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

Evaluate I = ∫₁⁴ ∫₀¹ (4xy² − 2) dy dx over the rectangle 1 ≤ x ≤ 4, 0 ≤ y ≤ 1.

f = 4xy² − 21 ≤ x ≤ 40 ≤ y ≤ 1

I =

2.Easy

Evaluate I = ∫₀² ∫₀³ (5y − 3) dy dx over the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 3.

f = 5y − 30 ≤ x ≤ 20 ≤ y ≤ 3

I =

3.Medium

The tangent plane to z = −3xy² − 3x³ + 3y² − 2xy at the point where (x, y) = (2, 3) can be written as z = px + qy + c. Find c.

f(x, y) = −3xy² − 3x³ + 3y² − 2xy(a, b) = (2, 3)

c =

4.Medium

For f(x, y) = y·e^2xy, find the directional derivative D_uf at (2, 0.5) in the direction of v = (−4, 4).

f(x, y) = y·e^2xyP = (2, 0.5)v = (−4, 4)

D_uf =

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

f(x, y) = −x³ + 12x − y² − 6y has a critical point at (2, −3). Compute the discriminant D = f_xxf_yy − f_xy² there.

f(x, y) = −x³ + 12x − y² − 6yP = (2, −3)

D =

Answer key

1. I = 4
2. I = 27
3. c = 141
4. D_uf = 13.06
5. f_max = 20
6. D = 24

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