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

For f(x, y) = 2 ln(x² + 3y²), find the mixed second partial derivative ∂²f/∂x∂y at (1, −1).

f(x, y) = 2 ln(x² + 3y²)P = (1, −1)
fxy(1, −1) =
2.Easy
xyx = 1x = 4y = 4y = 21442

Evaluate I = ∫14 ∫24 (−3x²y + 5) dy dx over the rectangle 1 ≤ x ≤ 4, 2 ≤ y ≤ 4.

f = −3x²y + 51 ≤ x ≤ 42 ≤ y ≤ 4
I =
3.Medium

z = 2x²ey with x = −3t − 2 and y = −2t² + 1. Use the chain rule to find dz/dt at t = 1.

z = 2x²eyx = −3t − 2y = −2t² + 1t = 1
dz/dt =
4.Medium

z = 3x³y² + 3x³ − 5y³ − 2xy with x = t − 1 and y = t² − 1. Use the chain rule to find dz/dt at t = −1.

z = 3x³y² + 3x³ − 5y³ − 2xyx = t − 1y = t² − 1t = −1
dz/dt =
5.Hard
xy02−2−101(1, −1)

Find the critical points of f(x, y) = x² − 3xy + 3y² − 5x + 9y − 3, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x² − 3xy + 3y² − 5x + 9y − 3
fmin =
6.Hard
xy−202−202ABC

The marked points are the critical points of f(x, y) = x⁴ + y⁴ − 4xy. Click the saddle point.

f(x, y) = x⁴ + y⁴ − 4xy
  1. 1A (1, 1)
  2. 2B (0, 0)
  3. 3C (−1, −1)

Answer key

  1. 1. fxy(1, −1) = 1.5
  2. 2. I = −348
  3. 3. dz/dt = −51.5
  4. 4. dz/dt = 28
  5. 5. fmin = −10
  6. 6. B (0, 0)

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