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
xy23P

For f(x, y) = −2x³y² + 2x² + 2y³ + xy, find the partial derivative ∂f/∂y at the point (2, 3).

f(x, y) = −2x³y² + 2x² + 2y³ + xyP = (2, 3)
fy(2, 3) =
2.Easy
xyx = 3y = 2y = 1321

Evaluate I = ∫03 ∫12 (2x·e−y) dy dx over the rectangle 0 ≤ x ≤ 3, 1 ≤ y ≤ 2.

f = 2x·e−y0 ≤ x ≤ 31 ≤ y ≤ 2
I =
3.Medium
xy024−20PQ

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

f(x, y) = 5 sin(x + 3y)P = (2, −1)Q = (5, 0)
Duf =
4.Medium
xy2−2∇fP

For f(x, y) = −2x³y − 4x³ + y³ + xy, find the maximum rate of change of f at the point (2, −2), that is |∇f|.

f(x, y) = −2x³y − 4x³ + y³ + xyP = (2, −2)
|∇f| =
5.Hard
xy05−2024AB

The marked points are the critical points of f(x, y) = x³ + y³ − 9xy. Click the saddle point.

f(x, y) = x³ + y³ − 9xy
  1. 1A (0, 0)
  2. 2B (3, 3)
6.Hard
xyx = 0.8y = x0.8

The integral I = ∫00.8 ∫y0.8 1/(1 + x²)² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + x²)²0 ≤ y ≤ x ≤ 0.8
I =

Answer key

  1. 1. fy(2, 3) = −40
  2. 2. I = 2.093
  3. 3. Duf = 5.126
  4. 4. |∇f| = 2.828
  5. 5. A (0, 0)
  6. 6. I = 0.1951

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