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
xy3−1P

For f(x, y) = 2xy³ − 5x² − 3y² − 2xy, find the partial derivative ∂f/∂y at the point (3, −1).

f(x, y) = 2xy³ − 5x² − 3y² − 2xyP = (3, −1)
fy(3, −1) =
2.Easy

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

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

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

z = 2 sin(−3x − 3y)x = −2t + 2y = −2t²t = 1
dz/dt =
4.Medium
xy−10123−4−20PQ

For f(x, y) = 3 sin(3x − 2y), find the directional derivative Duf at P(1, −1) in the direction towards Q(0, −5).

f(x, y) = 3 sin(3x − 2y)P = (1, −1)Q = (0, −5)
Duf =
5.Hard
xy−202−202(0, 0)(2, 2)(−2, −2)

Find the critical points of f(x, y) = x⁴ + y⁴ − 16xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x⁴ + y⁴ − 16xy
fmin =
6.Hard
xyx = 1y = x11

Evaluate I = ∬T (x + y) dA over the triangle T bounded by y = 0, x = 1 and y = x.

f = x + y0 ≤ x ≤ 10 ≤ y ≤ x
I =

Answer key

  1. 1. fy(3, −1) = 18
  2. 2. fxy(2, 1) = −11.83
  3. 3. dz/dt = 34.57
  4. 4. Duf = 1.032
  5. 5. fmin = −32
  6. 6. I = 0.5

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