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 = ∫₂⁴ ∫₁³ (3x·e^0.5y) dy dx over the rectangle 2 ≤ x ≤ 4, 1 ≤ y ≤ 3.

f = 3x·e^0.5y2 ≤ x ≤ 41 ≤ y ≤ 3

I =

2.Easy

For f(x, y) = xy² − 5x³ + y² + xy, find the second partial derivative ∂²f/∂y² at (−2, 2).

f(x, y) = xy² − 5x³ + y² + xyP = (−2, 2)

f_yy(−2, 2) =

3.Medium

z = 4x²y³ − 4x² + 4y³ + 2xy with x = 3t and y = −t² + 2. Use the chain rule to find dz/dt at t = −1.

z = 4x²y³ − 4x² + 4y³ + 2xyx = 3ty = −t² + 2t = −1

dz/dt =

4.Medium

For f(x, y) = −4x³y² − 2x² − 3y² − 4xy, find the directional derivative D_uf at (3, 3) in the direction of v = (−2, −1).

f(x, y) = −4x³y² − 2x² − 3y² − 4xyP = (3, 3)v = (−2, −1)

D_uf =

5.Hard

Find the critical points of f(x, y) = −x³ + 27x − y² + 2y, classify them, and give the value f_max of f at its local maximum.

f(x, y) = −x³ + 27x − y² + 2y

f_max =

6.Hard

f(x, y) = x² − 2xy + 3y² + 4x − 16y + 4 has a critical point at (1, 3). Compute the discriminant D = f_xxf_yy − f_xy² there.

f(x, y) = x² − 2xy + 3y² + 4x − 16y + 4P = (1, 3)

D =

Answer key

1. I = 102
2. f_yy(−2, 2) = −2
3. dz/dt = 234
4. D_uf = 1194
5. f_max = 55
6. D = 8

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