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−1−1P

For f(x, y) = 3 ln(x² + y²), find the partial derivative ∂f/∂y at the point (−1, −1).

f(x, y) = 3 ln(x² + y²)P = (−1, −1)
fy(−1, −1) =
2.Easy

For f(x, y) = 3y·e3xy, find the mixed second partial derivative ∂²f/∂x∂y at (0.5, 1).

f(x, y) = 3y·e3xyP = (0.5, 1)
fxy(0.5, 1) =
3.Medium

z = −3xy² + x² + 2y³ − 4xy with x = 2t − 1 and y = −t² − 2. Use the chain rule to find dz/dt at t = −1.

z = −3xy² + x² + 2y³ − 4xyx = 2t − 1y = −t² − 2t = −1
dz/dt =
4.Medium
xy−4−202−4−20PQ

For f(x, y) = −3x²y + 3x³ + 3y³ + 3xy, find the directional derivative Duf at P(−1, −2) in the direction towards Q(−5, −3).

f(x, y) = −3x²y + 3x³ + 3y³ + 3xyP = (−1, −2)Q = (−5, −3)
Duf =
5.Hard
xy−202−202(0, 0)(1, 1)(−1, −1)

Find the critical points of f(x, y) = x⁴ + y⁴ − 4xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x⁴ + y⁴ − 4xy
fmin =
6.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 =

Answer key

  1. 1. fy(−1, −1) = −3
  2. 2. fxy(0.5, 1) = 141.2
  3. 3. dz/dt = −18
  4. 4. Duf = 1.455
  5. 5. fmin = −2
  6. 6. I = 2.667

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