Chapter 4 - Applications of the Derivative - 4.3 Graphing Functions - 4.3 Exercises - Page 268: 29

$$\eqalign{ & y{\text{ - intercept }}\left( {0,0} \right) \cr & x{\text{ - intercepts: }}\left( { - \pi ,0} \right),\left( {0,0} \right){\text{ and }}\left( {\pi ,0} \right) \cr & {\text{Local minimum at }}\left( { - \frac{{3\pi }}{4}, - \frac{{\sqrt 2 }}{2}{e^{\frac{{3\pi }}{4}}}} \right) \cr & {\text{Local maximum at }}\left( {\frac{\pi }{4},\frac{{\sqrt 2 }}{2}{e^{ - \frac{\pi }{4}}}} \right) \cr & {\text{Inflection points: }}\left( { - \frac{\pi }{2}, - {e^{ - \frac{\pi }{2}}}} \right){\text{ and }}\left( {\frac{\pi }{2},{e^{ - \frac{\pi }{2}}}} \right) \cr & {\text{No vertical asymptotes}} \cr & {\text{No horizontal asymptotes}} \cr} $$

$$\eqalign{ & g\left( t \right) = {e^{ - t}}\sin t{\text{ on }}\left[ { - \pi ,\pi } \right] \cr & {\text{Find the }}y{\text{ intercept}}{\text{, let }}t = 0 \cr & g\left( 0 \right) = {e^{ - 0}}\sin 0 \cr & g\left( 0 \right) = 0 \cr & y{\text{ - intercept }}\left( {0,0} \right) \cr & \cr & {\text{Find the }}t{\text{ intercepts}}{\text{, let }}g\left( t \right) = 0 \cr & {e^{ - t}}\sin t = 0 \cr & \sin t = 0 \cr & {\text{On the interval }}\left[ { - \pi ,\pi } \right]{\text{ }}\sin t = 0{\text{ at}} \cr & t = - \pi ,{\text{ }}t = 0{\text{ and }}t = \pi \cr & x{\text{ - intercepts: }}\left( { - \pi ,0} \right),\left( {0,0} \right){\text{ and }}\left( {\pi ,0} \right) \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & g'\left( t \right) = \frac{d}{{dt}}\left[ {{e^{ - t}}\sin t} \right] \cr & g'\left( t \right) = {e^{ - t}}\cos t - {e^{ - t}}\sin t \cr & {\text{Let }}g'\left( t \right) = 0{\text{ to find critical points}} \cr & {e^{ - t}}\cos t - {e^{ - t}}\sin t = 0 \cr & {e^{ - t}}\left( {\cos t - \sin t} \right) = 0 \cr & \cos t - \sin t = 0 \cr & {\text{On the interval }}\left[ { - \pi ,\pi } \right]{\text{ }}\cos t = \sin t{\text{ at}} \cr & t = - \frac{{3\pi }}{4}\,{\text{and }}t = \frac{\pi }{4} \cr & \cr & *{\text{Find the second derivative}} \cr & g''\left( t \right) = \frac{d}{{dt}}\left[ {{e^{ - t}}\left( {\cos t - \sin t} \right)} \right] \cr & g''\left( t \right) = - {e^{ - t}}\left( {\sin t + \cos t} \right) - {e^{ - t}}\left( {\cos t - \sin t} \right) \cr & g''\left( t \right) = - {e^{ - t}}\sin t - {e^{ - t}}\cos t - {e^{ - t}}\cos t + {e^{ - t}}\sin t \cr & g''\left( t \right) = - 2{e^{ - t}}\cos t \cr & \cr & {\text{Evaluate }}g''\left( t \right){\text{ at the critical points}} \cr & g''\left( { - \frac{{3\pi }}{4}} \right) = - 2{e^{ - \left( { - 3\pi /4} \right)}}\cos \left( { - \frac{{3\pi }}{4}} \right) > 0,{\text{ then}} \cr & {\text{There is a local minimum at }}\left( { - \frac{{3\pi }}{4},g\left( { - \frac{{3\pi }}{4}} \right)} \right) \cr & g\left( { - \frac{{3\pi }}{4}} \right) = {e^{ - \left( { - \frac{{3\pi }}{4}} \right)}}\sin \left( { - \frac{{3\pi }}{4}} \right) = - \frac{{\sqrt 2 }}{2}{e^{\frac{{3\pi }}{4}}} \cr & \to {\text{local minimum at }}\left( { - \frac{{3\pi }}{4}, - \frac{{\sqrt 2 }}{2}{e^{\frac{{3\pi }}{4}}}} \right) \cr & and \cr & g''\left( {\frac{\pi }{4}} \right) = - 2{e^{ - \pi /4}}\cos \left( {\frac{\pi }{4}} \right) < 0,{\text{ then}} \cr & {\text{There is a local maximum at }}\left( {\frac{\pi }{4},g\left( {\frac{\pi }{4}} \right)} \right) \cr & g\left( {\frac{\pi }{4}} \right) = {e^{ - \left( {\frac{\pi }{4}} \right)}}\sin \left( {\frac{\pi }{4}} \right) = \frac{{\sqrt 2 }}{2}{e^{ - \frac{\pi }{4}}} \cr & \to {\text{local maximum at }}\left( {\frac{\pi }{4},\frac{{\sqrt 2 }}{2}{e^{ - \frac{\pi }{4}}}} \right) \cr & \cr & {\text{*Find the Inflection points}}{\text{, set }}g''\left( t \right) = 0 \cr & - 2{e^{ - t}}\cos t = 0 \cr & \cos t = 0 \cr & {\text{On the interval }}\left[ { - \pi ,\pi } \right]{\text{ }}\cos t = 0{\text{ at}} \cr & t = - \frac{\pi }{2}{\text{ and }}t = \frac{\pi }{2} \cr & g\left( { - \frac{\pi }{2}} \right) = {e^{\frac{\pi }{2}}}\sin \left( { - \frac{\pi }{2}} \right) = - {e^{\frac{\pi }{2}}} \cr & g\left( {\frac{\pi }{2}} \right) = {e^{ - \frac{\pi }{2}}}\sin \left( {\frac{\pi }{2}} \right) = {e^{ - \frac{\pi }{2}}} \cr & {\text{The inflection point are at }}\left( { - \frac{\pi }{2}, - {e^{ - \frac{\pi }{2}}}} \right){\text{ and }}\left( {\frac{\pi }{2},{e^{ - \frac{\pi }{2}}}} \right) \cr & \cr & {\text{*Calculate the asymptotes}} \cr & {\text{No vertical asymptotes}}{\text{, the denominator is 1}}{\text{.}} \cr & {\text{No horizontal asymptotes}} \cr & \cr & {\text{Graph}} \cr} $$