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) = 5 sin(−2x + 3y), find the mixed second partial derivative ∂²f/∂x∂y at (1, −1).

f(x, y) = 5 sin(−2x + 3y)P = (1, −1)
fxy(1, −1) =
2.Easy
xy−3−3P

For f(x, y) = 3xy − 2y³, find the partial derivative ∂f/∂x at the point (−3, −3).

f(x, y) = 3xy − 2y³P = (−3, −3)
fx(−3, −3) =
3.Medium

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

z = −xy − 4x² + 4y³x = 2t − 2y = −t² − 2t = 2
dz/dt =
4.Medium
xy1−1∇fP

For f(x, y) = xy − x² − 2y³, find the maximum rate of change of f at the point (1, −1), that is |∇f|.

f(x, y) = xy − x² − 2y³P = (1, −1)
|∇f| =
5.Hard
xyy = 1.2y = x1.21.2

The integral I = ∫01.2 ∫x1.2 cos(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = cos(y²)0 ≤ x ≤ y ≤ 1.2
I =
6.Hard
xy0−4−202(2, −3)(−2, −3)

Find the critical points of f(x, y) = −x³ + 12x − y² − 6y, classify them, and give the value fmax of f at its local maximum.

f(x, y) = −x³ + 12x − y² − 6y
fmax =

Answer key

  1. 1. fxy(1, −1) = 28.77
  2. 2. fx(−3, −3) = −9
  3. 3. dz/dt = −1740
  4. 4. |∇f| = 5.831
  5. 5. I = 0.4957
  6. 6. fmax = 25

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