Difference Quotient

[nycmath]

Precalculus
Michael Sullivan
Edition 10
Chapter 2, Section 2.1

Find the difference quotient of f; that is find
[f(x + h) - f(x)]/h, h cannot be 0, for each function. Be sure to simplify.

89.f(x) = sqrt{x - 2}

90.f(x) = sqrt{x + 1}

Hint for 89 amd 90: Rationalize the numerator.

Note: TODAY IS OUR DIFFERENCE QUOTIENT DAY. HAPPY SUNDAY!

[Eigenvalue]

[tex]\frac{sqrt(x+h-2)-sqrt(x-2)}{h}[/tex]=[tex]\frac{(sqrt(x+h-2)-sqrt(x-2))((sqrt(x+h-2)+sqrt(x-2))}{h(sqrt(x+h-2)+sqrt(x-2)}[/tex]
=[tex]\frac{h}{h(sqrt(x+h-2)-sqrt(x-2}[/tex]
=[tex]\frac{1}{sqrt(x+h-2)+ sqrt(x-2)}[/tex]

[Eigenvalue]

[tex]\frac{sqrt(x+h-1)-sqrt(x-1)}{h}[/tex]
=[tex]\frac{(sqrt(x+h-1)-sqrt(x-1))(sqrt(x+h-1)+sqrt(x-1))}{h(sqrt(x+h-1)+sqrt(x-1))}[/tex]
=[tex]\frac{1}{sqrt(x+h-1)+sqrt(x-1)}[/tex]