(a) Explanation Given Information: A worn shock absorber on a car undergoes damped harmonic motion so that the height of the car frame after t seconds is given by the equation p ( t ) = 4.2 e − 0.5 t sin ( 2 π t + π 2 ) centimeters above the rest position. Formula Used: The product rule is used which can be represented as follows: d d x ( x ⋅ y ) = y ⋅ d d x ( x ) + x ⋅ d d x y The chain rule is used which is as follows: d d x ( sin u ) = d d x ( sin u ) ⋅ d u d x The following derivative formula is used: d d x ( sin x ) = cos x Calculation: Consider the function, p ( t ) = 4.2 e − 0.5 t sin ( 2 π t + π 2 ) Determine the derivative of the function. p ′ ( t ) = d d t 4.2 e − 0.5 t sin ( 2 π t + π 2 ) Take out the constant term at the left side of derivative. p ′ ( t ) = d d t 4.2 e − 0.5 t sin ( 2 π t + π 2 ) p ′ ( t ) = 4.2 d d t e − 0.5 t sin ( 2 π t + π 2 ) Apply the product rule. p ′ ( t ) = 4.2 d d t e − 0.5 t sin ( 2 π t + π 2 ) p ′ ( t ) = 4.2 [ sin ( 2 π t + π 2 ) ⋅ d d t e − 0.5 t + e − 0.5 t ⋅ d d t sin ( 2 π t + π 2 ) ] p ′ ( t ) = 4.2 [ − 0.5 e − 0.5 t ⋅ sin ( 2 π t + π 2 ) + e − 0.5 t cos ( 2 π t + π 2 ) ⋅ 2 π ] Take out e − 0.1 t as a common term. p ′ ( t ) = 4.2 e − 0.5 t [ − 0.5 sin ( 2 π t + π 2 ) + 2 π ⋅ cos ( 2 π t + π 2 ) ] Substitute t = 0 in the derivative function (b) To determine To graph: The functions p and p ′ for t for 0 ≤ t ≤ 10 and also for 0 ≤ t ≤ 2 and then estimate to the nearest hundredth of a second the time at which the velocity of the car is greatest in upward direction using graphs and graphing technology.

7th Edition

ISBN: 9781337291248

Author: Waner, Stefan.

Publisher: Cengage Learning,

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Question

Chapter 9.2, Problem 70E

(a)

To determine

To calculate: The speed of the top of the moving car and the direction of its motion at times t=0 and t=0.25 if a worn shock absorber on a car experiences damped harmonic motion at t=0 such that the height of the car frame after t seconds is given by the equation p(t)=4.2e−0.5tsin(2πt+π2) centimeters above the rest position.

(b)

To determine

To graph: The functions p and p′ for t for 0≤t≤10 and also for 0≤t≤2 and then estimate to the nearest hundredth of a second the time at which the velocity of the car is greatest in upward direction using graphs and graphing technology.

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Chapter 9 Solutions

Applied Calculus

Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E

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Solution Summary: The author calculates the speed of the top of a moving car and the direction of its motion at times t=0 and 0.25.

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To graph: The functions p and p′ for t for 0≤t≤10 and also for 0≤t≤2 and then estimate to the nearest hundredth of a second the time at which the velocity of the car is greatest in upward direction using graphs and graphing technology.

Chapter 9 Solutions