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) = −2x²e2y, find the mixed second partial derivative ∂²f/∂x∂y at (−1, 2).

f(x, y) = −2x²e2yP = (−1, 2)
fxy(−1, 2) =
2.Easy
xy0.51P

For f(x, y) = −2x²e3y, find the partial derivative ∂f/∂y at the point (0.5, 1).

f(x, y) = −2x²e3yP = (0.5, 1)
fy(0.5, 1) =
3.Medium
xy0.50.5∇fP

For f(x, y) = ln(x² + 2y²), find the maximum rate of change of f at the point (0.5, 0.5), that is |∇f|.

f(x, y) = ln(x² + 2y²)P = (0.5, 0.5)
|∇f| =
4.Medium
xy0.5−1∇fP

For f(x, y) = xe2y, find the maximum rate of change of f at the point (0.5, −1), that is |∇f|.

f(x, y) = xe2yP = (0.5, −1)
|∇f| =
5.Hard
xy1.5

Use polar coordinates to evaluate I = ∬R (x² + y²) dA, where R is the upper half of the disc x² + y² ≤ 2.25 (y ≥ 0).

f = x² + y²0 ≤ r ≤ 1.50 ≤ θ ≤ π
I =
6.Hard
xy0−2024(2, 3)(−2, 3)

f(x, y) = x³ − 12x + y² − 6y has a critical point at (−2, 3). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ − 12x + y² − 6yP = (−2, 3)
D =

Answer key

  1. 1. fxy(−1, 2) = 436.8
  2. 2. fy(0.5, 1) = −30.13
  3. 3. |∇f| = 2.981
  4. 4. |∇f| = 0.1914
  5. 5. I = 3.976
  6. 6. D = −24

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