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
xy−10.5P

For f(x, y) = 3x³e−y, find the partial derivative ∂f/∂x at the point (−1, 0.5).

f(x, y) = 3x³e−yP = (−1, 0.5)
fx(−1, 0.5) =
2.Easy

For f(x, y) = 3 sin(x + 2y), find the mixed second partial derivative ∂²f/∂x∂y at (1, −1).

f(x, y) = 3 sin(x + 2y)P = (1, −1)
fxy(1, −1) =
3.Medium
xy024012PQ

For f(x, y) = ln(x² + y²), find the directional derivative Duf at P(1, 1) in the direction towards Q(5, 0).

f(x, y) = ln(x² + y²)P = (1, 1)Q = (5, 0)
Duf =
4.Medium
xy05024PQ

For f(x, y) = 3x²y − 3x² + 5y² − 2xy, find the directional derivative Duf at P(3, 2) in the direction towards Q(2, 1).

f(x, y) = 3x²y − 3x² + 5y² − 2xyP = (3, 2)Q = (2, 1)
Duf =
5.Hard
xyy = 1.5y = x1.51.5

The integral I = ∫01.5 ∫x1.5 cos(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = cos(y²)0 ≤ x ≤ y ≤ 1.5
I =
6.Hard
xy−20202(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 =

Answer key

  1. 1. fx(−1, 0.5) = 5.459
  2. 2. fxy(1, −1) = 5.049
  3. 3. Duf = 0.7276
  4. 4. Duf = −38.89
  5. 5. I = 0.389
  6. 6. D = −12

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