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−23P

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

f(x, y) = −3xy + 2x² − 4y³P = (−2, 3)
fx(−2, 3) =
2.Easy

For f(x, y) = −2xe0.5y, find the mixed second partial derivative ∂²f/∂x∂y at (1, −1).

f(x, y) = −2xe0.5yP = (1, −1)
fxy(1, −1) =
3.Medium
xy−20012vP

For f(x, y) = −2xy − 4x² + y³, find the directional derivative Duf at (−1, 1) in the direction of v = (−2, 1).

f(x, y) = −2xy − 4x² + y³P = (−1, 1)v = (−2, 1)
Duf =
4.Medium
xy−21∇fP

For f(x, y) = −x³y − x² + 5y², find the maximum rate of change of f at the point (−2, 1), that is |∇f|.

f(x, y) = −x³y − x² + 5y²P = (−2, 1)
|∇f| =
5.Hard
xyx = 2y = 0.5x21

Evaluate I = ∬T (2x + y) dA over the triangle T bounded by y = 0, x = 2 and y = 0.5x.

f = 2x + y0 ≤ x ≤ 20 ≤ y ≤ 0.5x
I =
6.Hard
xyx = 3y = x33

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

f = x + y0 ≤ x ≤ 30 ≤ y ≤ x
I =

Answer key

  1. 1. fx(−2, 3) = −17
  2. 2. fxy(1, −1) = −0.6065
  3. 3. Duf = −3.13
  4. 4. |∇f| = 19.7
  5. 5. I = 3
  6. 6. I = 13.5

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