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
xy22P

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

f(x, y) = 2 ln(x² + 4y²)P = (2, 2)
fy(2, 2) =
2.Easy
xy−10.5P

For f(x, y) = 2 sin(3x + 3y), find the partial derivative ∂f/∂y at the point (−1, 0.5).

f(x, y) = 2 sin(3x + 3y)P = (−1, 0.5)
fy(−1, 0.5) =
3.Medium
xy−4−20−2−10PQ

For f(x, y) = −3x²y³ − 5x² + 3y³ − 4xy, find the directional derivative Duf at P(−1, −1) in the direction towards Q(−5, −1).

f(x, y) = −3x²y³ − 5x² + 3y³ − 4xyP = (−1, −1)Q = (−5, −1)
Duf =
4.Medium
xy−4−20−4−202vP

For f(x, y) = 4x³y − 5x³ − 2xy, find the directional derivative Duf at (−2, −2) in the direction of v = (0, 4).

f(x, y) = 4x³y − 5x³ − 2xyP = (−2, −2)v = (0, 4)
Duf =
5.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 =
6.Hard
xyx = 1.2y = x1.2

The integral I = ∫01.2 ∫y1.2 sin(x²) dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = sin(x²)0 ≤ y ≤ x ≤ 1.2
I =

Answer key

  1. 1. fy(2, 2) = 1.6
  2. 2. fy(−1, 0.5) = 0.4244
  3. 3. Duf = −8
  4. 4. Duf = −28
  5. 5. I = 1.047
  6. 6. I = 0.4348

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