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) = 3x²e^3y, find the partial derivative ∂f/∂x at the point (−1, 0.5).

f(x, y) = 3x²e^3yP = (−1, 0.5)

f_x(−1, 0.5) =

2.Easy

Evaluate I = ∫₂⁴ ∫₀¹ (6x²y + 2) dy dx over the rectangle 2 ≤ x ≤ 4, 0 ≤ y ≤ 1.

f = 6x²y + 22 ≤ x ≤ 40 ≤ y ≤ 1

I =

3.Medium

For f(x, y) = xy + 4x³, find the directional derivative D_uf at (−1, 1) in the direction of v = (2, −1).

f(x, y) = xy + 4x³P = (−1, 1)v = (2, −1)

D_uf =

4.Medium

For f(x, y) = 2x³y + x² + 3y² + 4xy, find the maximum rate of change of f at the point (1, 2), that is |∇f|.

f(x, y) = 2x³y + x² + 3y² + 4xyP = (1, 2)

|∇f| =

5.Hard

Find the critical points of f(x, y) = −x³ + 12x − y² − 4y, classify them, and give the value f_max of f at its local maximum.

f(x, y) = −x³ + 12x − y² − 4y

f_max =

6.Hard

Find the critical points of f(x, y) = x³ + y³ − 6xy, classify them, and give the value f_min of f at its local minimum.

f(x, y) = x³ + y³ − 6xy

f_min =

Answer key

1. f_x(−1, 0.5) = −26.89
2. I = 60
3. D_uf = 12.07
4. |∇f| = 28.43
5. f_max = 20
6. f_min = −8

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