Questions: Curve (y=frac9x^2+4(-2 textdm<x<2 textdm)), along which lines must the supports be directed if they are located at (x=-1, x=0), and (x=1) ?

$Curve (y=frac9x^2+4(-2 textdm<x<2 textdm)), along which lines must the supports be directed if they are located at (x=-1, x=0), and (x=1) ?$

Transcript text: urve $y=\frac{9}{x^{2}+4}(-2 \mathrm{dm}

Solution

Solution Steps

To determine the direction of the supports located at $ x = -1 $, $ x = 0 $, and $ x = 1 $ for the curve $ y = \frac{9}{x^2 + 4} $, we need to find the slope of the tangent lines at these points. The slope of the tangent line at any point on the curve can be found by taking the derivative of the function $ y $ with respect to $ x $.

Step 1: Define the Function and Find its Derivative

Given the curve $ y = \frac{9}{x^2 + 4} $, we need to find the direction of the supports at $ x = -1 $, $ x = 0 $, and $ x = 1 $. To do this, we first find the derivative of the function to determine the slope of the tangent lines at these points.

The derivative of $ y $ with respect to $ x $ is: \[ \frac{dy}{dx} = -\frac{18x}{(x^2 + 4)^2} \]

Step 2: Evaluate the Derivative at $ x = -1 $

Substitute $ x = -1 $ into the derivative: \[ \left. \frac{dy}{dx} \right|_{x = -1} = -\frac{18(-1)}{((-1)^2 + 4)^2} = \frac{18}{25} \]

Step 3: Evaluate the Derivative at $ x = 0 $

Substitute $ x = 0 $ into the derivative: \[ \left. \frac{dy}{dx} \right|_{x = 0} = -\frac{18(0)}{(0^2 + 4)^2} = 0 \]

Step 4: Evaluate the Derivative at $ x = 1 $

Substitute $ x = 1 $ into the derivative: \[ \left. \frac{dy}{dx} \right|_{x = 1} = -\frac{18(1)}{(1^2 + 4)^2} = -\frac{18}{25} \]

Final Answer

\[ \boxed{\left. \frac{dy}{dx} \right|_{x = -1} = \frac{18}{25}, \left. \frac{dy}{dx} \right|_{x = 0} = 0, \left. \frac{dy}{dx} \right|_{x = 1} = -\frac{18}{25}} \]

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