Suppose the rental cost for a snowboard is $25 for the first d...

Question

Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).

e. For what values of t is f continuous? Explain.

Accepted Answer

Welcome back, everyone. A parking garage charges $10 for the first hour or any fraction of the first hour, and an additional $5 for each subsequent owner or a fraction of an hour. Let F of T be the total charge for parking and T be the number of hours. For what values of T is FFT continuous? Explain. Now, in order for us to test or to figure out the values of T for which F of T is continuous, the answer to it lies in this part of the problem where it tells us how the parking garage does its charging. OK. Now it says it charges $10 for the first order or any fraction of the first order and $5 for each subsequent order or a fraction of an hour. What does that mean? What does that look like? Well, let's try to define the behavior of the function FFT, OK? Now, let's say that we were looking for the total charge when T goes from 0 to 1, OK? Now, here, since it tells us it charges $10 for the first order or in a fraction of the first order, that means that the total charge for parking if of T is going to be equal to $10. Now, it doesn't matter here if T is 1, if T is 0.5. Once T is between 0 and 1, then the total charge is going to be $10. Likewise, when T is between 1 and 2, then it doesn't matter, OK, what fraction of the order we are at. We know it will be $10 for, for the first order plus $5 for an additional subsequent order or a fraction of that over, OK? So if we were to continue with T, let's say 2 T was between 2 and 3, then again, we know it would just be $10 for the first hour, $5 for the second hour, and $5 for any fraction of the hour between the second and the third hour. OK? I this kind of making a little bit of sense so far? So if we were to draw a graph, is let's just quickly sketch a graph of T versus F of T. Then if we go with T going from 0, 12, 3. And here we have FFTS 10. 15 and 20. Then we know that from 0 to 1. OK. We'd have FFT, OK, being at 10, OK. And we also know that at this point, then here when as soon as it switches from 1 to go to 2, there would be a jump. OK. Then we have our jump here and then it stops. And then as soon as it switches from 2 to 3, then we also know that there's gonna be a jump. OK, no matter what fraction of the over there is. So now, what is this telling us? No, we can tell that the total charge for parking increases by $5 at each integer value of T, creating a jump in the function. Now, a function is continuous at a point if there is no sudden change or jump in the value of the function at that point. But in this case, since it jumps when the parking increases by $5 at each integer value, that means that the FFT is discontinuous at integer values of T, OK? So let's, let's write that down. FFT is discontinuous. Add Integer Values of T. So if it's discontinuous at integer values of T, what else does that tell us? Well, this also tells us that at non-integer values of T, the function does not have jumps because the cost remains constant until the next full hour is reached. Therefore, the function F of T is continuous everywhere except that the integer values of T, since the total charge for parking jumps by $5 at each integer value of T. Thanks a lot for watching everyone. I hope this video helped.

Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).
e. For what values of t is f continuous? Explain.

Key Concepts

Continuity of Functions

Piecewise Functions

Limits

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