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

For f(x, y) = 2x³e^−y, find the partial derivative ∂f/∂x at the point (−1, 0.5).

f(x, y) = 2x³e^−yP = (−1, 0.5)

f_x(−1, 0.5) =

2.Easy

Evaluate I = ∫₀³ ∫₁³ (y + 4) dy dx over the rectangle 0 ≤ x ≤ 3, 1 ≤ y ≤ 3.

f = y + 40 ≤ x ≤ 31 ≤ y ≤ 3

I =

3.Medium

The tangent plane to z = −3xy² + 3x³ − 4y² at the point where (x, y) = (3, 3) can be written as z = px + qy + c. Find c.

f(x, y) = −3xy² + 3x³ − 4y²(a, b) = (3, 3)

c =

4.Medium

z = −x³y² − 4y³ − 3xy with x = 3t − 1 and y = 2t² − 2. Use the chain rule to find dz/dt at t = 1.

z = −x³y² − 4y³ − 3xyx = 3t − 1y = 2t² − 2t = 1

dz/dt =

5.Hard

The integral I = ∫₀¹ ∫_y¹ 1/(1 + x²)² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + x²)²0 ≤ y ≤ x ≤ 1

I =

6.Hard

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

f(x, y) = x³ − 27x + y²

f_min =

Answer key

1. f_x(−1, 0.5) = 3.639
2. I = 36
3. c = 36
4. dz/dt = −24
5. I = 0.25
6. f_min = −54

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