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 = 1y = 4y = 1141

Evaluate I = ∫01 ∫14 (x·ey) dy dx over the rectangle 0 ≤ x ≤ 1, 1 ≤ y ≤ 4.

f = x·ey0 ≤ x ≤ 11 ≤ y ≤ 4
I =
2.Easy
xyx = 2y = 4y = 1241

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

f = 2x·e−y0 ≤ x ≤ 21 ≤ y ≤ 4
I =
3.Medium
xy−23∇fP

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

f(x, y) = −3x²y² − 2y² − xyP = (−2, 3)
|∇f| =
4.Medium
xy−101202PQ

For f(x, y) = 5 sin(3x + y), find the directional derivative Duf at P(0.5, 0.5) in the direction towards Q(1.5, 2.5).

f(x, y) = 5 sin(3x + y)P = (0.5, 0.5)Q = (1.5, 2.5)
Duf =
5.Hard
xy−20−2−101AB

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

f(x, y) = x³ + y³ + 3xy
  1. 1A (−1, −1)
  2. 2B (0, 0)
6.Hard
xy3

Use polar coordinates to evaluate I = ∬R √(x² + y²) dA, where R is the disc x² + y² ≤ 9.

f = √(x² + y²)0 ≤ r ≤ 30 ≤ θ ≤ 2π
I =

Answer key

  1. 1. I = 25.94
  2. 2. I = 1.398
  3. 3. |∇f| = 133.2
  4. 4. Duf = −4.653
  5. 5. B (0, 0)
  6. 6. I = 56.55

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