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) = 3 ln(x² + 2y²), find the mixed second partial derivative ∂²f/∂x∂y at (2, −1).

f(x, y) = 3 ln(x² + 2y²)P = (2, −1)
fxy(2, −1) =
2.Easy
xyx = 3y = 4y = 2342

Evaluate I = ∫03 ∫24 (3x·e−y) dy dx over the rectangle 0 ≤ x ≤ 3, 2 ≤ y ≤ 4.

f = 3x·e−y0 ≤ x ≤ 32 ≤ y ≤ 4
I =
3.Medium
xy−11P

The tangent plane to z = ln(x² + 3y²) at the point where (x, y) = (−1, 1) can be written as z = px + qy + c. Find c.

f(x, y) = ln(x² + 3y²)(a, b) = (−1, 1)
c =
4.Medium
xy024−202PQ

For f(x, y) = 5x³y + 2x³ + 4y² + 2xy, find the directional derivative Duf at P(2, −1) in the direction towards Q(4, 3).

f(x, y) = 5x³y + 2x³ + 4y² + 2xyP = (2, −1)Q = (4, 3)
Duf =
5.Hard
xy1.5

Use polar coordinates to evaluate I = ∬R √(x² + y²) dA, where R is the disc x² + y² ≤ 2.25.

f = √(x² + y²)0 ≤ r ≤ 1.50 ≤ θ ≤ 2π
I =
6.Hard
xyy = 3xy = x²39

Find the area I = ∬R 1 dA of the region R between the parabola y = x² and the line y = 3x.

y = x²y = 3xf = 1
I =

Answer key

  1. 1. fxy(2, −1) = 1.333
  2. 2. I = 1.58
  3. 3. c = −0.6137
  4. 4. Duf = 15.21
  5. 5. I = 7.069
  6. 6. I = 4.5

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