + y²)0 ≤ r ≤ 10 ≤ θ ≤ π
I =

Answer key

  1. 1. I = 87
  2. 2. I = 69
  3. 3. Duf = 22.74
  4. 4. |∇f| = 8.147
  5. 5. A (1, 1)
  6. 6. I = 0.9929

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