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−12P

For f(x, y) = 2xy − 4x³ − 5y³, find the partial derivative ∂f/∂y at the point (−1, 2).

f(x, y) = 2xy − 4x³ − 5y³P = (−1, 2)
fy(−1, 2) =
2.Easy
xy−11P

For f(x, y) = −x²y³ − 5x² − 4y² + 4xy, find the partial derivative ∂f/∂x at the point (−1, 1).

f(x, y) = −x²y³ − 5x² − 4y² + 4xyP = (−1, 1)
fx(−1, 1) =
3.Medium
xy−4−20024PQ

For f(x, y) = 2 ln(x² + 3y²), find the directional derivative Duf at P(−1, 1) in the direction towards Q(−4, 4).

f(x, y) = 2 ln(x² + 3y²)P = (−1, 1)Q = (−4, 4)
Duf =
4.Medium
xy−4−202−1012PQ

For f(x, y) = −y·exy, find the directional derivative Duf at P(0.5, 0.5) in the direction towards Q(−3.5, −0.5).

f(x, y) = −y·exyP = (0.5, 0.5)Q = (−3.5, −0.5)
Duf =
5.Hard
xy−20−2−101AB

The marked points are the critical points of f(x, y) = x³ + y³ + 3xy. Click the saddle point.

f(x, y) = x³ + y³ + 3xy
  1. 1A (0, 0)
  2. 2B (−1, −1)
6.Hard
xy−202−202(0, 0)(2, 2)(−2, −2)

Find the critical points of f(x, y) = x⁴ + y⁴ − 16xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x⁴ + y⁴ − 16xy
fmin =

Answer key

  1. 1. fy(−1, 2) = −62
  2. 2. fx(−1, 1) = 16
  3. 3. Duf = 2.828
  4. 4. Duf = 0.7007
  5. 5. A (0, 0)
  6. 6. fmin = −32

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