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 = 1x = 3y = 3y = 11331

Evaluate I = ∫13 ∫13 (−2xy + 2) dy dx over the rectangle 1 ≤ x ≤ 3, 1 ≤ y ≤ 3.

f = −2xy + 21 ≤ x ≤ 31 ≤ y ≤ 3
I =
2.Easy

For f(x, y) = 2 ln(x² + y²), find the mixed second partial derivative ∂²f/∂x∂y at (2, 2).

f(x, y) = 2 ln(x² + y²)P = (2, 2)
fxy(2, 2) =
3.Medium

z = 3x²y² − 4x² + y³ + 2xy with x = −t + 1 and y = 2t² − 2. Use the chain rule to find dz/dt at t = 2.

z = 3x²y² − 4x² + y³ + 2xyx = −t + 1y = 2t² − 2t = 2
dz/dt =
4.Medium
xy10∇fP

For f(x, y) = −2x³e−y, find the maximum rate of change of f at the point (1, 0), that is |∇f|.

f(x, y) = −2x³e−yP = (1, 0)
|∇f| =
5.Hard
xyy = 3xy = x²39

Evaluate I = ∬R xy dA, where R is the region between the parabola y = x² and the line y = 3x.

y = x²y = 3xf = xy
I =
6.Hard
xy3

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

f = √(x² + y²)0 ≤ r ≤ 30 ≤ θ ≤ π
I =

Answer key

  1. 1. I = −24
  2. 2. fxy(2, 2) = −0.5
  3. 3. dz/dt = 1332
  4. 4. |∇f| = 6.325
  5. 5. I = 30.38
  6. 6. I = 28.27

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