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) = −2xey, find the mixed second partial derivative ∂²f/∂x∂y at (2, 2).

f(x, y) = −2xeyP = (2, 2)
fxy(2, 2) =
2.Easy

For f(x, y) = −4x²y³ − 2x³ − 5y³ − 4xy, find the second partial derivative ∂²f/∂y² at (1, −1).

f(x, y) = −4x²y³ − 2x³ − 5y³ − 4xyP = (1, −1)
fyy(1, −1) =
3.Medium
xy32∇fP

For f(x, y) = −xy³ − 3x³ − 4y² + 4xy, find the direction of steepest ascent at (3, 2) as the angle θ from the positive x-axis, in degrees between −180° and 180°.

f(x, y) = −xy³ − 3x³ − 4y² + 4xyP = (3, 2)
θ = °
4.Medium
xy−4−202−4−202vP

For f(x, y) = −2xy³ − 5x³ − 5y³ − 2xy, find the directional derivative Duf at (−1, −2) in the direction of v = (4, 4).

f(x, y) = −2xy³ − 5x³ − 5y³ − 2xyP = (−1, −2)v = (4, 4)
Duf =
5.Hard
xy−202−202(0, 0)(1, 1)(−1, −1)

Find the critical points of f(x, y) = x⁴ + y⁴ − 4xy, classify them, and give the value fmin of f at its local minimum.

f(x, y) = x⁴ + y⁴ − 4xy
fmin =
6.Hard
xy−20202(1, 2)(−1, 2)

f(x, y) = −x³ + 3x − y² + 4y has a critical point at (−1, 2). Compute the discriminant D = fxxfyy − fxy² there.

f(x, y) = −x³ + 3x − y² + 4yP = (−1, 2)
D =

Answer key

  1. 1. fxy(2, 2) = −14.78
  2. 2. fyy(1, −1) = 54
  3. 3. θ = −153.7 °
  4. 4. Duf = −20.51
  5. 5. fmin = −2
  6. 6. D = −12

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