Calculus Chapter 12.3 Question 5

Evaluate \(f_x(x,y)\) and \(f_y(x,y)\) at the indicated point.

  1. \(f(x) = x^2y - x + 2y + 3\) at (1,2).

Partially differentiate in respect to x: \[\frac{\partial f}{\partial x} = 2xy - 1\]

\[f_x(1,2) = 2*1*2 - 1 = 3\]

Partially differentiate in respect to y: \[\frac{\partial f}{\partial y} = x^2 + 2\]

\[f_y(1,2) = (1)^2 + 2 = 3\]

Calculus Chapter 12.3 Question 9

Find \(f_x, f_y, f_{xx}, f_{yy}, f_{xy}\), and \(f_{yx}\).

  1. \(f(x,y) = x^2y + 3x^2 + 4y - 5\)

\[f_x = 2xy + 6x\] \[f_y = x^2 + 4\] \[f_{xx} = 2y + 6\] \[f_{yy} = 0\] \[f_{xy} = 2x\] \[f_{yx} = 2x\]