use the function below to answer parts (a)-(c).\n$f(x)=x^{2}-3$\n(a) use the formal definition to find the…

use the function below to answer parts (a)-(c).\n$f(x)=x^{2}-3$\n(a) use the formal definition to find the derivative of $y = f(x)$ at $x = 1$.\n(b) find $f(1)$ and find the equation of the tangent line at the point $(1,f(1))$.\n(c) graph $y = f(x)$ and the tangent line at the point $(1,f(1))$ in the same coordinate system.\n(a) the derivative of a function $f$ at $x$, denoted by $f(x)$, is $f(x)=limlimits_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}$ provided that the limit exists. use the\nof the derivative of $f$ at $x$ to find the derivative of the given function $f(x)$ when $x = 1$.\n$f(1)=limlimits_{h\rightarrow0}\frac{(1 + h)^{2}-3-(1)^{2}-3}{h}$\nevaluate the limit expression to find $f(1)$.\n$f(1)=\\square$ (type an integer or a fraction.)

use the function below to answer parts (a)-(c).\n$f(x)=x^{2}-3$\n(a) use the formal definition to find the derivative of $y = f(x)$ at $x = 1$.\n(b) find $f(1)$ and find the equation of the tangent line at the point $(1,f(1))$.\n(c) graph $y = f(x)$ and the tangent line at the point $(1,f(1))$ in the same coordinate system.\n(a) the derivative of a function $f$ at $x$, denoted by $f(x)$, is $f(x)=limlimits_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}$ provided that the limit exists. use the\nof the derivative of $f$ at $x$ to find the derivative of the given function $f(x)$ when $x = 1$.\n$f(1)=limlimits_{h\rightarrow0}\frac{(1 + h)^{2}-3-(1)^{2}-3}{h}$\nevaluate the limit expression to find $f(1)$.\n$f(1)=\\square$ (type an integer or a fraction.)

Answer

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

$$f^{\prime}(1)=\lim_{h\rightarrow0}(2 + h)$$ As (h\rightarrow0), we substitute (h = 0) into (2+h)

Answer:

(2)