Chapter 4 - Applications of the Derivative - 4.3 Graphing Functions - 4.3 Exercises - Page 268: 36

$$\eqalign{ & {\text{Domain }}\left( { - \infty ,0} \right) \cup \left( {0,\infty } \right) \cr & {\text{No }}y{\text{ - intercepts}} \cr & {\text{No }}x{\text{ - intercepts}} \cr & {\text{No local extrema}} \cr & {\text{No Inflection points}} \cr & {\text{vertical asymptote }}x = 0 \cr & {\text{Horizontal asymptote }}y = 0{\text{ and }}y = - 1 \cr} $$

$$\eqalign{ & g\left( x \right) = \frac{1}{{{e^{ - x}} - 1}} \cr & {\text{Rewrite the function}} \cr & g\left( x \right) = \frac{{{e^x}}}{{{e^x}}}\left( {\frac{1}{{{e^{ - x}} - 1}}} \right) \cr & g\left( x \right) = \frac{{{e^x}}}{{1 - {e^x}}} \cr & {\text{Let }}1 - {e^x} \ne 0 \to x \ne 0 \cr & {\text{Domain }}\left( { - \infty ,0} \right) \cup \left( {0,\infty } \right) \cr & {\text{Find the }}y{\text{ intercept}}{\text{, let }}x = 0,\,{\text{ }}x = 0{\text{ is not in the domain}} \cr & {\text{No }}y{\text{ - intercepts}} \cr & {\text{Find the }}x{\text{ intercept}}{\text{, let }}g\left( x \right) = 0 \cr & \frac{{{e^x}}}{{1 - {e^x}}} = 0 \cr & {e^x} = 0 \cr & {e^x} > 0{\text{ for all real number }}x,{\text{ then}} \cr & {\text{No }}x{\text{ - intercepts}} \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & g'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{{e^x}}}{{1 - {e^x}}}} \right] \cr & g'\left( x \right) = \frac{{\left( {1 - {e^x}} \right){e^x} - {e^x}\left( { - {e^x}} \right)}}{{{{\left( {1 - {e^x}} \right)}^2}}} \cr & g'\left( x \right) = \frac{{{e^x} - {e^{2x}} + {e^{2x}}}}{{{{\left( {1 - {e^x}} \right)}^2}}} \cr & g'\left( x \right) = \frac{{{e^x}}}{{{{\left( {1 - {e^x}} \right)}^2}}} \cr & \cr & {\text{Let }}g'\left( x \right) = 0{\text{ to find critical points}} \cr & \frac{{{e^x}}}{{{{\left( {1 - {e^x}} \right)}^2}}} = 0 \cr & {e^x} = 0 \cr & {e^x} > 0{\text{ for all real number }}x,{\text{ no solution}} \cr & {\text{There are no relative extrema}} \cr & \cr & *{\text{Find the second derivative}} \cr & g''\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{{e^x}}}{{{{\left( {1 - {e^x}} \right)}^2}}}} \right] \cr & g''\left( x \right) = \frac{{{{\left( {1 - {e^x}} \right)}^2}{e^x} - {e^x}\left( 2 \right)\left( {1 - {e^x}} \right)\left( { - {e^x}} \right)}}{{{{\left( {1 - {e^x}} \right)}^4}}} \cr & g''\left( x \right) = \frac{{\left( {1 - {e^x}} \right){e^x} - {e^x}\left( 2 \right)\left( { - {e^x}} \right)}}{{{{\left( {1 - {e^x}} \right)}^3}}} \cr & g''\left( x \right) = \frac{{{e^x} - {e^{2x}} + 2{e^{2x}}}}{{{{\left( {1 - {e^x}} \right)}^3}}} \cr & g''\left( x \right) = \frac{{{e^{2x}} + {e^x}}}{{{{\left( {1 - {e^x}} \right)}^3}}} \cr & \cr & {\text{*Find the Inflection points}}{\text{, set }}g''\left( x \right) = 0 \cr & {e^{2x}} + {e^x} = 0 \cr & {e^x}\left( {{e^x} + 1} \right) = 0 \cr & {e^x} > 0{\text{ for all real number }}x \cr & {e^x} + 1 > 0{\text{ for all real number }}x,{\text{ there are no solutions}}{\text{, then}} \cr & {\text{No inflection points}} \cr & \cr & f\left( x \right) = \frac{{{e^x}}}{{1 - {e^x}}} \cr & {\text{Let the denominator equal to }}0 \cr & 1 - {e^x} = 0 \cr & {e^x} = 1 \cr & x = 0 \cr & {\text{vertical asymptotes }}x = 0 \cr & \mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{e^x}}}{{1 - {e^x}}}} \right) = - 1 \cr & \mathop {\lim }\limits_{x \to - \infty } \left( {\frac{{{e^x}}}{{1 - {e^x}}}} \right) = 0 \cr & {\text{Horizontal asymptote }}y = 0{\text{ and }}y = - 1 \cr & \cr & {\text{Graph}} \cr} $$