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 ∫₂⁴ (5y² + 1) dy dx over the rectangle 2 ≤ x ≤ 4, 2 ≤ y ≤ 4.

f = 5y² + 12 ≤ x ≤ 42 ≤ y ≤ 4

I =

2.Easy

Evaluate I = ∫₁³ ∫₀³ (2x·e^−y) dy dx over the rectangle 1 ≤ x ≤ 3, 0 ≤ y ≤ 3.

f = 2x·e^−y1 ≤ x ≤ 30 ≤ y ≤ 3

I =

3.Medium

The tangent plane to z = 3x³y² − 3x² at the point where (x, y) = (−1, −2) can be written as z = px + qy + c. Find c.

f(x, y) = 3x³y² − 3x²(a, b) = (−1, −2)

c =

4.Medium

z = 2 sin(x + 2y) with x = −t and y = −2t² + 2. Use the chain rule to find dz/dt at t = 1.

z = 2 sin(x + 2y)x = −ty = −2t² + 2t = 1

dz/dt =

5.Hard

f(x, y) = x³ − 12x + y² + 2y has a critical point at (−2, −1). Compute the discriminant D = f_xxf_yy − f_xy² there.

f(x, y) = x³ − 12x + y² + 2yP = (−2, −1)

D =

6.Hard

f(x, y) = 2x² + 5xy + 2y² − 3x − 6y − 3 has a critical point at (2, −1). Compute the discriminant D = f_xxf_yy − f_xy² there.

f(x, y) = 2x² + 5xy + 2y² − 3x − 6y − 3P = (2, −1)

D =

Answer key

1. I = 190.7
2. I = 7.602
3. c = 51
4. dz/dt = −9.725
5. D = −24
6. D = −9

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