Chapter 3 - Derivatives - 3.6 Derivatives as Rates of Change - 3.6 Exercises - Page 181: 16

$$\eqalign{ & \left( a \right){\text{Graphs}} \cr & \left( b \right){\text{Stationary at }}t = 2{\text{ and }}t = 5 \cr & {\text{Is moving to the righ on: }}\left[ {0,2} \right) \cr & {\text{Is moving to the left on: }}\left[ {5,6} \right) \cr & \left( c \right){\text{ }}v\left( 1 \right) = 24,{\text{ }}a\left( 1 \right) = - 30 \cr & \left( d \right){\text{ }}a\left( 5 \right) = 18 \cr & \left( e \right){\text{ Increasing on: }}\left( {1,2} \right){\text{ and }}\left( {3,4} \right] \cr} $$

$$\eqalign{ & f\left( t \right) = - 6{t^3} + 36{t^2} - 54t;{\text{ 0}} \leqslant t \leqslant 4 \cr & \cr & \left( a \right){\text{ Graph below}} \cr & \cr & \left( b \right){\text{ }} \cr & {\text{Position }}s = - 6{t^3} + 36{t^2} - 54t \cr & v = \frac{{ds}}{{dt}} \cr & v = \frac{d}{{dt}}\left[ { - 6{t^3} + 36{t^2} - 54t} \right] \cr & v = - 18{t^2} + 72t - 54 \cr & {\text{The object is stationary when }}v = 0 \cr & - 18{t^2} + 72t - 54 = 0 \cr & {t^2} - 4t + 3 = 0 \cr & \left( {t - 3} \right)\left( {t - 1} \right) = 0 \cr & t = 1,{\text{ }}t = 3 \cr & {\text{Stationary at }}t = 1{\text{ and }}t = 3 \cr & - 18{t^2} + 72t - 54 > 0 \cr & \left( {t - 3} \right)\left( {t - 1} \right) > 0 \cr & {\text{Solving}} \cr & 1 < t < 3 \cr & {\text{Is moving to the righ on: }}\left( {1,3} \right){\text{ and is movingto the left}} \cr & {\text{into the interval }}\left[ {0,1} \right){\text{ and }}\left( {3,4} \right] \cr & \cr & \left( c \right) \cr & v = - 18{t^2} + 72t - 54 \cr & a = \frac{{dv}}{{dt}} = - 36t + 72 \cr & {\text{at }}t = 1 \cr & v\left( 1 \right) = - 18{\left( 1 \right)^2} + 72\left( 1 \right) - 54 \cr & v\left( 1 \right) = 0 \cr & a\left( 1 \right) = - 36\left( 1 \right) + 72 \cr & a\left( 1 \right) = 36 \cr & \cr & \left( d \right){\text{ The velocity is 0 at }}t = 1,{\text{ }}t = 3 \cr & a\left( 1 \right) = 36 \cr & a\left( 3 \right) = - 36 \cr & \cr & \left( e \right){\text{ The speed is}} \cr & {\text{speed}} = \left| v \right| = \left| { - 18{t^2} + 72t - 54} \right| \cr & {\text{From the graph:}} \cr & {\text{Increasing on: }}\left( {1,2} \right){\text{ and }}\left( {3,4} \right] \cr} $$