Chapter 3 - Applications of Differentiation - 3.5 Exercises - Page 203: 74

$$\eqalign{ & {\text{Domain:}}\left( { - \infty , - 2} \right) \cup \left( {2,\infty } \right) \cr & {\text{Intercepts: None}} \cr & {\text{No relative extrema}} \cr & {\text{Symmetry:Origin}} \cr & {\text{Vertical asymptotes at: }}x = \pm 2 \cr & {\text{Horizontal asymptote at }}y = 0 \cr} $$

$$\eqalign{ & y = \frac{x}{{\sqrt {{x^2} - 4} }} \cr & {\text{The domain is given for }}{x^2} - 4 > 0,{\text{ Then}} \cr & \left( {x + 2} \right)\left( {x - 2} \right) > 0 \cr & {\text{Domain: }}\left( { - \infty , - 2} \right) \cup \left( {2,\infty } \right) \cr & \cr & {\text{Find the intercepts}} \cr & *{\text{For }}y = 0 \cr & 0 = \frac{x}{{\sqrt {{x^2} - 4} }},{\text{ No solution in the domain}} \cr & {\text{Then there are no }}x{\text{ intercept}}{\text{.}} \cr & *{\text{ }}x = 0,{\text{It is not in the domain}}{\text{, then there are no }}y{\text{ intercept}}{\text{.}} \cr & {\text{Intercepts: none}} \cr & \cr & {\text{Let }}f\left( x \right) = \frac{x}{{\sqrt {{x^2} - 4} }} \cr & {\text{The domain are all real numbers with }}x \ne 0. \cr & f\left( { - x} \right) = \frac{{\left( { - x} \right)}}{{\sqrt {{{\left( { - x} \right)}^2} - 4} }} \cr & f\left( { - x} \right) = \frac{{ - x}}{{\sqrt {{x^2} - 4} }} \cr & f\left( { - x} \right) = - \frac{x}{{\sqrt {{x^2} - 4} }} \cr & f\left( { - x} \right) = - f\left( x \right),{\text{ The function is odd}}{\text{, then the graph is}} \cr & {\text{symmetric with respect to the origin}}{\text{.}} \cr & \cr & {\text{Find the relative extrema}} \cr & f\left( x \right) = \frac{x}{{\sqrt {{x^2} - 4} }} \cr & f'\left( x \right) = - \frac{4}{{{{\left( {{x^2} - 4} \right)}^{3/2}}}} \cr & f'\left( x \right) = 0 \cr & {\text{There are no values at which }}f'\left( x \right) = 0,{\text{ then}} \cr & {\text{No relative extrema}} \cr & \cr & {\text{The denominator is zero at }}x = - 2,{\text{ }}x = 2,{\text{ then}} \cr & {\text{Vertical asymptotes at: }}x = \pm 2 \cr & \mathop {\lim }\limits_{x \to \infty } \frac{x}{{\sqrt {{x^2} - 4} }} = 0 \cr & {\text{Horizontal asymptote at }}y = 0 \cr & \cr & {\text{Summary:}} \cr & {\text{Domain:}}\left( { - \infty , - 2} \right) \cup \left( {2,\infty } \right) \cr & {\text{Intercepts: None}} \cr & {\text{No relative extrema}} \cr & {\text{Symmetry:Origin}} \cr & {\text{Vertical asymptotes at: }}x = \pm 2 \cr & {\text{Horizontal asymptote at }}y = 0 \cr & \cr & {\text{Graph}} \cr} $$