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
xyx = 1x = 2y = 2y = 11221

Evaluate I = ∫12 ∫12 (−3xy² − 1) dy dx over the rectangle 1 ≤ x ≤ 2, 1 ≤ y ≤ 2.

f = −3xy² − 11 ≤ x ≤ 21 ≤ y ≤ 2
I =
2.Easy
xy3−1P

For f(x, y) = −2x³y − x³ + 3y³, find the partial derivative ∂f/∂y at the point (3, −1).

f(x, y) = −2x³y − x³ + 3y³P = (3, −1)
fy(3, −1) =
3.Medium

z = 3x²y³ − x² + 3y³ with x = −3t + 2 and y = −t². Use the chain rule to find dz/dt at t = −1.

z = 3x²y³ − x² + 3y³x = −3t + 2y = −t²t = −1
dz/dt =
4.Medium
xy−23∇fP

For f(x, y) = −3x²y² + x² − 5y² − 3xy, find the direction of steepest ascent at (−2, 3) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = −3x²y² + x² − 5y² − 3xyP = (−2, 3)
θ = °
5.Hard
xy−202−20(1, −2)(−1, −2)

f(x, y) = x³ − 3x + y² + 4y has a critical point at (−1, −2). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ − 3x + y² + 4yP = (−1, −2)
D =
6.Hard
xy1

Use polar coordinates to evaluate I = ∬R (x² + y²) dA, where R is the disc x² + y² ≤ 1.

f = x² + y²0 ≤ r ≤ 10 ≤ θ ≤ 2π
I =

Answer key

  1. 1. I = −11.5
  2. 2. fy(3, −1) = −45
  3. 3. dz/dt = 588
  4. 4. θ = −45.3 °
  5. 5. D = −12
  6. 6. I = 1.571

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