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 = ∫₀³ ∫₁³ (−x²y²) dy dx over the rectangle 0 ≤ x ≤ 3, 1 ≤ y ≤ 3.

f = −x²y²0 ≤ x ≤ 31 ≤ y ≤ 3

I =

2.Easy

Evaluate I = ∫₀² ∫₁⁴ (2x·e^−y) dy dx over the rectangle 0 ≤ x ≤ 2, 1 ≤ y ≤ 4.

f = 2x·e^−y0 ≤ x ≤ 21 ≤ y ≤ 4

I =

3.Medium

The tangent plane to z = −x³y² − 5x³ + y³ − 3xy at the point where (x, y) = (−2, 1) can be written as z = px + qy + c. Find c.

f(x, y) = −x³y² − 5x³ + y³ − 3xy(a, b) = (−2, 1)

c =

4.Medium

z = −2xe^y with x = 2t − 2 and y = −2t² − 1. Use the chain rule to find dz/dt at t = −1.

z = −2xe^yx = 2t − 2y = −2t² − 1t = −1

dz/dt =

5.Hard

Find the critical points of f(x, y) = x³ + y³ − 9xy, classify them, and give the value f_min of f at its local minimum.

f(x, y) = x³ + y³ − 9xy

f_min =

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 = −78
2. I = 1.398
3. c = −120
4. dz/dt = 1.394
5. f_min = −27
6. D = −36

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