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−2−1P

For f(x, y) = xy² + 2x³ + y³ − 4xy, find the partial derivative ∂f/∂x at the point (−2, −1).

f(x, y) = xy² + 2x³ + y³ − 4xyP = (−2, −1)
fx(−2, −1) =
2.Easy

For f(x, y) = ln(x² + 4y²), find the mixed second partial derivative ∂²f/∂x∂y at (−1, 1).

f(x, y) = ln(x² + 4y²)P = (−1, 1)
fxy(−1, 1) =
3.Medium
xy10.5∇fP

For f(x, y) = 2x²e−y, find the direction of steepest ascent at (1, 0.5) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = 2x²e−yP = (1, 0.5)
θ = °
4.Medium

z = −4xy − 2x³ + 2y² with x = −3t + 1 and y = t². Use the chain rule to find dz/dt at t = −1.

z = −4xy − 2x³ + 2y²x = −3t + 1y = t²t = −1
dz/dt =
5.Hard
xy−202−4−202(1, −3)(−1, −3)

f(x, y) = −x³ + 3x − y² − 6y has a critical point at (1, −3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = −x³ + 3x − y² − 6yP = (1, −3)
D =
6.Hard
xy−202−202ABC

The marked points are the critical points of f(x, y) = x⁴ + y⁴ − 16xy. Click the saddle point.

f(x, y) = x⁴ + y⁴ − 16xy
  1. 1A (−2, −2)
  2. 2B (2, 2)
  3. 3C (0, 0)

Answer key

  1. 1. fx(−2, −1) = 29
  2. 2. fxy(−1, 1) = 0.64
  3. 3. θ = −26.57 °
  4. 4. dz/dt = 324
  5. 5. D = 12
  6. 6. C (0, 0)

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