Chapter 3 - Applications of Differentiation - 3.5 Exercises - Page 203: 66

$$\eqalign{ & {\text{Intercepts: none}} \cr & {\text{Symmetry: }}x{\text{ - axis}} \cr & {\text{There are no relative extrema}} \cr & {\text{Vertical asymptote at }}x = 0 \cr & {\text{Horizontal asymptote }}y = 0 \cr} $$

$$\eqalign{ & {x^2}y = 9 \cr & y = \frac{9}{{{x^2}}} \cr & {\text{Find the intercepts}} \cr & *{\text{For }}y = 0 \cr & 0 = \frac{9}{{{x^2}}},{\text{ Then there is no }}x{\text{ intercept}}{\text{.}} \cr & *{\text{For }}x = 0 \cr & y = \frac{9}{{{0^2}}},{\text{ Then there is no }}y{\text{ intercept}}{\text{.}} \cr & {\text{Intercepts: none}} \cr & \cr & {\text{Let }}f\left( x \right) = \frac{9}{{{x^2}}} \cr & {\text{The domain is all real numbers with }}x \ne 0. \cr & f\left( { - x} \right) = \frac{9}{{{{\left( { - x} \right)}^2}}} \cr & f\left( { - x} \right) = \frac{9}{{{x^2}}} \cr & f\left( { - x} \right) = f\left( x \right),{\text{ The function is even, so the graph is}} \cr & {\text{symmetric with respect to the }}y{\text{ axis}}{\text{.}} \cr & \cr & {\text{Find the relative extrema}} \cr & f'\left( x \right) = - \frac{{18}}{{{x^3}}} \cr & f'\left( x \right) = 0 \cr & - \frac{{18}}{{{x^3}}} = 0,{\text{ No solution, so there are no relative extrema}}{\text{.}} \cr & \cr & {\text{*Calculate the asymptotes}} \cr & f\left( x \right) = \frac{9}{{{x^2}}} \cr & y = 0 \cr & {\text{Undefined at }}x = 0 \cr & {\text{Vertical asymptote at }}x = 0 \cr & \mathop {\lim }\limits_{x \to \infty } \frac{9}{{{x^2}}} = 0 \cr & {\text{Horizontal asymptote }}y = 0 \cr & \cr & {\text{Graph}} \cr} $$