consider the curve defined by ( 9x - 36y + y^{3}=0 ) (see figure)\na. find the coordinates of the y…

consider the curve defined by ( 9x - 36y + y^{3}=0 ) (see figure)\na. find the coordinates of the y intercepts of the curve\nb. find ( \frac{dy}{dx} )\nc. find the slope of the curve at each point where ( x = 0 )\na. the coordinates of the y - intercepts are (type ordered pairs. use a comma to separate answers as needed.)\nb. ( \frac{dy}{dx}=)\nc. there are three points on the curve at which ( x = 0 ). at the topmost point, the slope of the curve is. at the middle point, the slope of the curve is. at the bottommost point, the slope of the curve is

consider the curve defined by ( 9x - 36y + y^{3}=0 ) (see figure)\na. find the coordinates of the y intercepts of the curve\nb. find ( \frac{dy}{dx} )\nc. find the slope of the curve at each point where ( x = 0 )\na. the coordinates of the y - intercepts are (type ordered pairs. use a comma to separate answers as needed.)\nb. ( \frac{dy}{dx}=)\nc. there are three points on the curve at which ( x = 0 ). at the topmost point, the slope of the curve is. at the middle point, the slope of the curve is. at the bottommost point, the slope of the curve is

Answer

Explanation:

Step1: Find y - intercepts

For y - intercepts, set (x = 0) in the equation (9x-36y + y^{3}=0). We get (y^{3}-36y=0), factor out (y): (y(y^{2}-36)=0), then (y(y - 6)(y + 6)=0). So (y=0), (y = 6), (y=-6). The coordinates of y - intercepts are ((0,0)), ((0,6)), ((0,-6)).

Step2: Differentiate implicitly

Differentiate (9x-36y + y^{3}=0) with respect to (x). Using the sum rule and chain rule: (\frac{d}{dx}(9x)-\frac{d}{dx}(36y)+\frac{d}{dx}(y^{3})=\frac{d}{dx}(0)). (9-36\frac{dy}{dx}+3y^{2}\frac{dy}{dx}=0). Group the terms with (\frac{dy}{dx}): ((3y^{2}-36)\frac{dy}{dx}=-9). Then (\frac{dy}{dx}=\frac{-9}{3y^{2}-36}=\frac{3}{12 - y^{2}}).

Step3: Find slopes when (x = 0)

When (x = 0), (y=-6), (y = 0), (y = 6). For (y = 6): (\frac{dy}{dx}=\frac{3}{12-36}=-\frac{1}{8}). For (y = 0): (\frac{dy}{dx}=\frac{3}{12-0}=\frac{1}{4}). For (y=-6): (\frac{dy}{dx}=\frac{3}{12 - 36}=-\frac{1}{8}).

Answer:

a. ((0,0)), ((0,6)), ((0,-6)) b. (\frac{3}{12 - y^{2}}) c. At the top - most point ((y = 6)): (-\frac{1}{8}); at the middle point ((y = 0)): (\frac{1}{4}); at the bottom - most point ((y=-6)): (-\frac{1}{8})