Chapter 3 - Applications of Differentiation - 3.6 Exercises - Page 212: 5

Graph

$$\eqalign{ & y = \frac{1}{{x - 2}} - 3 \cr & {\text{Find the }}y{\text{ intercept, let }}x = 0 \cr & y = \frac{1}{{0 - 2}} - 3 \cr & y = - \frac{7}{2} \cr & y{\text{ - intercept }}\left( {0, - \frac{7}{2}} \right) \cr & {\text{Find the }}x{\text{ intercept, let }}y = 0 \cr & 0 = \frac{1}{{x - 2}} - 3 \cr & 3 = \frac{1}{{x - 2}} \cr & x - 2 = \frac{1}{3} \cr & x = \frac{7}{3} \cr & x{\text{ - intercept }}\left( {\frac{7}{3},0} \right) \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & y' = \frac{d}{{dx}}\left[ {\frac{1}{{x - 2}} - 3} \right] \cr & y' = - \frac{1}{{{{\left( {x - 2} \right)}^2}}} - 0 \cr & y' = - \frac{1}{{{{\left( {x - 2} \right)}^2}}} \cr & {\text{Let }}y' = 0 \cr & - \frac{1}{{{{\left( {x - 2} \right)}^2}}} = 0,{\text{ there are no values at which }}y' = 0, \cr & {\text{No relative extrema}}{\text{.}} \cr & \cr & *{\text{Find the Inflection points}} \cr & y'' = \frac{d}{{dx}}\left[ { - \frac{1}{{{{\left( {x - 2} \right)}^2}}}} \right] \cr & y'' = - \left( { - 2} \right){\left( {x - 2} \right)^{ - 3}} \cr & y'' = \frac{2}{{{{\left( {x - 2} \right)}^3}}} \cr & {\text{Let }}y'' = 0 \cr & \frac{2}{{{{\left( {x - 2} \right)}^3}}} = 0 \cr & {\text{There are no values at which }}y'' = 0. \cr & {\text{No inflection points}}{\text{.}} \cr & \cr & {\text{*Calculate the asymptotes}} \cr & \frac{1}{{x - 2}} - 3 \cr & x - 2 = 0 \to x = 2 \cr & {\text{Vertical asymptote at }}x = 2 \cr & \mathop {\lim }\limits_{x \to \infty } \left( {\frac{1}{{x - 2}} - 3} \right) = - 3 \cr & \mathop {\lim }\limits_{x \to - \infty } \left( {\frac{1}{{x - 2}} - 3} \right) = - 3 \cr & {\text{Horizontal asymptote }}y = - 3 \cr & \cr & {\text{Graph}} \cr} $$