3. find $f(x)$ by determining $lim_{h o 0}\frac{f(x + h)-f(x)}{h}$, given $f(x)=x^{2}-2x$.

Question

3. find $f(x)$ by determining $lim_{h 	o 0}\frac{f(x + h)-f(x)}{h}$, given $f(x)=x^{2}-2x$.

Accepted Answer

# Explanation:
## Step1: Find $f(x + h)$
Substitute $x+h$ into $f(x)$:
$$f(x + h)=(x + h)^2-2(x + h)=x^{2}+2xh+h^{2}-2x - 2h$$
## Step2: Calculate $f(x + h)-f(x)$
$$
\begin{align*}
f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-2x - 2h)-(x^{2}-2x)\
&=x^{2}+2xh+h^{2}-2x - 2h - x^{2}+2x\
&=2xh+h^{2}-2h
\end{align*}
$$
## Step3: Calculate $\frac{f(x + h)-f(x)}{h}$
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-2h}{h}\
&=\frac{h(2x + h-2)}{h}\
&=2x+h - 2
\end{align*}
$$
## Step4: Find the limit as $h
ightarrow0$
$$
\begin{align*}
\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}&=\lim_{h
ightarrow0}(2x+h - 2)\
&=2x- 2
\end{align*}
$$
# Answer:
$f'(x)=2x - 2$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Find \(f(x + h)\)

Step2: Calculate \(f(x + h)-f(x)\)

Step3: Calculate \(\frac{f(x + h)-f(x)}{h}\)

Step4: Find the limit as \(h

Answer: