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JulsSmile [24]
3 years ago
11

Use the four-step process to find f'(x) and then find f'(1), f'(2), and f'(3).

Mathematics
1 answer:
Soloha48 [4]3 years ago
5 0

Step 1: evaluate f(x+h) and f(x)

We have

f(x+h) = -(x+h)^2+6(x+h)-5 = -(x^2+2xh+h^2)+6x+6h-5

= -x^2-2xh-h^2+6x+6h-5

And, of course,

f(x)=-x^2+6x-5

Step 2: evaluate f(x+h)-f(x)

f(x+h)-f(x)=-x^2-2xh-h^2+6x+6h-5-(-x^2+6x-5)=-2xh-h^2+6h

Step 3: evaluate (f(x+h)-f(x))/h

\dfrac{f(x+h)-f(x)}{h}=-2x-h+6

Step 4: evaluate the limit of step 3 as h->0

f'(x) = \displaystyle \lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h}=-2x+6

So, we have

f'(1) = -2\cdot 1+6 = 4,\quad f'(2) = -2\cdot 2+6 = 2,\quad f'(3) = -2\cdot 3+6 = 0

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