Explanation The average velocity does not describe the motion of the particular instant of time during which time interval, the average velocity is calculated. Hence, to know how fast the particle moves at a particular instant of time, we define the instantaneous velocity. Given: The position of a particle at time t is s ( t ) = t 3 + t Formula Used: The average velocity over [ t 0 , t 1 ] is Δ s Δ t = s ( t 1 ) − s ( t 0 ) t 1 − t 0 Calculation: a. The average velocity over the time interval [ 1 , 4 ] is Δ s Δ t = s ( 4 ) − s ( 1 ) 4 − 1 = ( 4 ) 3 + 4 − [ ( 1 ) 3 + 1 ] 3 = 64 + 4 − 2 3 = 66 3 = 22 Thus, the average velocity over time interval [ 1 , 4 ] is 22 unit / sec b. To compute the instantaneous rate of change at t = 1 by computing the average rate of change over intervals to the left and right of t = 1 ▪ Compute Δ s Δ t for Left-hand side at t = 1 • Δ s Δ t for the interval [ 0.5 , 1 ] Δ s Δ t = s ( 1 ) − s ( 0.5 ) 1 − 0.5 = ( 1 ) 3 + 1 − [ ( 0.5 ) 3 + 0.5 ] 0.5 = 2 − 0.625 0.5 = 1.375 0.5 = 2.75 Thus, the average rate of change for the interval [ 0.5 , 1 ] is 2.75 • Δ s Δ t for the interval [ 0.7 , 1 ] Δ s Δ t = s ( 1 ) − s ( 0.7 ) 1 − 0.7 = ( 1 ) 3 + 1 − [ ( 0.7 ) 3 + 0.7 ] 0.3 = 2 − 1.043 0.3 = 0.957 0.3 = 3.19 Thus, the average rate of change for the interval [ 0.7 , 1 ] is 3.19 • Δ s Δ t for the interval [ 0.9 , 1 ] Δ s Δ t = s ( 1 ) − s ( 0.9 ) 1 − 0.9 = ( 1 ) 3 + 1 − [ ( 0.9 ) 3 + 0.9 ] 0.1 = 2 − 1.629 0.1 = 0.371 0.1 = 3.71 Thus, the average rate of change for the interval [ 0

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 2.1, Problem 8E

To determine

a. The average velocity over the time interval [1,4]

b. The instantaneous velocity at t=1

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Chapter 2.1, Problem 7E

Chapter 2.1, Problem 9E

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A particle moving along a straight line has acceleration a(t) = 12t + 10 (measured in m/s2) and initial velocity v(0) = -4 m/s. (a) , Find the velocity at time t. Show your work. (b) Find the distance traveled during the time interval [1,3] . Show your work. (c) Find the average velocity during the time interval [1,3] . Show your work. (d) .. Sketch a possible graph of the position function. Annotate your plot.

Chapter 2 Solutions

Calculus - Standalone book

Ch. 2.1 - Prob. 1PQ Ch. 2.1 - Prob. 2PQ Ch. 2.1 - Prob. 3PQ Ch. 2.1 - Prob. 4PQ Ch. 2.1 - Prob. 5PQ Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E

Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.2 - Prob. 1PQ Ch. 2.2 - Prob. 2PQ Ch. 2.2 - Prob. 3PQ Ch. 2.2 - Prob. 4PQ Ch. 2.2 - Prob. 5PQ Ch. 2.2 - Prob. 6PQ Ch. 2.2 - Prob. 7PQ Ch. 2.2 - Prob. 1E Ch. 2.2 - Prob. 2E Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E Ch. 2.2 - Prob. 11E Ch. 2.2 - Prob. 12E Ch. 2.2 - Prob. 13E Ch. 2.2 - 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Prob. 56E Ch. 2.5 - Prob. 57E Ch. 2.5 - Prob. 58E Ch. 2.5 - Prob. 59E Ch. 2.6 - Prob. 1PQ Ch. 2.6 - Prob. 2PQ Ch. 2.6 - Prob. 3PQ Ch. 2.6 - Prob. 1E Ch. 2.6 - Prob. 2E Ch. 2.6 - Prob. 3E Ch. 2.6 - Prob. 4E Ch. 2.6 - Prob. 5E Ch. 2.6 - Prob. 6E Ch. 2.6 - Prob. 7E Ch. 2.6 - Prob. 8E Ch. 2.6 - Prob. 9E Ch. 2.6 - Prob. 10E Ch. 2.6 - Prob. 11E Ch. 2.6 - Prob. 12E Ch. 2.6 - Prob. 13E Ch. 2.6 - Prob. 14E Ch. 2.6 - Prob. 15E Ch. 2.6 - Prob. 16E Ch. 2.6 - Prob. 17E Ch. 2.6 - Prob. 18E Ch. 2.6 - Prob. 19E Ch. 2.6 - Prob. 20E Ch. 2.6 - Prob. 21E Ch. 2.6 - Prob. 22E Ch. 2.6 - Prob. 23E Ch. 2.6 - Prob. 24E Ch. 2.6 - Prob. 25E Ch. 2.6 - Prob. 26E Ch. 2.6 - Prob. 27E Ch. 2.6 - Prob. 28E Ch. 2.6 - Prob. 29E Ch. 2.6 - Prob. 30E Ch. 2.6 - Prob. 31E Ch. 2.6 - Prob. 32E Ch. 2.6 - Prob. 33E Ch. 2.6 - Prob. 34E Ch. 2.6 - Prob. 35E Ch. 2.6 - Prob. 36E Ch. 2.6 - Prob. 37E Ch. 2.6 - Prob. 38E Ch. 2.6 - Prob. 39E Ch. 2.6 - Prob. 40E Ch. 2.6 - Prob. 41E Ch. 2.6 - Prob. 42E Ch. 2.6 - Prob. 43E Ch. 2.6 - Prob. 44E Ch. 2.6 - 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a. The average velocity over the time interval [ 1 , 4 ] b. The instantaneous velocity at t = 1

Solution Summary: The author calculates the instantaneous velocity over the time interval left[1,4right] by computing the average rate of change.

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a. The average velocity over the time interval [1,4]

b. The instantaneous velocity at t=1

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Chapter 2 Solutions

a. The average velocity over the time interval [ 1 , 4 ] b. The instantaneous velocity at t = 1

Solution Summary: The author calculates the instantaneous velocity over the time interval left[1,4right] by computing the average rate of change.

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a. The average velocity over the time interval [1,4] b. The instantaneous velocity at t=1

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Chapter 2 Solutions

a. The average velocity over the time interval [1,4]

b. The instantaneous velocity at t=1