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 = 2y = 3y = 21232

Evaluate I = ∫12 ∫23 (−2y + 2) dy dx over the rectangle 1 ≤ x ≤ 2, 2 ≤ y ≤ 3.

f = −2y + 21 ≤ x ≤ 22 ≤ y ≤ 3
I =
2.Easy

For f(x, y) = −y·e2xy, find the mixed second partial derivative ∂²f/∂x∂y at (0.5, 2).

f(x, y) = −y·e2xyP = (0.5, 2)
fxy(0.5, 2) =
3.Medium
xy1−1P

The tangent plane to z = 2x³y + 4x³ − 3xy at the point where (x, y) = (1, −1) can be written as z = px + qy + c. Find c.

f(x, y) = 2x³y + 4x³ − 3xy(a, b) = (1, −1)
c =
4.Medium
xy0.5−1P

The tangent plane to z = 3 sin(x + 3y) at the point where (x, y) = (0.5, −1) can be written as z = px + qy + c. Find c.

f(x, y) = 3 sin(x + 3y)(a, b) = (0.5, −1)
c =
5.Hard
xy−202−202(0, 0)(2, 2)(−2, −2)

Find the critical points of f(x, y) = x⁴ + y⁴ − 16xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x⁴ + y⁴ − 16xy
fmin =
6.Hard
xy−505−202(3, 0)(−3, 0)

Find the critical points of f(x, y) = x³ − 27x + y², classify them, and give the value fmin of f at its local minimum.

f(x, y) = x³ − 27x + y²
fmin =

Answer key

  1. 1. I = −3
  2. 2. fxy(0.5, 2) = −118.2
  3. 3. c = −5
  4. 4. c = −7.804
  5. 5. fmin = −32
  6. 6. fmin = −54

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