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−3−3P

For f(x, y) = −3x³y − 4x³ + 3xy, find the partial derivative ∂f/∂x at the point (−3, −3).

f(x, y) = −3x³y − 4x³ + 3xyP = (−3, −3)
fx(−3, −3) =
2.Easy
xyx = 2x = 5y = 4y = 12541

Evaluate I = ∫25 ∫14 (3y − 1) dy dx over the rectangle 2 ≤ x ≤ 5, 1 ≤ y ≤ 4.

f = 3y − 12 ≤ x ≤ 51 ≤ y ≤ 4
I =
3.Medium

z = 3x³y + x³ − 3y³ with x = −2t + 2 and y = 2t² + 1. Use the chain rule to find dz/dt at t = −1.

z = 3x³y + x³ − 3y³x = −2t + 2y = 2t² + 1t = −1
dz/dt =
4.Medium
xy0.50.5∇fP

For f(x, y) = 3y·e3xy, find the maximum rate of change of f at the point (0.5, 0.5), that is |∇f|.

f(x, y) = 3y·e3xyP = (0.5, 0.5)
|∇f| =
5.Hard
xy−202−202(0, 0)(2, 2)(−2, −2)

f(x, y) = x⁴ + y⁴ − 16xy has a critical point at (2, 2). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x⁴ + y⁴ − 16xyP = (2, 2)
D =
6.Hard
xy−202−20AB

The marked points are the critical points of f(x, y) = x³ − 3x + y² + 4y. Click the saddle point.

f(x, y) = x³ − 3x + y² + 4y
  1. 1A (1, −2)
  2. 2B (−1, −2)

Answer key

  1. 1. fx(−3, −3) = 126
  2. 2. I = 58.5
  3. 3. dz/dt = −1404
  4. 4. |∇f| = 12.09
  5. 5. D = 2048
  6. 6. B (−1, −2)

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