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) = 5x³y² − 2x³ − 2y² − 4xy, find the partial derivative ∂f/∂x at the point (1, 2).

f(x, y) = 5x³y² − 2x³ − 2y² − 4xyP = (1, 2)

f_x(1, 2) =

2.Easy

Evaluate I = ∫₂³ ∫₀² (4y − 3) dy dx over the rectangle 2 ≤ x ≤ 3, 0 ≤ y ≤ 2.

f = 4y − 32 ≤ x ≤ 30 ≤ y ≤ 2

I =

3.Medium

The tangent plane to z = −x²y² − 5x² + 5y³ − 2xy at the point where (x, y) = (−1, 3) can be written as z = px + qy + c. Find c.

f(x, y) = −x²y² − 5x² + 5y³ − 2xy(a, b) = (−1, 3)

c =

4.Medium

For f(x, y) = −3x²y + 2x² − y³ − 3xy, find the directional derivative D_uf at P(3, 2) in the direction towards Q(2, −1).

f(x, y) = −3x²y + 2x² − y³ − 3xyP = (3, 2)Q = (2, −1)

D_uf =

5.Hard

The integral I = ∫₀² ∫_x² 1/(1 + y²)² dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + y²)²0 ≤ x ≤ y ≤ 2

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. f_x(1, 2) = 46
2. I = 2
3. c = −244
4. D_uf = 55.02
5. I = 0.4
6. D = −16

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