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 ∫₁⁴ (−x²y + 1) dy dx over the rectangle 1 ≤ x ≤ 2, 1 ≤ y ≤ 4.

f = −x²y + 11 ≤ x ≤ 21 ≤ y ≤ 4

I =

2.Easy

For f(x, y) = −4x²y² + 2x² + y² + 3xy, find the partial derivative ∂f/∂x at the point (2, −3).

f(x, y) = −4x²y² + 2x² + y² + 3xyP = (2, −3)

f_x(2, −3) =

3.Medium

The tangent plane to z = 2y·e^xy at the point where (x, y) = (0.5, 1) can be written as z = px + qy + c. Find c.

f(x, y) = 2y·e^xy(a, b) = (0.5, 1)

c =

4.Medium

z = −3x²y + 2x³ − 3y³ with x = t + 1 and y = −2t² + 1. Use the chain rule to find dz/dt at t = 1.

z = −3x²y + 2x³ − 3y³x = t + 1y = −2t² + 1t = 1

dz/dt =

5.Hard

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

f = sin(x²)0 ≤ y ≤ x ≤ 1

I =

6.Hard

f(x, y) = x⁴ + y⁴ − 4xy has a critical point at (0, 0). Compute the discriminant D = f_xxf_yy − f_xy² there.

f(x, y) = x⁴ + y⁴ − 4xyP = (0, 0)

D =

Answer key

1. I = −14.5
2. f_x(2, −3) = −145
3. c = −3.297
4. dz/dt = 120
5. I = 0.2298
6. D = −16

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