Respuesta :
Answer:
Option C.
Step-by-step explanation:
The given function is
[tex]y=\dfrac{2x+1}{x^2}[/tex]
We need to find the slope of the function when x=4.
Differentiate the given function w.r.t. x.
[tex]\dfrac{dy}{dx}=\dfrac{x^2\dfrac{d}{dx}(2x+1)-(2x+1)\dfrac{d}{dx}x^2}{(x^2)^2}[/tex] (Using quotient rule)
[tex]\dfrac{dy}{dx}=\dfrac{x^2(2+0)-(2x+1)(2x)}{x^4}[/tex]
[tex]\dfrac{dy}{dx}=\dfrac{2x^2-4x^2-2x}{x^4}[/tex]
[tex]\dfrac{dy}{dx}=\dfrac{-2x^2-2x}{x^4}[/tex]
[tex]\dfrac{dy}{dx}=\dfrac{-2x(x+1)}{x^4}[/tex]
[tex]\dfrac{dy}{dx}=\dfrac{-2(x+1)}{x^3}[/tex]
Now substitute x=4 in the above equation.
[tex]\dfrac{dy}{dx}_{x=4}=\dfrac{-2(4+1)}{4^3}[/tex]
[tex]\dfrac{dy}{dx}_{x=4}=\dfrac{-2(5)}{64}[/tex]
[tex]\dfrac{dy}{dx}_{x=4}=\dfrac{-10}{64}[/tex]
[tex]\dfrac{dy}{dx}_{x=4}=-0.15625[/tex]
[tex]\dfrac{dy}{dx}_{x=4}\approx -0.16[/tex]
Therefore, the correct option is C.
