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) = y·exy, find the mixed second partial derivative ∂²f/∂x∂y at (0.5, 0.5).

f(x, y) = y·exyP = (0.5, 0.5)
fxy(0.5, 0.5) =
2.Easy
xy31P

For f(x, y) = x³y² + 4x² + 3y² + 4xy, find the partial derivative ∂f/∂x at the point (3, 1).

f(x, y) = x³y² + 4x² + 3y² + 4xyP = (3, 1)
fx(3, 1) =
3.Medium
xy0505vP

For f(x, y) = 5x³y + 4x² − 3y³, find the directional derivative Duf at (1, 3) in the direction of v = (4, −4).

f(x, y) = 5x³y + 4x² − 3y³P = (1, 3)v = (4, −4)
Duf =
4.Medium
xy−22∇fP

For f(x, y) = 5xy³ − x³ + 3y³ − 4xy, find the maximum rate of change of f at the point (−2, 2), that is |∇f|.

f(x, y) = 5xy³ − x³ + 3y³ − 4xyP = (−2, 2)
|∇f| =
5.Hard
xy05−2024(0, 0)(3, 3)

f(x, y) = x³ + y³ − 9xy has a critical point at (0, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ + y³ − 9xyP = (0, 0)
D =
6.Hard
xy05−2024(0, 0)(3, 3)

Find the critical points of f(x, y) = x³ + y³ − 9xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x³ + y³ − 9xy
fmin =

Answer key

  1. 1. fxy(0.5, 0.5) = 1.445
  2. 2. fx(3, 1) = 55
  3. 3. Duf = 91.22
  4. 4. |∇f| = 78.59
  5. 5. D = −81
  6. 6. fmin = −27

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