Question

use the function below to answer parts (a)-(c).
$f(x)=x^{2}-3$
(a) use the formal definition to find the derivative of $y = f(x)$ at $x = 1$.
(b) find $f(1)$ and find the equation of the tangent line at the point $(1,f(1))$.
(c) graph $y = f(x)$ and the tangent line at the point $(1,f(1))$ in the same coordinate system.
(a) the derivative of a function $f$ at $x$, denoted by $f(x)$, is $f(x)=limlimits_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}$ provided that the limit exists. use the
of the derivative of $f$ at $x$ to find the derivative of the given function $f(x)$ when $x = 1$.
$f(1)=limlimits_{h
ightarrow0}\frac{(1 + h)^{2}-3-(1)^{2}-3}{h}$
evaluate the limit expression to find $f(1)$.
$f(1)=\square$ (type an integer or a fraction.)

Explanation:

Step1: Expand the numerator

$$\begin{align*} &\frac{[(1 + h)^2-3]-[(1)^2 - 3]}{h}\\ =&\frac{(1 + 2h+h^2-3)-(1 - 3)}{h}\\ =&\frac{1 + 2h+h^2-3 - 1+3}{h}\\ =&\frac{2h+h^2}{h} \end{align*}$$

Step2: Simplify the fraction

$$\frac{2h+h^2}{h}=\frac{h(2 + h)}{h}=2 + h$$

Step3: Evaluate the limit

$$f^{\prime}(1)=\lim_{h ightarrow0}(2 + h)$$

As \(h
ightarrow0\), we substitute \(h = 0\) into \(2+h\)

Answer:

\(2\)