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) = 5 sin(2x + y), find the mixed second partial derivative ∂²f/∂x∂y at (1, 0.5).

f(x, y) = 5 sin(2x + y)P = (1, 0.5)
fxy(1, 0.5) =
2.Easy

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

f(x, y) = −y·exyP = (−1, 1)
fxy(−1, 1) =
3.Medium
xy2−1∇fP

For f(x, y) = −xy + x² + 4y², find the maximum rate of change of f at the point (2, −1), that is |∇f|.

f(x, y) = −xy + x² + 4y²P = (2, −1)
|∇f| =
4.Medium
xy0.52∇fP

For f(x, y) = 3 ln(x² + 3y²), find the direction of steepest ascent at (0.5, 2) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = 3 ln(x² + 3y²)P = (0.5, 2)
θ = °
5.Hard
xyx = 1.5y = x1.5

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

f = sin(x²)0 ≤ y ≤ x ≤ 1.5
I =
6.Hard
xyx = 2y = 0.5x21

Evaluate I = ∬T (2x) dA over the triangle T bounded by y = 0, x = 2 and y = 0.5x.

f = 2x0 ≤ x ≤ 20 ≤ y ≤ 0.5x
I =

Answer key

  1. 1. fxy(1, 0.5) = −5.985
  2. 2. fxy(−1, 1) = −0.3679
  3. 3. |∇f| = 11.18
  4. 4. θ = 85.24 °
  5. 5. I = 0.8141
  6. 6. I = 2.667

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