Difference Quotient

[nycmath]

See attachment.

[Eigenvalue]

A. Use the difference quotient formula: [tex]\frac{-2(x+h)²+(x+h)+1-(2x²+x+1)}{h}[/tex]

Simplify: [tex]\frac{-4xh-2h²+h}{h}[/tex]
=4x-2h+1

[Eigenvalue]

B. The Signum function, sgn(x), has 3 different cases:
[tex]\begin{cases} 1, x>0 \\ 0, x=0 \\-1, x<0 \end{cases}[/tex]

Sidenote: sgn(x)=[tex]\frac{|x|}{x}[/tex]=[tex]\frac{x}{|x|}[/tex]

List the separate cases for the question:
x>0, x+h>0
x>0, x+h<0
x<0, x+h>0
x<0, x+h<0
x=0,x+h>0
x=0, x+h<0
x<0, x+h=0
x>0, x+h=0


Apply the distance quotient formula to the cases:

When x>0, x+h>0
[tex]\frac{sgn(x+h)-sgn(x)}{h}[/tex]
=[tex]\frac{1-1}{h}[/tex]
=[tex]\frac{0}{h}[/tex]=0

When x>0, x+h<0
[tex]\frac{sgn(x+h)-sgn(x)}{h}[/tex]
=[tex]\frac{-1-1}{h}[/tex]
=[tex]\frac{-2}{h}[/tex]

When x<0, x+h>0
[tex]\frac{sgn(x+h)-sgn(x)}{h}[/tex]
=[tex]\frac{1-(-1)}{h}[/tex]
=[tex]\frac{2}{h}[/tex]

When x<0, x+h<0
[tex]\frac{sgn(x+h)-sgn(x)}{h}[/tex]
=[tex]\frac{-1-(-1)}{h}[/tex]
=0

When x=0,x+h>0
[tex]\frac{sgn(x+h)-sgn(x)}{h}[/tex]
=[tex]\frac{1}{h}[/tex]

When x=0, x+h<0
[tex]\frac{sgn(x+h)-sgn(x)}{h}[/tex]
=[tex]\frac{-1}{h}[/tex]

When x<0, x+h=0
[tex]\frac{sgn(x+h)-sgn(x)}{h}[/tex]
=[tex]\frac{0-(-1)}{h}[/tex]
=[tex]\frac{1}{h}[/tex]

When x>0, x+h=0
[tex]\frac{0-1}{h}[/tex]
=[tex]\frac{-1}{h}[/tex]