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
xy−3−1P

For f(x, y) = x³y³ + x³ − y² + 3xy, find the partial derivative ∂f/∂y at the point (−3, −1).

f(x, y) = x³y³ + x³ − y² + 3xyP = (−3, −1)
fy(−3, −1) =
2.Easy
xyx = 2x = 5y = 3253

Evaluate I = ∫25 ∫03 (x·e−y) dy dx over the rectangle 2 ≤ x ≤ 5, 0 ≤ y ≤ 3.

f = x·e−y2 ≤ x ≤ 50 ≤ y ≤ 3
I =
3.Medium
xy−20024PQ

For f(x, y) = 2x²y³ − x³ + 3y² + 2xy, find the directional derivative Duf at P(−1, 1) in the direction towards Q(1, 4).

f(x, y) = 2x²y³ − x³ + 3y² + 2xyP = (−1, 1)Q = (1, 4)
Duf =
4.Medium
xy11P

The tangent plane to z = −2x³e2y at the point where (x, y) = (1, 1) can be written as z = px + qy + c. Find c.

f(x, y) = −2x³e2y(a, b) = (1, 1)
c =
5.Hard
xy−202−202(0, 0)(1, 1)(−1, −1)

Find the critical points of f(x, y) = x⁴ + y⁴ − 4xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x⁴ + y⁴ − 4xy
fmin =
6.Hard
xy−505−202(3, 0)(−3, 0)

f(x, y) = x³ − 27x + y² has a critical point at (3, 0). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x³ − 27x + y²P = (3, 0)
D =

Answer key

  1. 1. fy(−3, −1) = −88
  2. 2. I = 9.977
  3. 3. Duf = 5.547
  4. 4. c = 59.11
  5. 5. fmin = −2
  6. 6. D = 36

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