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
xy23P

For f(x, y) = x²y + y³ − xy, find the partial derivative ∂f/∂y at the point (2, 3).

f(x, y) = x²y + y³ − xyP = (2, 3)
fy(2, 3) =
2.Easy
xyx = 2y = 2y = 1221

Evaluate I = ∫02 ∫12 (x²y² + 1) dy dx over the rectangle 0 ≤ x ≤ 2, 1 ≤ y ≤ 2.

f = x²y² + 10 ≤ x ≤ 21 ≤ y ≤ 2
I =
3.Medium
xy−2−2∇fP

For f(x, y) = 4xy² + 3x³ + y³ − xy, find the maximum rate of change of f at the point (−2, −2), that is |∇f|.

f(x, y) = 4xy² + 3x³ + y³ − xyP = (−2, −2)
|∇f| =
4.Medium
xy2−1∇fP

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

f(x, y) = ln(x² + 2y²)P = (2, −1)
|∇f| =
5.Hard
xy−2002(−2, 2)

f(x, y) = x² + 4xy + y² − 4x + 4y − 5 has a critical point at (−2, 2). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x² + 4xy + y² − 4x + 4y − 5P = (−2, 2)
D =
6.Hard
xyy = 4xy = x²416

Evaluate I = ∬R xy dA, where R is the region between the parabola y = x² and the line y = 4x.

y = x²y = 4xf = xy
I =

Answer key

  1. 1. fy(2, 3) = 29
  2. 2. I = 8.222
  3. 3. |∇f| = 70.94
  4. 4. |∇f| = 0.9428
  5. 5. D = −12
  6. 6. I = 170.7

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