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
xy10.5P

For f(x, y) = ln(x² + 4y²), find the partial derivative ∂f/∂y at the point (1, 0.5).

f(x, y) = ln(x² + 4y²)P = (1, 0.5)
fy(1, 0.5) =
2.Easy
xy0.50.5P

For f(x, y) = 2y·e3xy, find the partial derivative ∂f/∂x at the point (0.5, 0.5).

f(x, y) = 2y·e3xyP = (0.5, 0.5)
fx(0.5, 0.5) =
3.Medium
xy20.5∇fP

For f(x, y) = 3x³e2y, find the direction of steepest ascent at (2, 0.5) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = 3x³e2yP = (2, 0.5)
θ = °
4.Medium
xy0.5−1∇fP

For f(x, y) = 5 sin(−3x + 2y), find the maximum rate of change of f at the point (0.5, −1), that is |∇f|.

f(x, y) = 5 sin(−3x + 2y)P = (0.5, −1)
|∇f| =
5.Hard
xy−505−4−202(3, −2)(−3, −2)

Find the critical points of f(x, y) = −x³ + 27x − y² − 4y, classify them, and give the value fmax of f at its local maximum.

f(x, y) = −x³ + 27x − y² − 4y
fmax =
6.Hard
xy2

Use polar coordinates to evaluate I = ∬R √(x² + y²) dA, where R is the upper half of the disc x² + y² ≤ 4 (y ≥ 0).

f = √(x² + y²)0 ≤ r ≤ 20 ≤ θ ≤ π
I =

Answer key

  1. 1. fy(1, 0.5) = 2
  2. 2. fx(0.5, 0.5) = 3.176
  3. 3. θ = 53.13 °
  4. 4. |∇f| = 16.88
  5. 5. fmax = 58
  6. 6. I = 8.378

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