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
xy0.51P

For f(x, y) = 3x²e−y, find the partial derivative ∂f/∂y at the point (0.5, 1).

f(x, y) = 3x²e−yP = (0.5, 1)
fy(0.5, 1) =
2.Easy
xy−10P

For f(x, y) = 3x³e2y, find the partial derivative ∂f/∂y at the point (−1, 0).

f(x, y) = 3x³e2yP = (−1, 0)
fy(−1, 0) =
3.Medium
xy33∇fP

For f(x, y) = −3xy² − x² + xy, find the maximum rate of change of f at the point (3, 3), that is |∇f|.

f(x, y) = −3xy² − x² + xyP = (3, 3)
|∇f| =
4.Medium
xy−1−1∇fP

For f(x, y) = ln(x² + 4y²), find the maximum rate of change of f at the point (−1, −1), that is |∇f|.

f(x, y) = ln(x² + 4y²)P = (−1, −1)
|∇f| =
5.Hard
xyy = 2xy = x²24

Evaluate I = ∬R x dA, where R is the region between the parabola y = x² and the line y = 2x.

y = x²y = 2xf = x
I =
6.Hard
xyx = 4y = 0.5x42

Evaluate I = ∬T (3x + 4y) dA over the triangle T bounded by y = 0, x = 4 and y = 0.5x.

f = 3x + 4y0 ≤ x ≤ 40 ≤ y ≤ 0.5x
I =

Answer key

  1. 1. fy(0.5, 1) = −0.2759
  2. 2. fy(−1, 0) = −6
  3. 3. |∇f| = 59.17
  4. 4. |∇f| = 1.649
  5. 5. I = 1.333
  6. 6. I = 42.67

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