1. (5 points) An object moves in a straight line and its position is given by a

Question

1. (5 points) An object moves in a straight line and its position is given by a function s(t) where t is the time in seconds and s(t) is the number of meters from its starting point. Let v(t) denote the velocity of the object given in meters per second. Assume that s(t) and v(t) are differentiable functions.

We are told that s(2) = 7, s(4) = 13, and s(9) = 38. Use the Intermediate Value Theorem and the Mean Value Theorem to carefully explain why the object is moving exactly 4 meters per second at some moment between t = 2 and t = 9.

2. (5 points) Two runners run in a straight line and their positions are given by functions g(t) and h(t), where t is the time in seconds, and g(t) is the number of meters from the starting point for the first runner and h(t) is the number of meters from the starting point for the second runner. Assume that g(t) and h(t) are differentiable functions.

Suppose that the runners begin a race at the same moment and end the race in a tie. Carefully explain why at some moment during the race they have the same velocity. (Suggestion: Consider the function f(t) = g(t) h(t) and use Rolle’s Theorem.)

Explanation / Answer

s(2) = 7, s(4) = 13, and s(9) = 38

Using (2,7) and (4,13),
the average velocity over that time interval is
(13 - 7) / (4 -2)
= 6 /2
=3 m/s

Using (4,13) and (9,38) :
avg vel = (38- 13) / (9 - 4)
= 25 / 5
= 5 m/s

Now, the average speed from t = 2 to 4 is 3 m/s
And average speed from t = 4 to t = 9 is 5 m/s

The mean value theorem says that if a function is continuous, and at end points has the value a and b, then every value between a and b is expressed somewhere in the range.

So, since the function for velocity is continuous,
and endvalues are 3 and 5, there has to be a particular timeframe
over which the velocity is ALL values between 3 and 5 including endvalues 3 and 5

So, proved that at some point,
the velocity of the object is exactly 4 meters per second

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.