I'm very confused on how I'm supposed to substitute. Can someone fill me in on the process?
b. As a consequence, f(x+h) - f(x) = x^2 + 2hx + h^2 + x + h - (x^2 + x) = 2hx + h^2 + h
I don't get where this part comes from ------------------------********
So if there are 2 x's in the equation then I just sub the f(x + h) in for the first x and - f(x) in for the second x?
I'm unsure as to how -(x^2 +x) fits into it, where does that come from? How do you put - f(x) into the equation?
"f" is just a machine that says: "Take this thing, and add its square to it."
"- f(x)" means (in this context) "... and subtract from that what you get from adding the square of x to x."
I understand it a little better thanks, so in this example.
f(x) = 2x^2 + 3 and find f(x + h) - f(x)
I would first sub in (x + h) yielding f(x) = 2(x^2 + 2hx + h^2) + 3
then distribute the 2 giving me f(x) = 2x^2 + 4hx + 2h^2 + 3
what would I do next then? I don't see how I could simplify it further and now I'm supposed to subtract that from itself?
where is defined as .
What you've done is:
which is saying which is wrong.
Let's call it or something:
What you're trying to do is work out what is by unpacking the two instances of : one of those instances is , and the other (which you have to subtract from the first) is just .
So you work out what is, you've done that - it's which works out as .
Then you work out what is. You don't need to "work that out" because it's already in exactly the form you want it to be in:
So you substitute the expressions you've just worked out for and into your expression for :
Is this any clearer?