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 = 4y = 3y = 11431

Evaluate I = ∫14 ∫13 (x²y − 3) dy dx over the rectangle 1 ≤ x ≤ 4, 1 ≤ y ≤ 3.

f = x²y − 31 ≤ x ≤ 41 ≤ y ≤ 3
I =
2.Easy
xy2−1P

For f(x, y) = sin(2x − 2y), find the partial derivative ∂f/∂x at the point (2, −1).

f(x, y) = sin(2x − 2y)P = (2, −1)
fx(2, −1) =
3.Medium
xy−2−2P

The tangent plane to z = 3x²y − 5x² + 3y³ + 2xy at the point where (x, y) = (−2, −2) can be written as z = px + qy + c. Find c.

f(x, y) = 3x²y − 5x² + 3y³ + 2xy(a, b) = (−2, −2)
c =
4.Medium
xy10.5P

Use the tangent plane of f(x, y) = 3 ln(x² + 2y²) at (1, 0.5) to approximate f(1.2, 0.4). Give the linear approximation L.

f(x, y) = 3 ln(x² + 2y²)(a, b) = (1, 0.5)(x, y) = (1.2, 0.4)
L =
5.Hard
xy05−4−202(3, −3)

f(x, y) = −2x² + 2xy − 3y² + 18x − 24y + 5 has a critical point at (3, −3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = −2x² + 2xy − 3y² + 18x − 24y + 5P = (3, −3)
D =
6.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 =

Answer key

  1. 1. I = 66
  2. 2. fx(2, −1) = 1.92
  3. 3. c = 108
  4. 4. L = 1.616
  5. 5. D = 20
  6. 6. I = 1.988

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