What do these terms mean? v(a)+R(h)

Kade Rosales

Kade Rosales

Open question

2022-08-13

What do these terms mean? v ( a ) + R ( h )
In regards to the differentiability of a function at a.
I know that is it differentiable if the limit of f ( a + h ) f ( a ) h exists.
Then I am given the following equation, supposing f is differentiable:
f ( a + h ) f ( a ) = h v ( a ) + R ( h ) with l i m h 0 R ( h ) | | h | |
What is v(a) and R(h). Where did they come from?

Answer & Explanation

Brooks Hogan

Brooks Hogan

Beginner2022-08-14Added 18 answers

Supposing f is differentiable, then you know that the limit
lim h 0 f ( a + h ) f ( a ) h
exists. Then you can
v ( a ) = lim h 0 f ( a + h ) f ( a ) h
which is nothing but the derivative of f at a and
R ( h ) = f ( a + h ) f ( a ) v ( a ) h
which is called remainder.
With these definitions in mind, you can see that
lim h 0 R ( h ) h = 0

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