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
xy3−1P

For f(x, y) = x²y² + 4x² − 2y² + 2xy, find the partial derivative ∂f/∂x at the point (3, −1).

f(x, y) = x²y² + 4x² − 2y² + 2xyP = (3, −1)
fx(3, −1) =
2.Easy
xy13P

For f(x, y) = 5xy³ − 4x³ − xy, find the partial derivative ∂f/∂y at the point (1, 3).

f(x, y) = 5xy³ − 4x³ − xyP = (1, 3)
fy(1, 3) =
3.Medium
xy05024vP

For f(x, y) = xy³ + 5y³ + 4xy, find the directional derivative Duf at (3, 2) in the direction of v = (3, 0).

f(x, y) = xy³ + 5y³ + 4xyP = (3, 2)v = (3, 0)
Duf =
4.Medium
xy2−1∇fP

For f(x, y) = −2x³y² − x³ − 4y² − 3xy, find the maximum rate of change of f at the point (2, −1), that is |∇f|.

f(x, y) = −2x³y² − x³ − 4y² − 3xyP = (2, −1)
|∇f| =
5.Hard
xy−202−202(0, 0)(2, 2)(−2, −2)

f(x, y) = x⁴ + y⁴ − 16xy has a critical point at (0, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x⁴ + y⁴ − 16xyP = (0, 0)
D =
6.Hard
xy2

Use polar coordinates to evaluate I = ∬R e−(x² + y²) dA, where R is the upper half of the disc x² + y² ≤ 4 (y ≥ 0).

f = e−(x² + y²)0 ≤ r ≤ 20 ≤ θ ≤ π
I =

Answer key

  1. 1. fx(3, −1) = 28
  2. 2. fy(1, 3) = 134
  3. 3. Duf = 16
  4. 4. |∇f| = 47.38
  5. 5. D = −256
  6. 6. I = 1.542

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