find [f(a + h) - f(a)] / h for f(x) = -x^2 + x + 5 for h not 0

[Mswen]

Find: f(a+h) - f(a)
_________
h
if h does not equal zero

f(x) = -x^2 + x +5


The very first thing is supposed to be all over h

 

[JeffM]

Find: f(a+h) - f(a)
_________
h
if h does not equal zero

f(x) = -x^2 + x +5


The very first thing is supposed to be all over h

\(\displaystyle \dfrac{f(x + h) - f(x)}{h},\ h \ne 0\) is a template, a recipe, into which you put whatever function is specified.

In fact, that template is a way to define a new kind of function from a given function.

In this case, \(\displaystyle f(x) = -x^2 + x + 5 \implies f(x + h) = - (x + h)^2 + (x + h) + 5.\) Simple, no?

So \(\displaystyle \dfrac{f(x + h) - f(x)}{h} = \dfrac{- (x + h)^2 + (x + h) + 5 - (-x^2 + x + 5)}{h}.\)

The rest is just algebra. Sometimes the algebra is hard; sometimes it is easy.