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) = −2x³e0.5y, find the partial derivative ∂f/∂x at the point (2, −1).

f(x, y) = −2x³e0.5yP = (2, −1)
fx(2, −1) =
2.Easy

For f(x, y) = 3y·e−xy, find the mixed second partial derivative ∂²f/∂x∂y at (−1, 0.5).

f(x, y) = 3y·e−xyP = (−1, 0.5)
fxy(−1, 0.5) =
3.Medium
xy13∇fP

For f(x, y) = 5xy³ − 3x³ − 4y³ + 3xy, find the maximum rate of change of f at the point (1, 3), that is |∇f|.

f(x, y) = 5xy³ − 3x³ − 4y³ + 3xyP = (1, 3)
|∇f| =
4.Medium

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

z = −xy − 5y²x = 3t + 2y = −2t²t = 2
dz/dt =
5.Hard
xyx = 0.8y = x0.8

The integral I = ∫00.8 ∫y0.8 ex² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = ex²0 ≤ y ≤ x ≤ 0.8
I =
6.Hard
xy3

Use polar coordinates to evaluate I = ∬R √(x² + y²) dA, where R is the quarter of the disc x² + y² ≤ 9 in the first quadrant.

f = √(x² + y²)0 ≤ r ≤ 30 ≤ θ ≤ π/2
I =

Answer key

  1. 1. fx(2, −1) = −14.56
  2. 2. fxy(−1, 0.5) = −6.183
  3. 3. |∇f| = 138.3
  4. 4. dz/dt = −552
  5. 5. I = 0.4482
  6. 6. I = 14.14

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