Pre-Calc question?

Question

f(a+h) - f(a)
-------------- <- Formula given in the book
h

Simplify x^2 + 1 (using the above)

Also, what does the F(a+h).... equation mean?

Step by step procedure please :-)

Could someone explain what f(a), f(h) ect mean as well?

Answer 1

f(a) is the value of the function f when the input is a. (a,f(a)) is a point on the graph. h is a little bitty value, so a+h is just a tiny bit bigger than a. f(a+h) is the function value at that point, so that (a+h, f(a+h)) is another point on the graph. picture a line through those points (called a secant line). f(a+h) - f(a) is the change in y, the rise, and h is the change in x, the run, and so the whole fraction is the rise over run, the slope of the line. when we let limits shrink h to 0, the line becomes tangent to the graph at (a, f(a)). for f(x) = x² + 1, that turns out to be 2x.

f(x+h) = (x+h)² + 1 = x² + 2xh + h² + 1
f(x) = x² + 1
so f(x+h) - f(x) = 2xh + h²
and [f(x+h) - f(x)]/h = 2x + h

as h→0, that becomes 2x.

Answer 2

I'm not at home right now to look up my old Pre-Calc homework, but I know that the above equation you wrote is
direct preparation for taking the derivative.

the "h" is a value you add to any x value.
IN other words, you pick any x-value on the x axis. and plug that into the function. then you plug another "new" x-value into the same equation (function) but the "new" x-value will be the "old" plus "h" thus you use a and a+ h subtract and divide by the "h".
This should remind you of the old equation for slope.
M = y2 - y1 / x2 - x1

The derivative by the way is considered a type of slope, instananeous slope.

So what they're trying to show you is that
you pick some x value plug it into the function
you pick another x value that is 'h' more than the first.
then plug that other value into your function,
subract them and divide by 'h'

hope this helps a little, I'm not at home to pull out my old Pre-Calc homework, I keep saying this becasue you just asked for help on one of the exact same questions that I had in pre-calc about two years ago!!