Question

If The Position Of The Object Moving Horizontally (Meters) At Time (Seconds) Is Given By The Function S(T)=41t4−3t3+12t2−16t+1, Time T≥0

Find the following. Show the work that leads to your answer. [10]
a. the velocity function v(t). The velocity function found in part a can be factored as follows: v(t)=(t−1)(t−4) 2
b. Find all time(s) when the object is at rest.
c. Find all time(s) when the object changes direction.
d. Find all time-interval(s) when the object is moving left, moving right.
e. Find the acceleration function, a(t).

Answer 1

a. The velocity function is v(t) = 164t^3 - 9t^2 + 24t - 16. b. The object is at rest when t = 1 or t = 4. c. The object changes direction at t = 1 and t = 4. d. The object is moving left when t < 1, moving right when 1 < t < 4, and moving left again when t > 4. e. The acceleration function is a(t) = 492t^2 - 18t + 24.

Step-by-step explanation:

a. To find the velocity function, we need to take the derivative of the position function with respect to time. Taking the derivative of S(t) = 41t^4 - 3t^3 + 12t^2 - 16t + 1, we get v(t) = 164t^3 - 9t^2 + 24t - 16.

b. To find when the object is at rest, we set the velocity function v(t) equal to zero and solve for t. By factoring v(t) = (t - 1)(t - 4)^2, we can see that the object is at rest when t = 1 or t = 4.

c. To find when the object changes direction, we look for where the velocity function changes sign. From the factored form of v(t), we can see that the object changes direction at t = 1 and t = 4.

d. To find when the object is moving left or right, we look at the sign of the velocity function. When t < 1, the object is moving left. When 1 < t < 4, the object is moving right. When t > 4, the object is moving left again.

e. To find the acceleration function, we take the derivative of the velocity function with respect to time. Taking the derivative of v(t) = 164t^3 - 9t^2 + 24t - 16, we get a(t) = 492t^2 - 18t + 24.