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) = −xy² + 4x² − 4y² − xy, find the second partial derivative ∂²f/∂y² at (−2, −2).

f(x, y) = −xy² + 4x² − 4y² − xyP = (−2, −2)
fyy(−2, −2) =
2.Easy
xy20P

For f(x, y) = 2x³e0.5y, find the partial derivative ∂f/∂y at the point (2, 0).

f(x, y) = 2x³e0.5yP = (2, 0)
fy(2, 0) =
3.Medium
xy3−2P

The tangent plane to z = −3x²y³ − 3x² + 3y² − 4xy at the point where (x, y) = (3, −2) can be written as z = px + qy + c. Find c.

f(x, y) = −3x²y³ − 3x² + 3y² − 4xy(a, b) = (3, −2)
c =
4.Medium
xy−2−1∇fP

For f(x, y) = 5xy + 5x³ − 3y³, find the maximum rate of change of f at the point (−2, −1), that is |∇f|.

f(x, y) = 5xy + 5x³ − 3y³P = (−2, −1)
|∇f| =
5.Hard
xy1

Use polar coordinates to evaluate I = ∬R y dA, where R is the upper half of the disc x² + y² ≤ 1 (y ≥ 0).

f = y0 ≤ r ≤ 10 ≤ θ ≤ π
I =
6.Hard
xy−202−2−101(1, −1)(−1, −1)

Find the critical points of f(x, y) = −x³ + 3x − y² − 2y, classify them, and give the value fmax of f at its local maximum.

f(x, y) = −x³ + 3x − y² − 2y
fmax =

Answer key

  1. 1. fyy(−2, −2) = −4
  2. 2. fy(2, 0) = 8
  3. 3. c = −873
  4. 4. |∇f| = 58.19
  5. 5. I = 0.6667
  6. 6. fmax = 3

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