Problem

Explain why each function is continuous or discon…

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Problem 9 Easy Difficulty

The toll $ T $ charged for driving on a certain stretch of a toll road is 5 dollars except during rush hours (between 7 AM and 10 AM and between 4 PM and 7 PM) when the toll is 7 dollars.
(a) Sketch a graph of $ T $ as a function of the time $ t $, measured in hours past midnight.
(b) Discuss the discontinuities of this function and their significance to someone who uses the road.

Answer

(a) The toll is $\$ 7$ between 7: 00 AM and 10: 00 AM and between 4: 00 PM and 7: 00 PM.
(b) The function $T$ has jump discontinuities at $t=7,10,16,$ and $19 .$ Their
significance to someone who uses the road is that, because of the sudden jumps in
the toll, they may want to avoid the higher rates between $t=7$ and $t=10$ and
between $t=16$ and $t=19$ if feasible.

Discussion

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Samira H.

June 23, 2021

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = x + 3x4 5 , a = ?1 lim x??1 f(x) = lim x??1 5 = lim x??1 5 by the power law =

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Video Transcript

In this problem there is a pro vote which costs $5 Except during peak hours where it costs eight. The peak hours are between 7- 10 a.m. And 4- seven p.m. And this is where it costs $8 And the rest of the time it cost five. What we wanted to determine is the graph that shows the toll. This is the price depending on time and that's hours past midnight. And how this graph will be important for people using that road. So we're gonna go ahead and start with. The normal cost Midnight is not peak. So it starts at five and it will be $5 up until seven a.m. Which is very yeah At that point it rises to eight instantaneously Until 10 which is about there. We'll drop back down from 10 to four p.m. sorry four p.m. which is 16 hours after midnight. This is still at $5 and then it'll go back up to eight between four and seven. That's right there and then until midnight it will be back and $5. So that's a graph. And what's most interesting is these discontinuities, it jumps up and down instantaneously multiple times a day Between $5.08 and two driver. This could actually be rather important as if they if they use the skirt often. So if they arriving at the word, let's say just as peak hours are going to end, let's say they're arriving at 9 59. I am. Then they could save $3 just by waiting and waiting an extra minute Team for if you grab just before four Which is before seven p.m. I bet. And also if they're arriving basically just Around 6:59 right before picking up his third, they have to get to the toll road within the next minute or else the price will get up. So that's all I hope this was helpful

Keyan S.

Topics

Limits

Derivatives

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 5

Continuity