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 = 1x = 2y = 3123

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

f = 3x·e−y1 ≤ x ≤ 20 ≤ y ≤ 3
I =
2.Easy
xyx = 1x = 3y = 5y = 21352

Evaluate I = ∫13 ∫25 (4xy² + 4) dy dx over the rectangle 1 ≤ x ≤ 3, 2 ≤ y ≤ 5.

f = 4xy² + 41 ≤ x ≤ 32 ≤ y ≤ 5
I =
3.Medium
xy−4−202024PQ

For f(x, y) = 2xy² − 2x² − 5y³ + 4xy, find the directional derivative Duf at P(−1, 2) in the direction towards Q(−4, 3).

f(x, y) = 2xy² − 2x² − 5y³ + 4xyP = (−1, 2)Q = (−4, 3)
Duf =
4.Medium
xy33∇fP

For f(x, y) = −3x²y² − 3x³ − 3y² + 2xy, find the maximum rate of change of f at the point (3, 3), that is |∇f|.

f(x, y) = −3x²y² − 3x³ − 3y² + 2xyP = (3, 3)
|∇f| =
5.Hard
xy21

Use polar coordinates to evaluate I = ∬R (x² + y²) dA, where R is the ring 1 ≤ x² + y² ≤ 4.

f = x² + y²1 ≤ r ≤ 20 ≤ θ ≤ 2π
I =
6.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 =

Answer key

  1. 1. I = 4.276
  2. 2. I = 648
  3. 3. Duf = −41.74
  4. 4. |∇f| = 294
  5. 5. I = 23.56
  6. 6. D = 2048

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