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 = 5y = 21252

Evaluate I = ∫12 ∫25 (xy + 3) dy dx over the rectangle 1 ≤ x ≤ 2, 2 ≤ y ≤ 5.

f = xy + 31 ≤ x ≤ 22 ≤ y ≤ 5
I =
2.Easy
xy−12P

For f(x, y) = 3y·exy, find the partial derivative ∂f/∂y at the point (−1, 2).

f(x, y) = 3y·exyP = (−1, 2)
fy(−1, 2) =
3.Medium
xy−23P

The tangent plane to z = 2xy² + 3x² − 4y² − 4xy at the point where (x, y) = (−2, 3) can be written as z = px + qy + c. Find c.

f(x, y) = 2xy² + 3x² − 4y² − 4xy(a, b) = (−2, 3)
c =
4.Medium
xy21P

Use the tangent plane of f(x, y) = −4x³y³ − x² − 3y³ − xy at (2, 1) to approximate f(1.9, 0.9). Give the linear approximation L.

f(x, y) = −4x³y³ − x² − 3y³ − xy(a, b) = (2, 1)(x, y) = (1.9, 0.9)
L =
5.Hard
xyy = 0.8y = x0.80.8

The integral I = ∫00.8 ∫x0.8 ey² dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = ey²0 ≤ x ≤ y ≤ 0.8
I =
6.Hard
xy−505−4−202(3, −3)(−3, −3)

f(x, y) = −x³ + 27x − y² − 6y has a critical point at (−3, −3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = −x³ + 27x − y² − 6yP = (−3, −3)
D =

Answer key

  1. 1. I = 24.75
  2. 2. fy(−1, 2) = −0.406
  3. 3. c = 72
  4. 4. L = −25
  5. 5. I = 0.4482
  6. 6. D = −36

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