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) = 4x²y³ − 4x³ + 4y² + 4xy, find the second partial derivative ∂²f/∂y² at (−2, 2).

f(x, y) = 4x²y³ − 4x³ + 4y² + 4xyP = (−2, 2)
fyy(−2, 2) =
2.Easy

For f(x, y) = x²ey, find the mixed second partial derivative ∂²f/∂x∂y at (−1, 1).

f(x, y) = x²eyP = (−1, 1)
fxy(−1, 1) =
3.Medium
xy2−1∇fP

For f(x, y) = 3x²e−y, find the maximum rate of change of f at the point (2, −1), that is |∇f|.

f(x, y) = 3x²e−yP = (2, −1)
|∇f| =
4.Medium
xy024−202PQ

For f(x, y) = 4x³y + 5x² − y² + 3xy, find the directional derivative Duf at P(1, −1) in the direction towards Q(5, 2).

f(x, y) = 4x³y + 5x² − y² + 3xyP = (1, −1)Q = (5, 2)
Duf =
5.Hard
xyy = xy = x²11

Find the area I = ∬R 1 dA of the region R between the parabola y = x² and the line y = x.

y = x²y = xf = 1
I =
6.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 =

Answer key

  1. 1. fyy(−2, 2) = 200
  2. 2. fxy(−1, 1) = −5.437
  3. 3. |∇f| = 46.13
  4. 4. Duf = 1.4
  5. 5. I = 0.1667
  6. 6. I = 56.55

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