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

use the function below to answer parts (a)-(c).\n$f(x)=x^{2}-2$\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(b) $f(1)=$ (type an integer or a fraction.)\nion of the tangent line at $(1,f(1))$ is $y = 2x + 1$. (type your answer in slope - intercept form.)\n(c) choose the correct graph below.

use the function below to answer parts (a)-(c).\n$f(x)=x^{2}-2$\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(b) $f(1)=$ (type an integer or a fraction.)\nion of the tangent line at $(1,f(1))$ is $y = 2x + 1$. (type your answer in slope - intercept form.)\n(c) choose the correct graph below.

Answer

Explanation:

Step1: Find ( f(1) )

Substitute ( x = 1 ) into ( f(x)=x^{2}-2 ). ( f(1)=1^{2}-2=-1 )

Step2: Find the derivative using the formal definition

The formal definition of the derivative is ( f^{\prime}(x)=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h} ). For ( f(x)=x^{2}-2 ), ( f(x + h)=(x + h)^{2}-2=x^{2}+2xh+h^{2}-2 ). Then ( \frac{f(x + h)-f(x)}{h}=\frac{(x^{2}+2xh+h^{2}-2)-(x^{2}-2)}{h}=\frac{2xh+h^{2}}{h}=2x + h ). Taking the limit as ( h\rightarrow0 ), ( f^{\prime}(x)=\lim_{h\rightarrow0}(2x + h)=2x ). At ( x = 1 ), ( f^{\prime}(1)=2\times1 = 2 ).

Step3: Find the equation of the tangent line

The point - slope form of a line is ( y - y_{1}=m(x - x_{1}) ), where ( (x_{1},y_{1})=(1,f(1))=(1,-1) ) and ( m = f^{\prime}(1)=2 ). ( y-(-1)=2(x - 1) ) ( y + 1=2x-2 ) ( y=2x-3 )

Answer:

( f(1)=-1 ), equation of the tangent line ( y = 2x-3 )