Question

1. (20 points) let $f(x)=x^{2}-7x - 1$ a) use the limit definition of the derivative to find $f(x)$.

Explanation:

Step1: Recall limit - definition of derivative

The limit - definition of the derivative is $f^{\prime}(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}$. Given $f(x)=x^{2}-7x - 1$, first find $f(x + h)$.

Step2: Calculate $f(x + h)$

$f(x + h)=(x + h)^{2}-7(x + h)-1=x^{2}+2xh+h^{2}-7x-7h - 1$.

Step3: Substitute $f(x + h)$ and $f(x)$ into the derivative formula

\[

$$\begin{align*} f^{\prime}(x)&=\lim_{h ightarrow0}\frac{(x^{2}+2xh+h^{2}-7x-7h - 1)-(x^{2}-7x - 1)}{h}\\ &=\lim_{h ightarrow0}\frac{x^{2}+2xh+h^{2}-7x-7h - 1 - x^{2}+7x + 1}{h}\\ &=\lim_{h ightarrow0}\frac{2xh+h^{2}-7h}{h} \end{align*}$$

\]

Step4: Simplify the expression

\[

$$\begin{align*} f^{\prime}(x)&=\lim_{h ightarrow0}\frac{h(2x + h-7)}{h}\\ &=\lim_{h ightarrow0}(2x+h - 7) \end{align*}$$

\]

Step5: Evaluate the limit

As $h
ightarrow0$, we have $f^{\prime}(x)=2x-7$.

Answer:

$f^{\prime}(x)=2x - 7$