(1) To estimate the position of the object after 2.5 seconds without using calculus, we can make use of the information we have at times t = 0 seconds and t = 2 seconds.
From the given information, we know that at t = 0 seconds, the object is at position x(t) = 0 feet. At t = 2 seconds, the object is 3 feet to the right of the origin and moving to the left at a speed of 8 feet per second.
Since we want to estimate the position at t = 2.5 seconds, we can assume that the object continues moving with the same speed and direction. Therefore, we can estimate that at t = 2.5 seconds, the object will have moved half of the distance it traveled between t = 0 seconds and t = 2 seconds.
The distance between t = 0 seconds and t = 2 seconds is 3 feet (since the object is 3 feet to the right of the origin at t = 2 seconds). Therefore, the object will have moved 1.5 feet at t = 2.5 seconds.
Since at t = 2 seconds, the object is moving to the left at a speed of 8 feet per second, we can estimate that at t = 2.5 seconds, the object will have moved another 0.5 seconds in the leftward direction. This means the object will have moved 0.5 * (-8) = -4 feet at t = 2.5 seconds.
Therefore, our estimate for the position of the object at t = 2.5 seconds is 1.5 feet to the left of the origin, which is at position x(t) = -1.5 feet.
(2) To solve the problem using calculus and the idea of linearization, we can start by finding the average rate of change of position between t = 0 seconds and t = 2 seconds.
The average rate of change of position is given by the formula:
average rate of change = (change in position) / (change in time)
Between t = 0 seconds and t = 2 seconds, the change in position is 3 feet (since the object is 3 feet to the right of the origin at t = 2 seconds). The change in time is 2 seconds.
Therefore, the average rate of change of position is 3 feet / 2 seconds = 1.5 feet per second.
Next, we can use this average rate of change to estimate the instantaneous rate of change of position at t = 2 seconds.
Since the object is moving to the left at a speed of 8 feet per second at t = 2 seconds, we can estimate the instantaneous rate of change of position as -8 feet per second.
Using the idea of linearization, we can approximate the change in position between t = 2 seconds and t = 2.5 seconds by multiplying the instantaneous rate of change of position (-8 feet per second) by the change in time (0.5 seconds):
change in position = (-8 feet per second) * (0.5 seconds) = -4 feet
Therefore, our estimate for the position of the object at t = 2.5 seconds is obtained by adding this change in position to the position at t = 2 seconds:
position at t = 2 seconds + change in position = 3 feet - 4 feet = -1 feet.
So, our estimate for the position of the object at t = 2.5 seconds is x(t) = -1 feet.