The displacement (in centimeters) of a particle moving back andforth along a str

Question

The displacement (in centimeters) of a particle moving back andforth along a straight line is given by the equation of motions = 3sin(?t) + 2cos(?t), where t ismeasured in seconds. (Round all answers to the nearesthundredth.)
(a) Find the average velocity during the time period [1, 2].
(b) Find the average velocity during the time period [1, 1.1].
(c) Find the average velocity during the time period [1, 1.01].
(d) Find the average velocity during the time period [1,1.001].
(e) Estimate the instantaneous velocity of the particle whent = 1.

Explanation / Answer

v_avg = (d final - d initial)/change in time

so just plug in the bounds of the period for each problem

a) v_avg = [d(2)-d(1)]/(2-1) = 4 cm/s

b) v_avg = [d(1.1)-d(1)]/(1.1-1) = -8.29 cm/s

c) v_avg = [d/(1.01)-d(1)]/(1.01-1) = -9.23 cm/s

d) v_avg = [d/(1.001)-d(1)]/(1.001-1) = -9.41 cm/s

e) so if you notice, this number is approaching -3pi cm/s as you get closer and closer to 1, which ends up being the derivative of the function at t = 1

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