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
xy3−3P

For f(x, y) = −2xy² + x³ + 4xy, find the partial derivative ∂f/∂x at the point (3, −3).

f(x, y) = −2xy² + x³ + 4xyP = (3, −3)
fx(3, −3) =
2.Easy
xyx = 2x = 3y = 4y = 12341

Evaluate I = ∫23 ∫14 (3y − 4) dy dx over the rectangle 2 ≤ x ≤ 3, 1 ≤ y ≤ 4.

f = 3y − 42 ≤ x ≤ 31 ≤ y ≤ 4
I =
3.Medium
xy024−4−20vP

For f(x, y) = −2x³y³ + x² − 5y³ + 2xy, find the directional derivative Duf at (2, −1) in the direction of v = (−2, −3).

f(x, y) = −2x³y³ + x² − 5y³ + 2xyP = (2, −1)v = (−2, −3)
Duf =
4.Medium
xy2−1P

The tangent plane to z = −xy² − 2x² + 2y² + 2xy at the point where (x, y) = (2, −1) can be written as z = px + qy + c. Find c.

f(x, y) = −xy² − 2x² + 2y² + 2xy(a, b) = (2, −1)
c =
5.Hard
xy1.5

Use polar coordinates to evaluate I = ∬R (x² + y²) dA, where R is the quarter of the disc x² + y² ≤ 2.25 in the first quadrant.

f = x² + y²0 ≤ r ≤ 1.50 ≤ θ ≤ π/2
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. fx(3, −3) = −3
  2. 2. I = 10.5
  3. 3. Duf = 34.67
  4. 4. c = 14
  5. 5. I = 1.988
  6. 6. D = 2048

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