1. Rate of Change and Accumulation Review

2. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine the direction the particle is moving?

3. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine the direction the particle is moving?
Evaluate the velocity at that time. If v(t) > 0, the particle is moving right (or up). If v(t) < 0, the particle is moving left (or down).

4. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
In which direction (left or right) is the particle moving at time t = 4 minutes?

5. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
In which direction (left or right) is the particle moving at time t = 4 minutes?
𝑣 4 =−0.677 inches/minute
Since v(4) < 0, the particle is moving to the left.

6. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine if the velocity is increasing or decreasing?

7. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine if the velocity is increasing or decreasing?
Evaluate the acceleration (the derivative of the velocity) at that time. If a(t) > 0, the velocity is increasing. If a(t) < 0, the velocity is decreasing.

8. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Is the velocity of the particle increasing or decreasing at time t = 4 minutes?

9. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Is the velocity of the particle increasing or decreasing at time t = 4 minutes?
𝑎 4 = 𝑣 ′ 4 =1.633 inches/minute/minute
Since a(4) > 0, the velocity of the particle is increasing.

10. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Evaluate 𝑣 ′ and explain what it represents in the context of the problem. Be sure to use appropriate units.

11. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Evaluate 𝑣 ′ and explain what it represents in the context of the problem. Be sure to use appropriate units.
𝑣 ′ 2.5 =𝑎 2.5 =−2.871 inches/minute/minute
The particle’s velocity is decreasing at a rate of inches per minute per minute. (The rate at which the rate is decreasing.)

12. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine if the particle changes direction?

13. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine if the particle changes direction?
If the velocity changes sign (from positive to negative or from negative to positive), the particle has changed direction.

14. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How many times does the particle change direction between t = 0 minutes and t = 6 minutes?

15. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How many times does the particle change direction between t = 0 minutes and t = 6 minutes?
The particle changes direction 2 times.

16. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Find the time(s) at which the particle changes direction between t = 0 minutes and t = 6 minutes.

17. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Find the time(s) at which the particle changes direction between t = 0 minutes and t = 6 minutes.
The particle changes direction at t = minutes (because the velocity changes from positive to negative) and at
t = minutes (because the velocity changes from negative to positive).

18. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Find the acceleration of the particle at the time(s) found in the previous problem.

19. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Find the acceleration of the particle at the time(s) found in the previous problem.
𝑣 ′ =𝑎 =−2.515 inches/minute/minute
𝑣 ′ =𝑎 =2.515 inches/minute/minute

20. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine the change in position of the particle from t = 0 minute to t = 4 minutes?

21. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine the change in position of the particle from t = 0 minute to t = 4 minutes?
Evaluate the definite integral of v(t) from t = 0 to t = 𝑣 𝑡 𝑑𝑡
This finds the difference between the distance from the particle’s starting position and the particle’s ending position.

22. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine the total distance traveled by the particle from t = 0 minutes to t = 4 minutes?

23. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
How can you determine the total distance traveled by the particle from t = 0 minutes to t = 4 minutes?
Evaluate the definite integral of the absolute value of v(t) from t = 0 to t = 𝑣 𝑡 𝑑𝑡
This takes into account the times when the particle was moving backwards (or downwards) and includes it as part of the distance traveled.

24. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Evaluate 0 4 𝑣 𝑡 𝑑𝑡 and explain what it represents in the context of the problem. Be sure to use appropriate units.

25. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Evaluate 0 4 𝑣 𝑡 𝑑𝑡 and explain what it represents in the context of the problem. Be sure to use appropriate units.
0 4 𝑣 𝑡 𝑑𝑡 =9.984 inches
The particle moved a distance of inches from 0 minutes to 4 minutes.

26. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Given that x(t) is the position of the particle at time t and that x(0) = -2, find y(4).

27. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Given that x(t) is the position of the particle at time t and that x(0) = -2, find x(4).
𝑥 4 =𝑥 𝑣 𝑡 𝑑𝑡 =− =7.984 inches
The particle traveled inches from it’s starting position of -2 and ended at a position of 7.984

28. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Find the total distance traveled by the particle from t = 0 to
t = 4.

29. A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4
A particle moves along the x-axis with velocity given by 𝑣 𝑡 =3+4.1 cos 0.9𝑡 measured in inches per minute.
Find the total distance traveled by the particle from t = 0 to
t = 4.
0 4 𝑣 𝑡 𝑑𝑡= inches
Even though the particle moved inches from it’s starting position, it traveled a total distance of inches.