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) = 3y·e3xy, find the mixed second partial derivative ∂²f/∂x∂y at (−1, 2).

f(x, y) = 3y·e3xyP = (−1, 2)
fxy(−1, 2) =
2.Easy

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

f(x, y) = 3 sin(2x + 2y)P = (2, −1)
fxy(2, −1) =
3.Medium

z = 3 ln(x² + 4y²) with x = −t − 1 and y = 2t². Use the chain rule to find dz/dt at t = 2.

z = 3 ln(x² + 4y²)x = −t − 1y = 2t²t = 2
dz/dt =
4.Medium
xy0246−2024PQ

For f(x, y) = 3x³e3y, find the directional derivative Duf at P(2, 0) in the direction towards Q(6, 4).

f(x, y) = 3x³e3yP = (2, 0)Q = (6, 4)
Duf =
5.Hard
xy−202−1012(1, 1)(−1, 1)

Find the critical points of f(x, y) = x³ − 3x + y² − 2y, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x³ − 3x + y² − 2y
fmin =
6.Hard
xyx = 4y = x44

Evaluate I = ∬T (3x + 2y) dA over the triangle T bounded by y = 0, x = 4 and y = x.

f = 3x + 2y0 ≤ x ≤ 40 ≤ y ≤ x
I =

Answer key

  1. 1. fxy(−1, 2) = −0.1785
  2. 2. fxy(2, −1) = −10.91
  3. 3. dz/dt = 5.864
  4. 4. Duf = 76.37
  5. 5. fmin = −3
  6. 6. I = 85.33

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