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1 Suppose that at t 4 the position of a particle is s(4)-8 m and its velocity is e(4) - 3 m/s. (a) Use an appropriate linearization L(t) to estimate the position of the particle at t-4.2. (b) Suppose that we know the particle's acceleration satisfies (a(t)| < 10 m/s² for all times. Determine the maximum possible value of the error (4.2) - L(4.2)

1 Suppose that at t 4 the position of a particle is s(4)-8 m and its velocity is e(4) - 3 m/s. (a) Use an appropriate linearization L(t) to estimate the position of the particle at t-4.2. (b) Suppose that we know the particle's acceleration satisfies (a(t)| < 10 m/s² for all times. Determine the maximum possible value of the error (4.2) - L(4.2)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Math 140** **Written Homework 8: 4.1, 4.1 supplement** **Page 1 of 4** **Question 1** Suppose that at \( t = 4 \) the position of a particle is \( a(4) = 8 \) m and its velocity is \( v(4) = 3 \) m/s. - (a) Use an appropriate linearization \( L(t) \) to estimate the position of the particle at \( t = 4.2 \). - (b) Suppose that we know the particle’s acceleration satisfies \( |a(t)| < 10 \, \text{m/s}^2 \) for all times. Determine the maximum possible value of the error \( |a(4.2) - L(4.2)| \). **Question 2** Oil is leaking from an uncapped well and polluting a lake. Ten days after the leak is discovered, environmental engineers measure the amount of oil in the water to be 200 gallons with a current inflow rate of 30 gallons per day. The leak is slowing so that on the tenth day, the inflow rate is decreasing by 5 gallons/day each day. Suppose \( A(t) \) is the amount of oil (in gallons) \( t \) days after the leak is discovered. - (a) Find the quadratic approximation for \( A(t) \) centered at \( t = 10 \). - (b) Use your answer in the previous part to estimate the amount of oil in the lake at \( t = 12 \). **Question 3** The period of a clock pendulum \( T \) is given by the equation \( T = 2\pi \sqrt{\frac{L}{g}} \) where the constant \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity. The period of the clock pendulum varies slightly depending on whether it is located on Earth’s surface, and on small changes in \( g \). - Does this correspond to the clock speeding up or slowing down? Explain your reasoning. - (a) Find the linear approximation for \( T(g) \) centered at \( g = 980 \, \text{cm/s}^2 \), if the length of the pendulum is held constant. - (b) Suppose \(
Transcribed Image Text:**Math 140** **Written Homework 8: 4.1, 4.1 supplement** **Page 1 of 4** **Question 1** Suppose that at \( t = 4 \) the position of a particle is \( a(4) = 8 \) m and its velocity is \( v(4) = 3 \) m/s. - (a) Use an appropriate linearization \( L(t) \) to estimate the position of the particle at \( t = 4.2 \). - (b) Suppose that we know the particle’s acceleration satisfies \( |a(t)| < 10 \, \text{m/s}^2 \) for all times. Determine the maximum possible value of the error \( |a(4.2) - L(4.2)| \). **Question 2** Oil is leaking from an uncapped well and polluting a lake. Ten days after the leak is discovered, environmental engineers measure the amount of oil in the water to be 200 gallons with a current inflow rate of 30 gallons per day. The leak is slowing so that on the tenth day, the inflow rate is decreasing by 5 gallons/day each day. Suppose \( A(t) \) is the amount of oil (in gallons) \( t \) days after the leak is discovered. - (a) Find the quadratic approximation for \( A(t) \) centered at \( t = 10 \). - (b) Use your answer in the previous part to estimate the amount of oil in the lake at \( t = 12 \). **Question 3** The period of a clock pendulum \( T \) is given by the equation \( T = 2\pi \sqrt{\frac{L}{g}} \) where the constant \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity. The period of the clock pendulum varies slightly depending on whether it is located on Earth’s surface, and on small changes in \( g \). - Does this correspond to the clock speeding up or slowing down? Explain your reasoning. - (a) Find the linear approximation for \( T(g) \) centered at \( g = 980 \, \text{cm/s}^2 \), if the length of the pendulum is held constant. - (b) Suppose \(
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