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) = −4x³y − x³ − y² − xy, find the second partial derivative ∂²f/∂x² at (−1, −1).

f(x, y) = −4x³y − x³ − y² − xyP = (−1, −1)
fxx(−1, −1) =
2.Easy
xyx = 1x = 4y = 3143

Evaluate I = ∫14 ∫03 (5xy² − 3) dy dx over the rectangle 1 ≤ x ≤ 4, 0 ≤ y ≤ 3.

f = 5xy² − 31 ≤ x ≤ 40 ≤ y ≤ 3
I =
3.Medium
xy2−1∇fP

For f(x, y) = 5 sin(−x + y), find the direction of steepest ascent at (2, −1) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = 5 sin(−x + y)P = (2, −1)
θ = °
4.Medium
xy0.52∇fP

For f(x, y) = 3 ln(x² + 3y²), find the maximum rate of change of f at the point (0.5, 2), that is |∇f|.

f(x, y) = 3 ln(x² + 3y²)P = (0.5, 2)
|∇f| =
5.Hard
xy−505−202(3, 0)(−3, 0)

f(x, y) = −x³ + 27x − y² has a critical point at (−3, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = −x³ + 27x − y²P = (−3, 0)
D =
6.Hard
xy1

Use polar coordinates to evaluate I = ∬R x dA, where R is the quarter of the disc x² + y² ≤ 1 in the first quadrant.

f = x0 ≤ r ≤ 10 ≤ θ ≤ π/2
I =

Answer key

  1. 1. fxx(−1, −1) = −18
  2. 2. I = 310.5
  3. 3. θ = −45 °
  4. 4. |∇f| = 2.949
  5. 5. D = −36
  6. 6. I = 0.3333

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