The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 sin πt + 3 cos πt, where t is measured in seconds. (Round your answers to two decimal places.)
(a) Find the average velocity during each time period?
i) 1,2
ii) 1,1.1
iii)1,1.01
iv)1, 1.001

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1. as you know, the average velocity is the integral divided by the time interval. Since the position is the integral of the velocity, the average velocity in [a,b] = (s(b)-s(a))/(b-a)

So, we have

i) (s(2)-s(1))/(2-1) = (2sin2π+3)-(2sinπ+3) = 0
ii) (s(1.1)-s(1))/(1.1-1) = ((2sin1.1π+3)-(2sin.1π+3)))/.1 = -6.18034
and so on

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2. a,b,c,d put all the value in the formula than divide the time :

a) figure out s(2) and s(1) than divide

s(2) = 3sin(pi*2) + 4cos(pi2) = 3(0) +4(1) = 4

s(1) = 3sin(pi1) +4cos(pi1) = 3(0) +4(-1) = -4

average velocity is ( s2-s1)/(2-1) = (4- (-4))/ 1 = 8 cm/s

b) {s(1.1) -s(1)}/(1.1 -1) = -7.3 cm/s

c) { s( 1.01)-s(1)}/(1.01-1) = -9.2 cm/ s

d) {s(1.004)-s(1)}/(1.004-1) = -9.3

e) take derivative then put t =1

3picos(pit)-4pisin(pit)

3picos(pi1)-4pisin(pi1)
3pi(-1) = -3pi = -9.42 cm/s

you can estimate t= 1 when they go small interval 1.0005 and so on they will approach to -9.4m/s bylooking

note part b,c,d are the same part a, i just give you an answer but you have to work out like part a to show your works. you just punching in your calculator to figure out your answer by yourself

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3. s = 4 sin(πt) + 3 cos(πt)

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