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
xy23P

For f(x, y) = 5x³y² − x³ − 3y² + 3xy, find the partial derivative ∂f/∂x at the point (2, 3).

f(x, y) = 5x³y² − x³ − 3y² + 3xyP = (2, 3)
fx(2, 3) =
2.Easy
xy−11P

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

f(x, y) = 2 sin(x − 2y)P = (−1, 1)
fy(−1, 1) =
3.Medium
xy−10.5P

The tangent plane to z = y·e−xy at the point where (x, y) = (−1, 0.5) can be written as z = px + qy + c. Find c.

f(x, y) = y·e−xy(a, b) = (−1, 0.5)
c =
4.Medium
xy−11∇fP

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

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

The marked points are the critical points of f(x, y) = −x³ + 3x − y² − 6y. Click the local maximum.

f(x, y) = −x³ + 3x − y² − 6y
  1. 1A (1, −3)
  2. 2B (−1, −3)
6.Hard
xyx = 3y = 2x36

Evaluate I = ∬T (x − 2y) dA over the triangle T bounded by y = 0, x = 3 and y = 2x.

f = x − 2y0 ≤ x ≤ 30 ≤ y ≤ 2x
I =

Answer key

  1. 1. fx(2, 3) = 537
  2. 2. fy(−1, 1) = 3.96
  3. 3. c = −0.8244
  4. 4. |∇f| = 8.485
  5. 5. A (1, −3)
  6. 6. I = −18

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