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−1−2P

For f(x, y) = −x²y² + 4x² − y³ − xy, find the partial derivative ∂f/∂y at the point (−1, −2).

f(x, y) = −x²y² + 4x² − y³ − xyP = (−1, −2)
fy(−1, −2) =
2.Easy

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

f(x, y) = 2 sin(−2x − y)P = (2, 2)
fxy(2, 2) =
3.Medium
xy−20012vP

For f(x, y) = xey, find the directional derivative Duf at (−1, 1) in the direction of v = (2, 0).

f(x, y) = xeyP = (−1, 1)v = (2, 0)
Duf =
4.Medium

z = −2x³e−y with x = t − 1 and y = −t² − 2. Use the chain rule to find dz/dt at t = −1.

z = −2x³e−yx = t − 1y = −t² − 2t = −1
dz/dt =
5.Hard
xyx = 2y = x22

Evaluate I = ∬T (x) dA over the triangle T bounded by y = 0, x = 2 and y = x.

f = x0 ≤ x ≤ 20 ≤ y ≤ x
I =
6.Hard
xy02−2−101(2, −1)

Find the critical points of f(x, y) = 2x² + xy + 3y² − 7x + 4y + 2, classify them, and give the value fmin of f at its local minimum.

f(x, y) = 2x² + xy + 3y² − 7x + 4y + 2
fmin =

Answer key

  1. 1. fy(−1, −2) = −7
  2. 2. fxy(2, 2) = −1.118
  3. 3. Duf = 2.718
  4. 4. dz/dt = −1125
  5. 5. I = 2.667
  6. 6. fmin = −7

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