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) = 3xe0.5y, find the mixed second partial derivative ∂²f/∂x∂y at (2, 0.5).

f(x, y) = 3xe0.5yP = (2, 0.5)
fxy(2, 0.5) =
2.Easy
xy−1−3P

For f(x, y) = −xy + 4x² − 2y³, find the partial derivative ∂f/∂x at the point (−1, −3).

f(x, y) = −xy + 4x² − 2y³P = (−1, −3)
fx(−1, −3) =
3.Medium
xy−4−202024vP

For f(x, y) = 5xy³ − 3x² − 5y² + 4xy, find the directional derivative Duf at (−1, 2) in the direction of v = (−1, −1).

f(x, y) = 5xy³ − 3x² − 5y² + 4xyP = (−1, 2)v = (−1, −1)
Duf =
4.Medium
xy1−1∇fP

For f(x, y) = 2 ln(x² + 3y²), find the maximum rate of change of f at the point (1, −1), that is |∇f|.

f(x, y) = 2 ln(x² + 3y²)P = (1, −1)
|∇f| =
5.Hard
xy02−20(2, −2)

Find the critical points of f(x, y) = 3x² + 4xy + 2y² − 4x + 5, classify them, and give the value fmin of f at its local minimum.

f(x, y) = 3x² + 4xy + 2y² − 4x + 5
fmin =
6.Hard
xy−202−202(2, 0)(−2, 0)

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

f(x, y) = x³ − 12x + y²
fmin =

Answer key

  1. 1. fxy(2, 0.5) = 1.926
  2. 2. fx(−1, −3) = −5
  3. 3. Duf = 21.21
  4. 4. |∇f| = 3.162
  5. 5. fmin = 1
  6. 6. fmin = −16

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