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) = −xy² + 4x³ − y² − 4xy, find the second partial derivative ∂²f/∂y² at (2, 1).

f(x, y) = −xy² + 4x³ − y² − 4xyP = (2, 1)
fyy(2, 1) =
2.Easy
xyx = 2x = 5y = 5y = 22552

Evaluate I = ∫25 ∫25 (x·e0.5y) dy dx over the rectangle 2 ≤ x ≤ 5, 2 ≤ y ≤ 5.

f = x·e0.5y2 ≤ x ≤ 52 ≤ y ≤ 5
I =
3.Medium
xy−4−202−2024vP

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

f(x, y) = sin(−2x + 2y)P = (−1, 2)v = (−2, −4)
Duf =
4.Medium
xy02−202vP

For f(x, y) = 3 ln(x² + 2y²), find the directional derivative Duf at (0.5, −1) in the direction of v = (2, 3).

f(x, y) = 3 ln(x² + 2y²)P = (0.5, −1)v = (2, 3)
Duf =
5.Hard
xy−202−202(2, −1)(−2, −1)

Find the critical points of f(x, y) = −x³ + 12x − y² − 2y, classify them, and give the value fmax of f at its local maximum.

f(x, y) = −x³ + 12x − y² − 2y
fmax =
6.Hard
xy−202−202(2, 0)(−2, 0)

f(x, y) = −x³ + 12x − y² has a critical point at (2, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = −x³ + 12x − y²P = (2, 0)
D =

Answer key

  1. 1. fyy(2, 1) = −6
  2. 2. I = 198.7
  3. 3. Duf = −0.8588
  4. 4. Duf = −3.698
  5. 5. fmax = 17
  6. 6. D = 24

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