An object at rest begins moving
horizontally on the x-axis, starting at position x(t) = 0 feet at time t = 0 seconds.Suppose, after exactly 2 seconds we measure the position of the
object to be 3 feet to the right of the origin AND measure the object to be moving to the leftward at a speed of 8 feet per second.
Caution: We do NOT know anything about the position or motion of the object, other than what we know at times t = 0
seconds and t = 2 seconds. So do NOT simply assume the object is moving at constant speed the whole time.
Goal of the Problem: Estimate the position of the object after 2.5 seconds:
(1) by writing a solution as if you were carefully and clearly
explaining it (without calculus) to someone who has NOT had
calculus AND
(2) by writing a solution (with calculus) using the ideas of
linearization.
(1) Without using calculus, we can estimate the position of the object after 2.5 seconds by considering its motion between 0 seconds and 2 seconds.
From the given information, we know that after exactly 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. This means that the object has traveled a distance of 3 feet in the first 2 seconds and is currently moving towards the left.
To estimate its position after 2.5 seconds, we can assume that the object continues to move at the same speed and direction as it did during the first 2 seconds. Therefore, in the next 0.5 seconds, the object would travel a distance of 0.5 * 8 = 4 feet towards the left.
Since the object was initially at position x(t) = 0 feet, it will move 4 feet to the left from there. Therefore, our estimate for the position of the object after 2.5 seconds is x(t) = 0 - 4 = -4 feet.
So, the estimated position of the object after 2.5 seconds is -4 feet.
(2) Using the ideas of linearization, we can estimate the position of the object after 2.5 seconds by approximating its motion with a straight line between 0 seconds and 2 seconds.
First, let's find the slope of the line. We know that the object moves 3 feet to the right in 2 seconds, which gives us a slope of (change in position)/(change in time) = (3 - 0)/(2 - 0) = 3/2 feet per second.
Now, using this slope, we can calculate the position of the object after 2.5 seconds. Since the time has changed from 0 seconds to 2.5 seconds, the change in time is 2.5 - 0 = 2.5 seconds.
Using the formula for a straight line (y = mx + b), where y represents the position, m represents the slope, x represents the time, and b represents the initial position, we can substitute the known values to find the estimated position after 2.5 seconds:
x(t) = (3/2) * 2.5 + 0 = 3.75 feet.
So, the estimated position of the object after 2.5 seconds, using linearization, is 3.75 feet