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

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

f(x, y) = 3xe−yP = (0.5, 2)
fxy(0.5, 2) =
2.Easy

For f(x, y) = sin(2x + 2y), find the mixed second partial derivative ∂²f/∂x∂y at (−1, −1).

f(x, y) = sin(2x + 2y)P = (−1, −1)
fxy(−1, −1) =
3.Medium
xy22P

Use the tangent plane of f(x, y) = −4x³y² + 3x³ − 4y² − 3xy at (2, 2) to approximate f(2.2, 2.2). Give the linear approximation L.

f(x, y) = −4x³y² + 3x³ − 4y² − 3xy(a, b) = (2, 2)(x, y) = (2.2, 2.2)
L =
4.Medium

z = 3x³y − 5x³ + 2y² + xy with x = t + 1 and y = t² − 1. Use the chain rule to find dz/dt at t = 2.

z = 3x³y − 5x³ + 2y² + xyx = t + 1y = t² − 1t = 2
dz/dt =
5.Hard
xyx = 1.5y = x1.5

The integral I = ∫01.5 ∫y1.5 1/(1 + x²)² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + x²)²0 ≤ y ≤ x ≤ 1.5
I =
6.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 =

Answer key

  1. 1. fxy(0.5, 2) = −0.406
  2. 2. fxy(−1, −1) = −3.027
  3. 3. L = −194.4
  4. 4. dz/dt = 495
  5. 5. I = 0.3462
  6. 6. I = 2.667

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