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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
xyx = 2x = 4y = 3y = 22432

Evaluate I = ∫24 ∫23 (y² + 6) dy dx over the rectangle 2 ≤ x ≤ 4, 2 ≤ y ≤ 3.

f = y² + 62 ≤ x ≤ 42 ≤ y ≤ 3
I =
2.Easy
xyx = 2x = 3y = 1231

Evaluate I = ∫23 ∫01 (4xy² + 1) dy dx over the rectangle 2 ≤ x ≤ 3, 0 ≤ y ≤ 1.

f = 4xy² + 12 ≤ x ≤ 30 ≤ y ≤ 1
I =
3.Medium
xy−202024vP

For f(x, y) = 2x³e2y, find the directional derivative Duf at (1, 1) in the direction of v = (−4, 4).

f(x, y) = 2x³e2yP = (1, 1)v = (−4, 4)
Duf =
4.Medium

z = 4x²y³ + 3x³ + 2y³ + 2xy with x = −2t − 1 and y = t² − 1. Use the chain rule to find dz/dt at t = −1.

z = 4x²y³ + 3x³ + 2y³ + 2xyx = −2t − 1y = t² − 1t = −1
dz/dt =
5.Hard
xy−50−4−202(0, 0)(−3, −3)

f(x, y) = x³ + y³ + 9xy has a critical point at (0, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ + y³ + 9xyP = (0, 0)
D =
6.Hard
xy0−4−202AB

The marked points are the critical points of f(x, y) = −x³ + 12x − y² − 6y. Click the local maximum.

f(x, y) = −x³ + 12x − y² − 6y
  1. 1A (2, −3)
  2. 2B (−2, −3)

Answer key

  1. 1. I = 24.67
  2. 2. I = 4.333
  3. 3. Duf = −10.45
  4. 4. dz/dt = −22
  5. 5. D = −81
  6. 6. A (2, −3)

Full step-by-step solutions: practise this topic on examproblems.com.