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−12P

For f(x, y) = 3y·e3xy, find the partial derivative ∂f/∂x at the point (−1, 2).

f(x, y) = 3y·e3xyP = (−1, 2)
fx(−1, 2) =
2.Easy
xyx = 1y = 313

Evaluate I = ∫01 ∫03 (2y + 2) dy dx over the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 3.

f = 2y + 20 ≤ x ≤ 10 ≤ y ≤ 3
I =
3.Medium
xy−4−20−4−20vP

For f(x, y) = −3x²y − 3x² − 5y³ − 4xy, find the directional derivative Duf at (−2, −2) in the direction of v = (−2, −3).

f(x, y) = −3x²y − 3x² − 5y³ − 4xyP = (−2, −2)v = (−2, −3)
Duf =
4.Medium
xy−11P

The tangent plane to z = xy³ − 4x² − 5y² − 2xy at the point where (x, y) = (−1, 1) can be written as z = px + qy + c. Find c.

f(x, y) = xy³ − 4x² − 5y² − 2xy(a, b) = (−1, 1)
c =
5.Hard
xy−20−2−101(−2, −1)

Find the critical points of f(x, y) = 3x² − xy + 3y² + 11x + 4y, classify them, and give the value fmin of f at its local minimum.

f(x, y) = 3x² − xy + 3y² + 11x + 4y
fmin =
6.Hard
xy−20−20AB

The marked points are the critical points of f(x, y) = x³ + y³ + 6xy. Click the saddle point.

f(x, y) = x³ + y³ + 6xy
  1. 1A (−2, −2)
  2. 2B (0, 0)

Answer key

  1. 1. fx(−1, 2) = 0.08924
  2. 2. I = 15
  3. 3. Duf = 55.47
  4. 4. c = 10
  5. 5. fmin = −13
  6. 6. B (0, 0)

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