#43 An object travels along a line so that its position s is s = t2+ 1 meters after t seconds.

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Question

An object travels along a line so that its position s is s = t2+ 1 meters after t seconds.

(a) What is its average velocity on the interval 2 ≤ t â‰¤ 3?

(b) What is its average velocity on the interval 2 ≤ t â‰¤ 2.003?

(c) What is its average velocity on the interval 2 ≤ t ≤ 2 + h?

 (d) Find its instantaneous velocity at t = 2.

Answer

Step 1 - We are told that we have an object whose position is given by the equation:

, where  is in seconds.

Step 2 - (a) We are first tasked with determining its average velocity for time between 2 and 3.

To find an average velocity, we need to know the time interval which has taken place and the distance traveled in that time interval.

In the present case, the time interval is 1 second.

So our task is to find the distance traveled.

Step 3 - We can plug each of our values for time into the equation to get the corresponding positions of the object:

And

Step 4 - So we can see that the total distance the object has traveled between times  and  is:

Step 5 - So our average velocity is the total distance traversed over the given time interval, or:

Step 6 - (b) Next, we are tasked with finding the average velocity between time  and time.

Step 7 - Again, to find an average velocity, we need to know the time interval which has taken place and the distance traveled in that time interval.

In the present case, the time interval is 0.003 seconds.

So our task is to find the distance traveled.

Step 8 - We found previously that the object’s position at time  is .

To find the object’s position at,

We use our provided equation:

Step 9 - So we can see that the total distance the object has traveled between time  and  is:

meters

Step 10 - So our average velocity is the total distance traversed over the given time interval, or:

Step 11 -(c)Next, we are tasked with finding the average velocity between time  and time .

Step 12 - Again, to find an average velocity, we need to know the time interval which has taken place and the distance traveled in that time interval.

In the present case, the time interval is  seconds.

So our task is to find the distance traveled.

Step 13 - We found previously that the object’s position at time  is 5 meters.

To find the object’s position at, we use our provided equation:

Step 14 - So we can see that the total distance the object has traveled between time  and  is:

meters

Step 15 - So our average velocity is the total distance traversed over the given time interval, or:

Step 16 -(d) Finally, we are tasked with finding the object’s instantaneous velocity when.

Step 17 - We recall the definition of instantaneous velocity as:

, where is the position function

Using this definition and our provided position function, we have:

Step 19 -