Chapter 3 - Applications of Differentiation - 3.5 Exercises - Page 203: 62

Graph

$$\eqalign{ & y = \frac{{2x}}{{9 - {x^2}}} \cr & {\text{Find the }}y{\text{ intercept, let }}x = 0 \cr & y = \frac{{2\left( 0 \right)}}{{9 - {{\left( 0 \right)}^2}}} \to y = 0 \cr & y{\text{ - intercept }}\left( {0,0} \right) \cr & {\text{Find the }}x{\text{ intercept, let }}y = 0 \cr & 0 = \frac{{2x}}{{9 - {x^2}}} \cr & x = 0 \cr & x{\text{ - intercept }}\left( {0,0} \right) \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & y' = \frac{d}{{dx}}\left[ {\frac{{2x}}{{9 - {x^2}}}} \right] \cr & y' = \frac{{\left( {9 - {x^2}} \right)\left( 2 \right) - 2x\left( { - 2x} \right)}}{{{{\left( {9 - {x^2}} \right)}^2}}} \cr & y' = \frac{{18 - 2{x^2} + 4{x^2}}}{{{{\left( {9 - {x^2}} \right)}^2}}} \cr & y' = \frac{{4{x^2} + 18}}{{{{\left( {9 - {x^2}} \right)}^2}}} \cr & {\text{Let }}y' = 0 \cr & 4{x^2} + 18 = 0:{\text{ there are no values at which }}y' = 0, \cr & {\text{No relative extrema}}{\text{.}} \cr & \cr & {\text{*Calculate the asymptotes}} \cr & \frac{{2x}}{{9 - {x^2}}} \cr & 9 - {x^2} = 0 \to x = \pm 3 \cr & {\text{Vertical asymptotes at }}x = \pm 3 \cr & \mathop {\lim }\limits_{x \to \infty } \frac{{2x}}{{9 - {x^2}}} = 0 \cr & \mathop {\lim }\limits_{x \to - \infty } \frac{{2x}}{{9 - {x^2}}} = 0 \cr & {\text{Horizontal asymptote }}y = 0 \cr & \cr & {\text{*Symmetry}} \cr & f\left( { - x} \right) = \frac{{ - x + 1}}{{{{\left( { - x} \right)}^2} - 4}} \cr & f\left( { - x} \right) = \frac{{ - x + 1}}{{{x^2} - 4}} \cr & {\text{There is no symmetry}}{\text{.}} \cr & \cr & {\text{Graph}} \cr} $$