1. We’ll need the product rule.
Part (a) u v
We’ll need the product rule.
They want us to find acceleration, which is the derivative of velocity.
u’ = -1
v’ = t cos (t2/2)
a(t) = v’(t) = (-1)(sin (t2/2)) + (-t-1)(t cos (t2/2))
a(2) = =
-sin cos 2
1.587 or 1.588

2. “Is the speed of the particle increasing at t=2 ? Why or why not ?”
v(2) = -(2+1) sin(4/2)
v(2) = -3 sin 2
Since a(2) was positive, but v(2) is negative, the SPEED of the particle is DECREASING at t=2.

3. The velocity changes direction when v(t) = 0.
Part (b)
The velocity changes direction when v(t) = 0.
-(t+1) sin (t2/2) = 0
t could be -1, since that would make the first parenthesis equal to zero. Unfortunately, -1 is not in the interval 0 < t < 3.

4. The velocity changes direction when v(t) = 0.
-(t+1) sin (t2/2) = 0
t could also be 0, since the sine of 0 is equal to zero. Unfortunately, 0 is outside the interval 0 < t < 3.

5. The velocity changes direction when v(t) = 0.
-(t+1) sin (t2/2) = 0
NOTE:
t = - 2p is not in the interval.
Since sin (t2/2) must be equal to 0, t2/2 has to be equal to p.
This works because sin p = 0.
t2/2 = p
t2 = 2p
t = 2p

6. Using the graphing calculator...
Part (c)
TD = (t+1) sin (t2/2) dt 3
TD = (t+1) sin (t2/2) dt (t+1) sin (t2/2) dt 2p 3
Using the graphing calculator...
1.069
-(-3.265) +
TD = or 4.334

7. Part (d)
The greatest distance from the particle to the origin occurs when s(t) is a MAXIMUM.
s(t) will be a maximum either at the endpoints (t = 0 & t =3), or when its derivative [v(t)] is zero. From Part (c), we know that v(t) = 0 at t = 2p. 2p
-(t+1) sin (t2/2) dt =
From Part (c)... 3
-(t+1) sin (t2/2) dt = 1.069 2p

8. Here’s what the actual movement of the particle would look like on the x-axis:
-(t+1) sin (t2/2) dt =
From Part (c)... 3
-(t+1) sin (t2/2) dt = 1.069 2p

9. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4
t = 0 x = 1 2p
-(t+1) sin (t2/2) dt = 3 2p

10. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4
t = 0 x = 1 2p
-(t+1) sin (t2/2) dt = 3 2p

11. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

12. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

13. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

14. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

15. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

16. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

17. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

18. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

19. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

20. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 2p
-(t+1) sin (t2/2) dt = 3 2p

21. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4
t = 2p x = 2p 3
-(t+1) sin (t2/2) dt =

22. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 3 2p

23. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 3 2p

24. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 3 2p

25. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 3 2p

26. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4 3 2p

27. Here’s what the actual movement of the particle would look like on the x-axis:-4 -2 2 4
t = 3 x = 3 2p

28. Therefore, the greatest distance from the origin would be 2.265 units.
As you could see, the greatest distance from the origin happened right here. -4 -2 2 4
Since the entire distance traveled by the particle was only 4.334, it will not make it all the way back to the origin..
t = 2p x = 3 2p