Question

f*(a)=lim as h->0 ([f(a+h)]^2-[f(a)]^2)/h We were asked to express it in terms of f'(a) =lim h->0 (f(a+h)-f(a))/h

Answer 1

This is the alternate definition of a derivative. I'll take a look at it and get back to you if I figure it out. If not, you can look in your textbook under the derivative section and look for the term alternate definition of a derivative for examples. They all solve in similar ways.

Answer 2

Thanks for the reply

Answer 3

Ok so the method to solve this is expand f(a+h)^2 and f(a)^2. The a^2 terms will go to zero. (a^2-a^2 = 0) leaving you with a higher degree H term in the numerator. Cancel all h's in numerator and denominator. Any terms that still have an H term go to 0. This leaves you with the derivative.

For your problem specifically:
= lim h-> 0 ((a^2 + 2ah + h^2) - a^2)/ (h)
= lim h-> 0 (2ah+h^2)/h
= lim h-> 0 (2a + h)/1
= (2a + 0)/1
= 2a
Note that I stopped writing the lim h->0 after I plugged it in, and I plugged it in after I removed h from the denominator.

Answer 4

So to express this in terms of f'(a) would be 2f(a) f'(a)

Answer 5

I'm not sure about that notation i'm sorry.
You asked for it to be expressed in terms of f'(a) and that is equal to 2a.
Perhaps the question was asking you to solve the question with the definition of a derivative, rather than in terms of the equation for the definition of a derivative.
And I apologize, I misinformed you. This is the regular definition of a derivative. Not the alternative definition. You'll probably cover that next. lim x->a (f(x)-f(a))/(x-a)