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 = 2122

Evaluate I = ∫12 ∫02 (y) dy dx over the rectangle 1 ≤ x ≤ 2, 0 ≤ y ≤ 2.

f = y1 ≤ x ≤ 20 ≤ y ≤ 2
I =
2.Easy
xy0.51P

For f(x, y) = 3 ln(x² + 2y²), find the partial derivative ∂f/∂y at the point (0.5, 1).

f(x, y) = 3 ln(x² + 2y²)P = (0.5, 1)
fy(0.5, 1) =
3.Medium

z = −3x²y² + 4x² + 5y³ + 4xy with x = 2t + 2 and y = 2t² + 1. Use the chain rule to find dz/dt at t = 1.

z = −3x²y² + 4x² + 5y³ + 4xyx = 2t + 2y = 2t² + 1t = 1
dz/dt =
4.Medium
xy31P

Use the tangent plane of f(x, y) = 5x³y − 5x³ + 4y³ + 4xy at (3, 1) to approximate f(3.05, 1.2). Give the linear approximation L.

f(x, y) = 5x³y − 5x³ + 4y³ + 4xy(a, b) = (3, 1)(x, y) = (3.05, 1.2)
L =
5.Hard
xy05−2024(0, 0)(3, 3)

Find the critical points of f(x, y) = x³ + y³ − 9xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x³ + y³ − 9xy
fmin =
6.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)

Answer key

  1. 1. I = 2
  2. 2. fy(0.5, 1) = 5.333
  3. 3. dz/dt = −892
  4. 4. L = 48
  5. 5. fmin = −27
  6. 6. A (1, −3)

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