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
xy11P

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

f(x, y) = 5 sin(−2x − 2y)P = (1, 1)
fy(1, 1) =
2.Easy
xy1−1P

For f(x, y) = 3x²ey, find the partial derivative ∂f/∂y at the point (1, −1).

f(x, y) = 3x²eyP = (1, −1)
fy(1, −1) =
3.Medium

z = 3xy² + y³ + 3xy with x = 2t + 1 and y = −2t² + 2. Use the chain rule to find dz/dt at t = 2.

z = 3xy² + y³ + 3xyx = 2t + 1y = −2t² + 2t = 2
dz/dt =
4.Medium

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

z = sin(−x − y)x = −3t + 1y = −2t² − 1t = 2
dz/dt =
5.Hard
xyy = 0.8y = x0.80.8

The integral I = ∫00.8 ∫x0.8 cos(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = cos(y²)0 ≤ x ≤ y ≤ 0.8
I =
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. fy(1, 1) = 6.536
  2. 2. fy(1, −1) = 1.104
  3. 3. dz/dt = 636
  4. 4. dz/dt = 1.504
  5. 5. I = 0.2986
  6. 6. B (0, 0)

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