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
xy0.52P

For f(x, y) = 3 sin(x − y), find the partial derivative ∂f/∂y at the point (0.5, 2).

f(x, y) = 3 sin(x − y)P = (0.5, 2)
fy(0.5, 2) =
2.Easy
xy22P

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

f(x, y) = 2 sin(−x − y)P = (2, 2)
fy(2, 2) =
3.Medium

z = 2x²e0.5y with x = 3t − 1 and y = −2t² − 2. Use the chain rule to find dz/dt at t = 1.

z = 2x²e0.5yx = 3t − 1y = −2t² − 2t = 1
dz/dt =
4.Medium
xy−500246vP

For f(x, y) = 2xy³ + 3x² + 5y³ + 3xy, find the directional derivative Duf at (−2, 3) in the direction of v = (1, 1).

f(x, y) = 2xy³ + 3x² + 5y³ + 3xyP = (−2, 3)v = (1, 1)
Duf =
5.Hard
xy−505−2024(3, 3)(−3, 3)

f(x, y) = x³ − 27x + y² − 6y has a critical point at (3, 3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ − 27x + y² − 6yP = (3, 3)
D =
6.Hard
xy−50−2024(−3, 2)

f(x, y) = 2x² − xy + y² + 14x − 7y + 1 has a critical point at (−3, 2). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = 2x² − xy + y² + 14x − 7y + 1P = (−3, 2)
D =

Answer key

  1. 1. fy(0.5, 2) = −0.2122
  2. 2. fy(2, 2) = 1.307
  3. 3. dz/dt = 1.083
  4. 4. Duf = 50.91
  5. 5. D = 36
  6. 6. D = 7

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