The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 4 sin ¦Ðt + 3 cos ¦Ðt, where t is measured in seconds.

(a)Find the average velocity during each time period.
(i) [1, 2]
(ii) [1, 1.1]
(iii) [1, 1.01]
(iv) [1, 1.001]
(b) Estimate the instantaneous velocity of the particle when t = 1.

On both my Mac and my PC your equation

s = 4 sin ¦Ðt + 3 cos ¦Ðt

came out with strange symbols that I could not read.
Are the π or √ or something else ?

To find the average velocity during a time period, you need to calculate the displacement and divide it by the time elapsed.

(a)
(i) Average velocity during the time period [1, 2]:
To find the displacement, substitute t = 2 and t = 1 into the equation of motion:
s(2) = 4sin(2¦Ð) + 3cos(2¦Ð)
s(1) = 4sin(¦Ð) + 3cos(¦Ð)
Displacement = s(2) - s(1)
Average Velocity = Displacement / Time elapsed = (s(2) - s(1)) / (2 - 1)
Now, calculate the values and find the average velocity.

(ii) Average velocity during the time period [1, 1.1]:
Repeat the same process as in (i), but substitute t = 1.1 and t = 1 into the equation of motion to find the displacement. Then divide the displacement by the time elapsed (0.1) to find the average velocity.

(iii) Average velocity during the time period [1, 1.01]:
Repeat the same process as in (i), but substitute t = 1.01 and t = 1 into the equation of motion to find the displacement. Then divide the displacement by the time elapsed (0.01) to find the average velocity.

(iv) Average velocity during the time period [1, 1.001]:
Repeat the same process as in (i), but substitute t = 1.001 and t = 1 into the equation of motion to find the displacement. Then divide the displacement by the time elapsed (0.001) to find the average velocity.

(b) To estimate the instantaneous velocity of the particle when t = 1, you can calculate the derivative of the equation of motion with respect to time. This will give you the velocity function. Then, substitute t = 1 into the velocity function to get the instantaneous velocity.