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) = 5x²y − 5x² − y³ + 4xy, find the partial derivative ∂f/∂y at the point (−3, −1).

f(x, y) = 5x²y − 5x² − y³ + 4xyP = (−3, −1)
fy(−3, −1) =
2.Easy
xy0.52P

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

f(x, y) = 3 sin(x + 3y)P = (0.5, 2)
fy(0.5, 2) =
3.Medium
xy12∇fP

For f(x, y) = 2 ln(x² + y²), find the maximum rate of change of f at the point (1, 2), that is |∇f|.

f(x, y) = 2 ln(x² + y²)P = (1, 2)
|∇f| =
4.Medium

z = x³y + 5x² + y² + 2xy with x = 3t − 1 and y = t². Use the chain rule to find dz/dt at t = 0.

z = x³y + 5x² + y² + 2xyx = 3t − 1y = t²t = 0
dz/dt =
5.Hard
xyy = 2xy = x²24

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

y = x²y = 2xf = xy
I =
6.Hard
xy−202−202(0, 0)(2, 2)(−2, −2)

f(x, y) = x⁴ + y⁴ − 16xy has a critical point at (−2, −2). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = x⁴ + y⁴ − 16xyP = (−2, −2)
D =

Answer key

  1. 1. fy(−3, −1) = 30
  2. 2. fy(0.5, 2) = 8.789
  3. 3. |∇f| = 1.789
  4. 4. dz/dt = −30
  5. 5. I = 2.667
  6. 6. D = 2048

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