Chapter 6 - Applications of Integration - 6.1 Velocity and Net Change - 6.1 Exercises - Page 408: 30

$$s\left( t \right) = 39 + 61t + {e^{ - t}}{\text{ and }}v\left( t \right) = 61 - {e^{ - t}}$$

$$\eqalign{ & a\left( t \right) = {e^{ - t}},\,\,\,\,v\left( 0 \right) = 60,\,\,\,\,s\left( 0 \right) = 40 \cr & \cr & {\text{Use theorem 6}}{\text{.2 }} \cr & {\text{Given the acceleration }}a\left( t \right){\text{ of an object moving along a line and its initial velocity }}v\left( 0 \right){\text{, }} \cr & {\text{the velocity of the object for future time }}t \geqslant 0{\text{ is }}v\left( t \right) = v\left( 0 \right) + \int_0^t {a\left( x \right)} dx \cr & {\text{let }}a\left( x \right) = {e^{ - x}}{\text{ and }}v\left( 0 \right) = 60.{\text{ then }}v\left( t \right){\text{ is}} \cr & v\left( t \right) = 60 + \int_0^t {{e^{ - x}}} dx \cr & {\text{integrate}} \cr & v\left( t \right) = 60 + \left( { - {e^{ - x}}} \right)_0^t \cr & {\text{evaluate the limits}} \cr & v\left( t \right) = 60 - \left( {{e^{ - t}} - {e^{ - 0}}} \right) \cr & {\text{simplify}} \cr & v\left( t \right) = 60 - {e^{ - t}} + 1 \cr & v\left( t \right) = 61 - {e^{ - t}} \cr & \cr & {\text{Use theorem 6}}{\text{.1 }} \cr & {\text{Given the velocity }}v\left( t \right){\text{ of an object moving along a line and its initial position }}s\left( 0 \right){\text{, }} \cr & {\text{the position function of the object for future time }}t \geqslant 0{\text{ is }}s\left( t \right) = s\left( 0 \right) + \int_0^t {v\left( x \right)} dx \cr & {\text{let }}v\left( x \right) = 61 - {e^{ - x}}{\text{ and }}s\left( 0 \right) = 40.{\text{ then }}s\left( t \right){\text{ is}} \cr & s\left( t \right) = 40 + \int_0^t {\left( {61 - {e^{ - x}}} \right)} dx \cr & {\text{integrate}} \cr & s\left( t \right) = 40 + \left( {61x + {e^{ - x}}} \right)_0^t \cr & {\text{evaluate the limits}} \cr & s\left( t \right) = 40 + \left( {61\left( t \right) + {e^{ - t}}} \right) - \left( {61\left( 0 \right) + {e^{ - 0}}} \right) \cr & {\text{simplify}} \cr & s\left( t \right) = 40 + \left( {61t + {e^{ - t}}} \right) - \left( 1 \right) \cr & s\left( t \right) = 39 + 61t + {e^{ - t}} \cr} $$