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.

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1.Easy
xy−2−2P

For f(x, y) = 2x²y² − 4x² + y² − 4xy, find the partial derivative ∂f/∂x at the point (−2, −2).

f(x, y) = 2x²y² − 4x² + y² − 4xyP = (−2, −2)
fx(−2, −2) =
2.Easy
xy3−1P

For f(x, y) = −x³y + 5x² + 2xy, find the partial derivative ∂f/∂x at the point (3, −1).

f(x, y) = −x³y + 5x² + 2xyP = (3, −1)
fx(3, −1) =
3.Medium
xy0.5−1∇fP

For f(x, y) = 2 sin(3x − y), find the maximum rate of change of f at the point (0.5, −1), that is |∇f|.

f(x, y) = 2 sin(3x − y)P = (0.5, −1)
|∇f| =
4.Medium

z = 2 ln(x² + y²) with x = −2t + 2 and y = t². Use the chain rule to find dz/dt at t = 0.

z = 2 ln(x² + y²)x = −2t + 2y = t²t = 0
dz/dt =
5.Hard
xy−202−20(1, −2)(−1, −2)

Find the critical points of f(x, y) = x³ − 3x + y² + 4y, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x³ − 3x + y² + 4y
fmin =
6.Hard
xy−50−4−202(0, 0)(−3, −3)

f(x, y) = x³ + y³ + 9xy has a critical point at (−3, −3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ + y³ + 9xyP = (−3, −3)
D =

Answer key

  1. 1. fx(−2, −2) = −8
  2. 2. fx(3, −1) = 55
  3. 3. |∇f| = 5.067
  4. 4. dz/dt = −4
  5. 5. fmin = −6
  6. 6. D = 243

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