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
xy0.51P

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

f(x, y) = 2y·e−xyP = (0.5, 1)
fy(0.5, 1) =
2.Easy
xy−10.5P

For f(x, y) = −2x²e3y, find the partial derivative ∂f/∂x at the point (−1, 0.5).

f(x, y) = −2x²e3yP = (−1, 0.5)
fx(−1, 0.5) =
3.Medium
xy−2−1∇fP

For f(x, y) = −4x²y³ − 3x³ + y³ − 4xy, find the maximum rate of change of f at the point (−2, −1), that is |∇f|.

f(x, y) = −4x²y³ − 3x³ + y³ − 4xyP = (−2, −1)
|∇f| =
4.Medium
xy1−2P

Use the tangent plane of f(x, y) = 5xy + 3x² + 3y² at (1, −2) to approximate f(1.05, −1.8). Give the linear approximation L.

f(x, y) = 5xy + 3x² + 3y²(a, b) = (1, −2)(x, y) = (1.05, −1.8)
L =
5.Hard
xyy = 3xy = x²39

Find the area I = ∬R 1 dA of the region R between the parabola y = x² and the line y = 3x.

y = x²y = 3xf = 1
I =
6.Hard
xyy = 2y = x22

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

f = cos(y²)0 ≤ x ≤ y ≤ 2
I =

Answer key

  1. 1. fy(0.5, 1) = 0.6065
  2. 2. fx(−1, 0.5) = 17.93
  3. 3. |∇f| = 60.61
  4. 4. L = 3.4
  5. 5. I = 4.5
  6. 6. I = −0.3784

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