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−2−3P

For f(x, y) = 3xy² + x² − 5y³ + 4xy, find the partial derivative ∂f/∂y at the point (−2, −3).

f(x, y) = 3xy² + x² − 5y³ + 4xyP = (−2, −3)
fy(−2, −3) =
2.Easy
xyx = 2x = 3y = 3y = 22332

Evaluate I = ∫23 ∫23 (5x²y² + 5) dy dx over the rectangle 2 ≤ x ≤ 3, 2 ≤ y ≤ 3.

f = 5x²y² + 52 ≤ x ≤ 32 ≤ y ≤ 3
I =
3.Medium
xy11∇fP

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

f(x, y) = 2 sin(x + 3y)P = (1, 1)
|∇f| =
4.Medium
xy−12∇fP

For f(x, y) = ln(x² + 4y²), find the maximum rate of change of f at the point (−1, 2), that is |∇f|.

f(x, y) = ln(x² + 4y²)P = (−1, 2)
|∇f| =
5.Hard
xy−202−20(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 =
6.Hard
xy−202−101(1, 0)(−1, 0)

f(x, y) = −x³ + 3x − y² has a critical point at (1, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = −x³ + 3x − y²P = (1, 0)
D =

Answer key

  1. 1. fy(−2, −3) = −107
  2. 2. I = 205.6
  3. 3. |∇f| = 4.134
  4. 4. |∇f| = 0.9485
  5. 5. D = 12
  6. 6. D = 12

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