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
xy12P

For f(x, y) = 3 ln(x² + 4y²), find the partial derivative ∂f/∂y at the point (1, 2).

f(x, y) = 3 ln(x² + 4y²)P = (1, 2)
fy(1, 2) =
2.Easy
xy1−1P

For f(x, y) = sin(2x − y), find the partial derivative ∂f/∂x at the point (1, −1).

f(x, y) = sin(2x − y)P = (1, −1)
fx(1, −1) =
3.Medium
xy−202−1012vP

For f(x, y) = −y·e3xy, find the directional derivative Duf at (−1, 0.5) in the direction of v = (3, −1).

f(x, y) = −y·e3xyP = (−1, 0.5)v = (3, −1)
Duf =
4.Medium
xy02−202PQ

For f(x, y) = 2 sin(−2x − 3y), find the directional derivative Duf at P(1, −1) in the direction towards Q(3, 2).

f(x, y) = 2 sin(−2x − 3y)P = (1, −1)Q = (3, 2)
Duf =
5.Hard
xyy = 3xy = x²39

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

y = x²y = 3xf = y
I =
6.Hard
xy1

Use polar coordinates to evaluate I = ∬R √(x² + y²) dA, where R is the upper half of the disc x² + y² ≤ 1 (y ≥ 0).

f = √(x² + y²)0 ≤ r ≤ 10 ≤ θ ≤ π
I =

Answer key

  1. 1. fy(1, 2) = 2.824
  2. 2. fx(1, −1) = −1.98
  3. 3. Duf = −0.194
  4. 4. Duf = −3.896
  5. 5. I = 16.2
  6. 6. I = 1.047

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