Question

use the function below to answer parts (a)-(c).
$f(x)=x^{2}+2$
(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 $f(x)=lim _{h
ightarrow 0} \frac{f(x+h)-f(x)}{h}$ 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 $x = 1$.

$$f^{prime}(1)=lim _{h ightarrow 0} \frac{left(1+h)^{2}+2 ight-left(1)^{2}+2 ight}{h}$$

evaluate the limit expression to find $f^{prime}(1)$.
$f^{prime}(1)=2$ (type an integer or a fraction.)
(b) $f(1)=\square$ (type an integer or a fraction.)

Explanation:

Step1: Substitute \(x = 1\) into \(f(x)\)

Given \(f(x)=x^{2}+2\), substitute \(x = 1\) into the function: \(f(1)=(1)^{2}+2\)

Step2: Calculate the value

\(f(1)=1 + 2\)

Answer:

\(f(1)=3\)