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
xyx = 1x = 3y = 2132

Evaluate I = ∫13 ∫02 (xy + 1) dy dx over the rectangle 1 ≤ x ≤ 3, 0 ≤ y ≤ 2.

f = xy + 11 ≤ x ≤ 30 ≤ y ≤ 2
I =
2.Easy

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

f(x, y) = 2 ln(x² + 4y²)P = (2, −1)
fxy(2, −1) =
3.Medium
xy−11P

The tangent plane to z = −x²y³ − 4x³ at the point where (x, y) = (−1, 1) can be written as z = px + qy + c. Find c.

f(x, y) = −x²y³ − 4x³(a, b) = (−1, 1)
c =
4.Medium
xy20.5∇fP

For f(x, y) = xey, find the maximum rate of change of f at the point (2, 0.5), that is |∇f|.

f(x, y) = xeyP = (2, 0.5)
|∇f| =
5.Hard
xy−202−4−202(1, −3)(−1, −3)

f(x, y) = x³ − 3x + y² + 6y has a critical point at (1, −3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ − 3x + y² + 6yP = (1, −3)
D =
6.Hard
xyy = 4xy = x²416

Evaluate I = ∬R xy dA, where R is the region between the parabola y = x² and the line y = 4x.

y = x²y = 4xf = xy
I =

Answer key

  1. 1. I = 12
  2. 2. fxy(2, −1) = 1
  3. 3. c = −4
  4. 4. |∇f| = 3.687
  5. 5. D = 12
  6. 6. I = 170.7

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