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−11P

For f(x, y) = 3y·e2xy, find the partial derivative ∂f/∂x at the point (−1, 1).

f(x, y) = 3y·e2xyP = (−1, 1)
fx(−1, 1) =
2.Easy
xy3−2P

For f(x, y) = 5xy − 2x³ − 2y³, find the partial derivative ∂f/∂y at the point (3, −2).

f(x, y) = 5xy − 2x³ − 2y³P = (3, −2)
fy(3, −2) =
3.Medium
xy0.50.5∇fP

For f(x, y) = 3x³e2y, find the direction of steepest ascent at (0.5, 0.5) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = 3x³e2yP = (0.5, 0.5)
θ = °
4.Medium
xy10.5∇fP

For f(x, y) = y·e−xy, find the maximum rate of change of f at the point (1, 0.5), that is |∇f|.

f(x, y) = y·e−xyP = (1, 0.5)
|∇f| =
5.Hard
xyy = 1y = x11

The integral I = ∫01 ∫x1 1/(1 + y²)² dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + y²)²0 ≤ x ≤ y ≤ 1
I =
6.Hard
xyx = 1.5y = x1.5

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

f = cos(x²)0 ≤ y ≤ x ≤ 1.5
I =

Answer key

  1. 1. fx(−1, 1) = 0.812
  2. 2. fy(3, −2) = −9
  3. 3. θ = 18.43 °
  4. 4. |∇f| = 0.3391
  5. 5. I = 0.25
  6. 6. I = 0.389

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