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
xyx = 1x = 3y = 3y = 11331

Evaluate I = ∫13 ∫13 (6x²y² + 5) dy dx over the rectangle 1 ≤ x ≤ 3, 1 ≤ y ≤ 3.

f = 6x²y² + 51 ≤ x ≤ 31 ≤ y ≤ 3
I =
2.Easy

For f(x, y) = 3y·e−xy, find the mixed second partial derivative ∂²f/∂x∂y at (0.5, 1).

f(x, y) = 3y·e−xyP = (0.5, 1)
fxy(0.5, 1) =
3.Medium
xy13P

The tangent plane to z = −3xy³ − 2x² + 3y² − 3xy at the point where (x, y) = (1, 3) can be written as z = px + qy + c. Find c.

f(x, y) = −3xy³ − 2x² + 3y² − 3xy(a, b) = (1, 3)
c =
4.Medium
xy11∇fP

For f(x, y) = 5xy + x² − y², find the maximum rate of change of f at the point (1, 1), that is |∇f|.

f(x, y) = 5xy + x² − y²P = (1, 1)
|∇f| =
5.Hard
xyx = 4y = 2x48

Evaluate I = ∬T (2x − y) dA over the triangle T bounded by y = 0, x = 4 and y = 2x.

f = 2x − y0 ≤ x ≤ 40 ≤ y ≤ 2x
I =
6.Hard
xy−202−101AB

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

f(x, y) = −x³ + 3x − y²
  1. 1A (1, 0)
  2. 2B (−1, 0)

Answer key

  1. 1. I = 470.7
  2. 2. fxy(0.5, 1) = −2.729
  3. 3. c = 227
  4. 4. |∇f| = 7.616
  5. 5. I = 42.67
  6. 6. B (−1, 0)

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