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 = ∫₁² ∫₂³ (3x·e^y) dy dx over the rectangle 1 ≤ x ≤ 2, 2 ≤ y ≤ 3.

f = 3x·e^y1 ≤ x ≤ 22 ≤ y ≤ 3

I =

2.Easy

Evaluate I = ∫₁³ ∫₂⁴ (−3x²y) dy dx over the rectangle 1 ≤ x ≤ 3, 2 ≤ y ≤ 4.

f = −3x²y1 ≤ x ≤ 32 ≤ y ≤ 4

I =

3.Medium

z = −3x²y² − 4x² − y² − 4xy with x = −t and y = −2t² + 2. Use the chain rule to find dz/dt at t = −1.

z = −3x²y² − 4x² − y² − 4xyx = −ty = −2t² + 2t = −1

dz/dt =

4.Medium

For f(x, y) = −xy³ + 5x³ − 2y³, find the directional derivative D_uf at P(1, 2) in the direction towards Q(−1, 3).

f(x, y) = −xy³ + 5x³ − 2y³P = (1, 2)Q = (−1, 3)

D_uf =

5.Hard

f(x, y) = x⁴ + y⁴ − 16xy has a critical point at (2, 2). Compute the discriminant D = f_xxf_yy − f_xy² there.

f(x, y) = x⁴ + y⁴ − 16xyP = (2, 2)

D =

6.Hard

f(x, y) = −x³ + 27x − y² + 6y has a critical point at (−3, 3). Compute the discriminant D = f_xxf_yy − f_xy² there.

f(x, y) = −x³ + 27x − y² + 6yP = (−3, 3)

D =

Answer key

1. I = 57.13
2. I = −156
3. dz/dt = −8
4. D_uf = −22.36
5. D = 2048
6. D = −36

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