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.

87. f(x) = 2x/(x + 3)

88. f(x) = 5x/(x - 4)

[Eigenvalue]

[tex]\frac{((2x+2h)/(x+h+3))-((2x)/(x+3))}{h}[/tex]

=[tex]\frac{((2x+2h)(x+3)-(2x)(x+h+3))/((x+3)(x+h+3)}{h}[/tex]

=[tex]\frac{2x²+6x+2hx+6h-(2x²+2hx+6x)}{h(x+3)(x+h+3)}[/tex]

=[tex]\frac{6h}{h(x+3)(x+h+3)}[/tex]

=[tex]\frac{6}{(x+3)(x+h)}[/tex]

[Eigenvalue]

[tex]\frac{((5x+5h)/(x-4))+(5x(x+h-4))}{h}[/tex]

=[tex]\frac{(5x+5h)(x-4)-(5x(x+h-4)}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{5x²-20x+5xh-20h-5x²-5xh-20x}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{-20h}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{-20}{(x-4)(x+h-4)}[/tex]

[nycmath]

[tex]\frac{((2x+2h)/(x+h+3))-((2x)/(x+3))}{h}[/tex]

=[tex]\frac{((2x+2h)(x+3)-(2x)(x+h+3))/((x+3)(x+h+3)}{h}[/tex]

=[tex]\frac{2x²+6x+2hx+6h-(2x²+2hx+6x)}{h(x+3)(x+h+3)}[/tex]

=[tex]\frac{6h}{h(x+3)(x+h+3)}[/tex]

=[tex]\frac{6}{(x+3)(x+h)}[/tex]

Math questions are now getting interesting.

[nycmath]

[tex]\frac{((5x+5h)/(x-4))+(5x(x+h-4))}{h}[/tex]

=[tex]\frac{(5x+5h)(x-4)-(5x(x+h-4)}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{5x²-20x+5xh-20h-5x²-5xh-20x}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{-20h}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{-20}{(x-4)(x+h-4)}[/tex]

The computation for the difference quotient of f can be tedious.

[Eigenvalue]

[tex]\frac{((5x+5h)/(x-4))+(5x(x+h-4))}{h}[/tex]

=[tex]\frac{(5x+5h)(x-4)-(5x(x+h-4)}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{5x²-20x+5xh-20h-5x²-5xh-20x}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{-20h}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{-20}{(x-4)(x+h-4)}[/tex]

The computation for the difference quotient of f can be tedious.

I find it interesting

[nycmath]

[tex]\frac{((5x+5h)/(x-4))+(5x(x+h-4))}{h}[/tex]

=[tex]\frac{(5x+5h)(x-4)-(5x(x+h-4)}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{5x²-20x+5xh-20h-5x²-5xh-20x}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{-20h}{h(x-4)(x+h-4)}[/tex]

=[tex]\frac{-20}{(x-4)(x+h-4)}[/tex]

The computation for the difference quotient of f can be tedious.

I find it interesting

I love it. I love anything math-related.