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

For f(x, y) = −2x³y + 4x³ + 4y² + 2xy, find the second partial derivative ∂²f/∂y² at (2, −1).

f(x, y) = −2x³y + 4x³ + 4y² + 2xyP = (2, −1)
fyy(2, −1) =
2.Easy
xy31P

For f(x, y) = x²y³ − 5x³ − 3xy, find the partial derivative ∂f/∂y at the point (3, 1).

f(x, y) = x²y³ − 5x³ − 3xyP = (3, 1)
fy(3, 1) =
3.Medium

z = 2x³y² + 2x³ + y² with x = −3t and y = 2t² + 2. Use the chain rule to find dz/dt at t = 1.

z = 2x³y² + 2x³ + y²x = −3ty = 2t² + 2t = 1
dz/dt =
4.Medium
xy−12∇fP

For f(x, y) = ln(x² + y²), find the maximum rate of change of f at the point (−1, 2), that is |∇f|.

f(x, y) = ln(x² + y²)P = (−1, 2)
|∇f| =
5.Hard
xy0202AB

The marked points are the critical points of f(x, y) = x³ + y³ − 6xy. Click the local minimum.

f(x, y) = x³ + y³ − 6xy
  1. 1A (0, 0)
  2. 2B (2, 2)
6.Hard
xyy = 1.5y = x1.51.5

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

f = cos(y²)0 ≤ x ≤ y ≤ 1.5
I =

Answer key

  1. 1. fyy(2, −1) = 8
  2. 2. fy(3, 1) = 18
  3. 3. dz/dt = −4450
  4. 4. |∇f| = 0.8944
  5. 5. B (2, 2)
  6. 6. I = 0.389

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