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
xy2−1P

For f(x, y) = ln(x² + 4y²), find the partial derivative ∂f/∂y at the point (2, −1).

f(x, y) = ln(x² + 4y²)P = (2, −1)
fy(2, −1) =
2.Easy
xy−33P

For f(x, y) = −xy² − 4y³ − 4xy, find the partial derivative ∂f/∂x at the point (−3, 3).

f(x, y) = −xy² − 4y³ − 4xyP = (−3, 3)
fx(−3, 3) =
3.Medium

z = −2xe3y with x = 2t + 1 and y = 2t² − 1. Use the chain rule to find dz/dt at t = −1.

z = −2xe3yx = 2t + 1y = 2t² − 1t = −1
dz/dt =
4.Medium

z = x³y + 2x² − 4y² + 2xy with x = −2t and y = −2t² + 2. Use the chain rule to find dz/dt at t = 0.

z = x³y + 2x² − 4y² + 2xyx = −2ty = −2t² + 2t = 0
dz/dt =
5.Hard
xyx = 2y = x2

The integral I = ∫02 ∫y2 cos(x²) dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = cos(x²)0 ≤ y ≤ x ≤ 2
I =
6.Hard
xy1.51

Use polar coordinates to evaluate I = ∬R (x² + y²) dA, where R is the ring 1 ≤ x² + y² ≤ 2.25.

f = x² + y²1 ≤ r ≤ 1.50 ≤ θ ≤ 2π
I =

Answer key

  1. 1. fy(2, −1) = −1
  2. 2. fx(−3, 3) = −21
  3. 3. dz/dt = −562.4
  4. 4. dz/dt = −8
  5. 5. I = −0.3784
  6. 6. I = 6.381

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