Chapter 3 - Applications of Differentiation - 3.5 Exercises - Page 203: 59

Graph

$$\eqalign{ & y = \frac{x}{{1 - x}} \cr & {\text{Find the }}y{\text{ intercept, let }}x = 0 \cr & y = \frac{0}{{1 - 0}} \to y = 0 \cr & y{\text{ - intercept }}\left( {0,0} \right) \cr & {\text{Find the }}x{\text{ intercept, let }}y = 0 \cr & 0 = \frac{x}{{1 - x}} \to x = 0 \cr & x{\text{ - intercept }}\left( {0,0} \right) \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & y' = \frac{d}{{dx}}\left[ {\frac{x}{{1 - x}}} \right] \cr & y' = \frac{{\left( {1 - x} \right)\left( 1 \right) - x\left( { - 1} \right)}}{{{{\left( {1 - x} \right)}^2}}} \cr & y' = \frac{{1 - x + x}}{{{{\left( {1 - x} \right)}^2}}} \cr & y' = \frac{1}{{{{\left( {1 - x} \right)}^2}}} \cr & {\text{Let }}y' = 0 \cr & \frac{1}{{{{\left( {1 - x} \right)}^2}}} = 0,{\text{ there are no values at which }}y' = 0. \cr & {\text{No relative extrema}}{\text{.}} \cr & \cr & {\text{*Calculate the asymptotes}} \cr & \frac{x}{{1 - x}} \cr & 1 - x = 0 \to x = 1 \cr & {\text{Vertical asymptote at }}x = 1 \cr & \mathop {\lim }\limits_{x \to \infty } \frac{x}{{1 - x}} = - 1 \cr & \mathop {\lim }\limits_{x \to - \infty } \frac{x}{{1 - x}} = - 1 \cr & {\text{Horizontal asymptote }}y = - 1 \cr & \cr & {\text{*Symmetry}} \cr & f\left( { - x} \right) = \frac{{\left( { - x} \right)}}{{1 - \left( { - x} \right)}} \cr & f\left( { - x} \right) = - \frac{x}{{1 + x}} \cr & {\text{Neither odd nor even, so there is no symmetry}}{\text{.}} \cr & \cr & {\text{Graph}} \cr} $$