The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 3sin(ðt) + 5cos(ðt), where t is measured in seconds. (Round all answers to the nearest hundredth.)

(a) Find the average velocity during the time period [1, 2].
cm/s

(b) Find the average velocity during the time period [1, 1.1].
cm/s

(c) Find the average velocity during the time period [1, 1.01].
cm/s

(d) Find the average velocity during the time period [1, 1.001].
cm/s

(e) Estimate the instantaneous velocity of the particle when t = 1.
cm/s

That should be sin(pi times t) not whatever symbol is up there.. sorry

Nevermind I figured it out

a) to d) are all done the same way

I will do b)

when t=1
s = 3sin(pi) + 5 cos (pi)
= -5
when t=1.1
s = 3sin(1.1pi) + 5cos(1.1pi)
= -5.682333

average velocity = (-5.682333 - (-5))/(1.1-1)
= -6.8233

ensure calculator is set to radians, not degrees

To find the average velocity during a given time period, we need to calculate the displacement of the particle at the beginning and end of that time period.

(a) To find the average velocity during the time period [1, 2], we need to find the displacement at time t=1 and t=2.

Plug in t=1 into the equation of motion:
s = 3sin(π(1)) + 5cos(π(1))
s = 3sin(π) + 5cos(π)
s = 3(0) + 5(-1)
s = -5

Similarly, plug in t=2 into the equation of motion:
s = 3sin(π(2)) + 5cos(π(2))
s = 3sin(2π) + 5cos(2π)
s = 3(0) + 5(1)
s = 5

The average velocity is given by the formula:
Average Velocity = (Displacement) / (Time)

So, the average velocity during the time period [1, 2] is:
Average Velocity = (-5 - 5) / (2 - 1)
Average Velocity = -10 cm/s

Therefore, the average velocity during the time period [1, 2] is -10 cm/s.

(b) To find the average velocity during the time period [1, 1.1], follow the same steps as above, but plug in t=1 and t=1.1 into the equation of motion. Calculate the displacement and find the average velocity using the formula.

(c) Repeat the steps for the time period [1, 1.01].

(d) Repeat the steps for the time period [1, 1.001].

(e) To estimate the instantaneous velocity when t=1, we can calculate the derivative of the displacement equation with respect to time, and then evaluate it at t=1. This will give us the instantaneous velocity.

The derivative of the displacement equation is:
v = d/dt (3sin(πt) + 5cos(πt))
v = 3πcos(πt) - 5πsin(πt)

Now, plug in t=1 into the derivative equation:
v = 3πcos(π(1)) - 5πsin(π(1))
v = 3πcos(π) - 5πsin(π)
v = 3π(-1) - 5π(0)
v = -3π

Therefore, the estimate of the instantaneous velocity when t=1 is -3π cm/s.