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
xy−2−2P

For f(x, y) = −x³y − 4x² + y² + 3xy, find the partial derivative ∂f/∂x at the point (−2, −2).

f(x, y) = −x³y − 4x² + y² + 3xyP = (−2, −2)
fx(−2, −2) =
2.Easy
xy21P

For f(x, y) = −4x²y − 3x² + 2y³ + 4xy, find the partial derivative ∂f/∂x at the point (2, 1).

f(x, y) = −4x²y − 3x² + 2y³ + 4xyP = (2, 1)
fx(2, 1) =
3.Medium
xy23∇fP

For f(x, y) = −4xy² − 4x³ + 5y³ − 3xy, find the maximum rate of change of f at the point (2, 3), that is |∇f|.

f(x, y) = −4xy² − 4x³ + 5y³ − 3xyP = (2, 3)
|∇f| =
4.Medium
xy−22∇fP

For f(x, y) = xy³ + 3x² − 2y² − 3xy, find the maximum rate of change of f at the point (−2, 2), that is |∇f|.

f(x, y) = xy³ + 3x² − 2y² − 3xyP = (−2, 2)
|∇f| =
5.Hard
xyy = 1.5y = x1.51.5

The integral I = ∫01.5 ∫x1.5 1/(1 + y²)² dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + y²)²0 ≤ x ≤ y ≤ 1.5
I =
6.Hard
xy−505−4−202AB

The marked points are the critical points of f(x, y) = x³ − 27x + y² + 6y. Click the local minimum.

f(x, y) = x³ − 27x + y² + 6y
  1. 1A (3, −3)
  2. 2B (−3, −3)

Answer key

  1. 1. fx(−2, −2) = 34
  2. 2. fx(2, 1) = −24
  3. 3. |∇f| = 123.3
  4. 4. |∇f| = 27.86
  5. 5. I = 0.3462
  6. 6. A (3, −3)

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