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P15 - the bumper returns to its original position(s becomes...

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Project 15 MAC 2311 YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!! 1. Find the slopes of the tangent lines to f ( x ) = tan( x ) , g ( x ) = sin( x ) , and h ( x ) = x at x = 0 and x = π 4 . Carefully graph the three functions below on the interval ( - π 2 , π 2 ). What angle, in radians, do the graphs of f ( x ) and g ( x ) make with the x -axis at ( 0 , 0 ) ? Prove to yourself that sin( x ) x tan( x ) on [ 0 , π 2 ) by considering the slopes and derivatives. 2. A car hits a pothole in the road, which causes the chassis to bounce up and down. Using differential equations, an engineer determines that the position of the bumper at time t seconds after hitting the pothole is s ( t ) = 36 sin( t ) e t centimeters from its original position s = 0. Thus, for example, s = 2 means the bumper has moved up 2 cm, while s = - 2 means the bumper has moved down 2 cm. What is s (0) ? Using squeeze theorem, show that as time passes (to infinity),
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Unformatted text preview: the bumper returns to its original position (s becomes zero again). What is the velocity of the bumper at the instant it begins to move (in-clude units with your answer)? and after 3 seconds (round to the nearest hundredth)? Is it moving upward or downward at these times? At what times does the bumper change the direction of its motion (from upward to downward or vice versa—think of the bumper as a particle moving in a straight line)? Find the position of the bumper the first four times that it changes direction (round your answers to the nearest hundredth)? In your opinion, are the shock absorbers on this vehicle working reasonably well (to keep the size of the oscillations small and to slow them quickly)?...
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