Mathematics: Sing the definition of the derivative, find f (x). then find f (1), f (2), and f (3)when the derivative exists. f(x)=-x^2+4x-9

Sing the definition of the derivative, find f - id3319791, alyonaprotopopova

amandajennings01

07.01.2022

By definition of the derivative, and with f(x)=-x^2+4x-9 , we have

$f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h$

In this case,

f(x+h)=-(x+h)^2+4(x+h)-9=-x^2-2xh-h^2+4x+4h-9=f(x)+(4-2x)h-h^2

Then

f(x+h)-f(x)=(4-2x)h-h^2

$\implies\dfrac{f(x+h)-f(x)}h=4-2x-h$

so that as $h\to0$ , we're left with

f'(x)=4-2x

and so

f'(1)=2

f'(2)=0

f'(3)=-2

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oomale

19.05.2020

D 8.5 ≥ 2.5p

Step-by-step explanation:

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