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
xy1−1P

For f(x, y) = 3x³e0.5y, find the partial derivative ∂f/∂y at the point (1, −1).

f(x, y) = 3x³e0.5yP = (1, −1)
fy(1, −1) =
2.Easy
xy3−3P

For f(x, y) = 3x²y³ + 5x³ − 5y³ + 2xy, find the partial derivative ∂f/∂x at the point (3, −3).

f(x, y) = 3x²y³ + 5x³ − 5y³ + 2xyP = (3, −3)
fx(3, −3) =
3.Medium
xy11∇fP

For f(x, y) = −3x³y³ + 3x² − 2y³ − xy, find the maximum rate of change of f at the point (1, 1), that is |∇f|.

f(x, y) = −3x³y³ + 3x² − 2y³ − xyP = (1, 1)
|∇f| =
4.Medium
xy−202012PQ

For f(x, y) = 5x³y − 5x² + 2y³ + 4xy, find the directional derivative Duf at P(−1, 1) in the direction towards Q(3, 0).

f(x, y) = 5x³y − 5x² + 2y³ + 4xyP = (−1, 1)Q = (3, 0)
Duf =
5.Hard
xyy = 0.8y = x0.80.8

The integral I = ∫00.8 ∫x0.8 cos(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = cos(y²)0 ≤ x ≤ y ≤ 0.8
I =
6.Hard
xy05−202(3, −1)

f(x, y) = x² − xy + 3y² − 7x + 9y − 3 has a critical point at (3, −1). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x² − xy + 3y² − 7x + 9y − 3P = (3, −1)
D =

Answer key

  1. 1. fy(1, −1) = 0.9098
  2. 2. fx(3, −3) = −357
  3. 3. |∇f| = 16.49
  4. 4. Duf = 28.86
  5. 5. I = 0.2986
  6. 6. D = 11

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