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.

83. f(x) = x^2 - x + 4

84. f(x) = 3x^2 - 2x + 6

[Eigenvalue]

83. [tex]\frac{(x+h)²-x-h+4-x²+x-4}{h}[/tex]

=[tex]\frac{x²+2xh+h²-x²+h}{h}[/tex]

=[tex]\frac{2xh+h²+h}{h}[/tex]

=2x+h+1

[Eigenvalue]

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

=[tex]\frac{3x²+6hx+3h²-2x-2h+6-3x²+2x-6}{h}[/tex]

=[tex]\frac{6hx+3h²-2h}{h}[/tex]

=6x+3h-2

[Eigenvalue]

Generalized difference quotient for ax²+bx+c
[tex]\frac{a(x+h)²+b(x+h)+c-(ax²+bx+c)}{h}[/tex]

=[tex]\frac{ax²+2hax+ah²+bx+bh+c+ax²-bx-c}{h}[/tex]

=[tex]\frac{2hax+ah²+bh}{h}[/tex]
=2ax+ah+b

[nycmath]

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

=[tex]\frac{3x²+6hx+3h²-2x-2h+6-3x²+2x-6}{h}[/tex]

=[tex]\frac{6hx+3h²-2h}{h}[/tex]

=6x+3h-2

Correct.

[nycmath]

Generalized difference quotient for ax²+bx+c
[tex]\frac{a(x+h)²+b(x+h)+c-(ax²+bx+c)}{h}[/tex]

=[tex]\frac{ax²+2hax+ah²+bx+bh+c+ax²-bx-c}{h}[/tex]

=[tex]\frac{2hax+ah²+bh}{h}[/tex]
=2ax+ah+b

You outdid yourself.