Determine whether the following statements are true and give a...

Question

Determine whether the following statements are true and give an explanation or counterexample.

c. It is impossible for the instantaneous velocity at all times a≤t≤b to equal the average velocity over the interval a≤t≤b.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine whether the given statement is true or false. A car travels at an average velocity of 35 MPH. On the interval, one is less than or equal to T and T is less than or equal to 3, where T is the time in hours. It is possible for this car to have an instantaneous velocity equal to the average velocity at every point on the same interval. 1 is less than or equal to T and T is less than or equal to 3. Awesome. So it appears for this particular problem, we're ultimately trying to figure out if this statement is true or false. So now that we know that we're ultimately trying to determine whether or not if it's a true or B, false, our first step in order to solve our final answer whether or not the statement is true or false, the first step that we need to take is we need to figure out and focus our attention on the average velocity. And we need to note that the average velocity, and let's denote this as AV, the average velocity over an interval of closed bracket, T1, T2. And these square brackets means that it's a closed interval, will be so essentially the average velocity over this specific interval will be calculated as the total displacement divided by the total time, and we could write this equation as the average, which once again, this is the average velocity, which is going to be equal to S of T. 2 minus S of T1. Divided by T2 minus T1, where SFT will be our position function. So once again, S of T denotes our position function, and that is our first step that we needed to take. Our second step now is we need to focus our attention now on the instantaneous velocity. And as we should recall a note, the instantaneous velocity at time T will be the derivative of the position function SFT with respect to time, and this equation will be denoted as VFT. Will be equal to S T, where this prime symbol denotes the first derivative. So this means that S T is the first derivative of our position function, so it's the first derivative of S of T. So now we need to note and recall that if the instantaneous velocity at all times is, and let's write this in blue, so once again we're focusing on the instantaneous velocity, which I'll denote as IV. So we're essentially saying if the instantaneous velocity at So we're trying to say the instantaneous velocity at all times of T1 is less than or equal to T and T is less than or equal to T2. This will be equal to the average velocity. And if this is equal to the average velocity, then the derivative ST will have to be a constant value, and it will have to be equal to the average velocity throughout the specific interval. As a result, this will mean that the position functions of T will be a linear function such that we'll be able to write that S of T will be equal to M multiplied by T plus C. Where M will be a constant slope which is equal to the average. Awesome. So that will mean, without a doubt, that the given scenario is indeed possible if the car travels at a constant velocity, that's very important, it has to be constant. That if it's not constant, then it won't work. So if the car travels at a constant velocity of 35 MPH throughout the interval of, and once again, this is the interval, 1 is less than or equal to T and T is less than or equal to 3. So once again, the given scenario will be possible if the car travels at a constant velocity of 35 MPH throughout this specific interval of 1 is less than or equal to T and T is less than or equal to 3. So therefore, without a doubt, at the constant velocity of 35 MPH, the instantaneous velocity would also be 35 MPH for every time value T within our interval of 1 is less than or equal to T and T is less than or equal to 3. So that will mean, without a doubt, our final answer has to be multiple choice answer letter A, which once again is true. So our final answer is true. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Determine whether the following statements are true and give an explanation or counterexample.
c. It is impossible for the instantaneous velocity at all times a≤t≤b to equal the average velocity over the interval a≤t≤b.

Key Concepts

Instantaneous Velocity

Average Velocity

Mean Value Theorem

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