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
xyx = 2x = 3y = 2232

Evaluate I = ∫23 ∫02 (3x²y + 5) dy dx over the rectangle 2 ≤ x ≤ 3, 0 ≤ y ≤ 2.

f = 3x²y + 52 ≤ x ≤ 30 ≤ y ≤ 2
I =
2.Easy

For f(x, y) = −x³y³ + 4y² + xy, find the second partial derivative ∂²f/∂x² at (−1, −1).

f(x, y) = −x³y³ + 4y² + xyP = (−1, −1)
fxx(−1, −1) =
3.Medium
xy−4−20−20vP

For f(x, y) = −xy + 5x² + 3y³, find the directional derivative Duf at (−2, −1) in the direction of v = (3, 0).

f(x, y) = −xy + 5x² + 3y³P = (−2, −1)v = (3, 0)
Duf =
4.Medium
xy2−1∇fP

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

f(x, y) = sin(2x + 3y)P = (2, −1)
|∇f| =
5.Hard
xy−4−202−4−202(−2, −3)

Find the critical points of f(x, y) = 3x² − xy + 2y² + 9x + 10y + 5, classify them, and give the value fmin of f at its local minimum.

f(x, y) = 3x² − xy + 2y² + 9x + 10y + 5
fmin =
6.Hard
xy3

Use polar coordinates to evaluate I = ∬R y dA, where R is the upper half of the disc x² + y² ≤ 9 (y ≥ 0).

f = y0 ≤ r ≤ 30 ≤ θ ≤ π
I =

Answer key

  1. 1. I = 48
  2. 2. fxx(−1, −1) = −6
  3. 3. Duf = −19
  4. 4. |∇f| = 1.948
  5. 5. fmin = −19
  6. 6. I = 18

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