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 sin(x + y), find the mixed second partial derivative ∂²f/∂x∂y at (1, 2).

f(x, y) = 3 sin(x + y)P = (1, 2)
fxy(1, 2) =
2.Easy
xy1−3P

For f(x, y) = −4xy² + 3x³ − 4y² + 2xy, find the partial derivative ∂f/∂y at the point (1, −3).

f(x, y) = −4xy² + 3x³ − 4y² + 2xyP = (1, −3)
fy(1, −3) =
3.Medium
xy13P

The tangent plane to z = 4x²y² + 3x³ at the point where (x, y) = (1, 3) can be written as z = px + qy + c. Find c.

f(x, y) = 4x²y² + 3x³(a, b) = (1, 3)
c =
4.Medium
xy02−20vP

For f(x, y) = −2xey, find the directional derivative Duf at (1, −1) in the direction of v = (2, 2).

f(x, y) = −2xeyP = (1, −1)v = (2, 2)
Duf =
5.Hard
xy1.5

Use polar coordinates to evaluate I = ∬R e−(x² + y²) dA, where R is the quarter of the disc x² + y² ≤ 2.25 in the first quadrant.

f = e−(x² + y²)0 ≤ r ≤ 1.50 ≤ θ ≤ π/2
I =
6.Hard
xy2

Use polar coordinates to evaluate I = ∬R √(x² + y²) dA, where R is the quarter of the disc x² + y² ≤ 4 in the first quadrant.

f = √(x² + y²)0 ≤ r ≤ 20 ≤ θ ≤ π/2
I =

Answer key

  1. 1. fxy(1, 2) = −0.4234
  2. 2. fy(1, −3) = 50
  3. 3. c = −114
  4. 4. Duf = −1.041
  5. 5. I = 0.7026
  6. 6. I = 4.189

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