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 = 2y = 11321

Evaluate I = ∫13 ∫12 (4x²y − 3) dy dx over the rectangle 1 ≤ x ≤ 3, 1 ≤ y ≤ 2.

f = 4x²y − 31 ≤ x ≤ 31 ≤ y ≤ 2
I =
2.Easy
xy−2−2P

For f(x, y) = x²y − x³ − 4xy, find the partial derivative ∂f/∂x at the point (−2, −2).

f(x, y) = x²y − x³ − 4xyP = (−2, −2)
fx(−2, −2) =
3.Medium

z = 5x³y² − x³ − 5y² + 3xy with x = t − 1 and y = −t² + 1. Use the chain rule to find dz/dt at t = 2.

z = 5x³y² − x³ − 5y² + 3xyx = t − 1y = −t² + 1t = 2
dz/dt =
4.Medium
xy11∇fP

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

f(x, y) = 5 sin(−x − 2y)P = (1, 1)
|∇f| =
5.Hard
xy3

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

f = √(x² + y²)0 ≤ r ≤ 30 ≤ θ ≤ 2π
I =
6.Hard
xyx = 2y = 3x26

Evaluate I = ∬T (x + 3y) dA over the triangle T bounded by y = 0, x = 2 and y = 3x.

f = x + 3y0 ≤ x ≤ 20 ≤ y ≤ 3x
I =

Answer key

  1. 1. I = 46
  2. 2. fx(−2, −2) = 4
  3. 3. dz/dt = 111
  4. 4. |∇f| = 11.07
  5. 5. I = 56.55
  6. 6. I = 44

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