Problem

Use the definition of continuity and the properti…

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Problem 10 Easy Difficulty

Explain why each function is continuous or discontinuous.
(a) The temperature at a specific location as a function of time
(b) The temperature at a specific time as a function of the distance due west from New York City
(c) The altitude above sea level as a function of the distance due west from New York City
(d) The cost of a taxi ride as a function of the distance traveled
(e) The current in the circuit for the lights in a room as a function of time

Answer

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Video Transcript

in this activity. We're being asked to explain why a given situation is represented by a continuous or discontinuous function. And the first situation we're given is the temperature at a specific location as a function of time. Well, yeah, I think about that graph. I've got time in seconds, minutes hours. I've got temperature in degrees Fahrenheit, kelvin Celsius and time is continuous time just happens. We're not going to all of a sudden jump from one time to another and temperature also has to be continuous. So as time passes the temperature may get warmer, it may get cooler, it may get warmer again, but there is never a circumstance where the temperature is going to start and then all of a sudden leap to a completely different temperature that will not happen. So temperature as a function of time is continuous. Yeah. And then we're asked for a situation where at a specific time the temperature is measured as a function of the distance from New york city going to west. Well, if I consider um a map, I've got new york city here and I am looking due west and as I travel further and further west, I am recording the temperature, I've got a similar situation. If I'm going to graph that what my input access is going to be, distance d my output is going to be temperature t I'm not all of a sudden going to jump from one place to another. So distances continuous. And as I travel Along, the temperature at one place is going to gradually change. So the temperature right at new york city may be fairly warm as I crossed the river, it may cool off as I head back into the mainland, it may get warmer again as I get higher in elevation, it may cool off but the temperature is going to change gradually again. There are not going to be any sudden jumps from one temperature to another without passing through all of the temperatures in between. So that is also continuous. Then we're being asked to look at a function, consider a function, the altitude above sea level as a function of distance due west from new york city. So I can think about my same map. If I am starting at new york and I am traveling due west, I don't have a map handy, but if I were to graph that D is my distance, I'm going to use E for elevation above sea level. New york city is about right at sea level. And as I had further inland I'm going to get higher and higher in elevation. I may hit a mountain, I may have to climb over the mountain, I may hit a valley but there's not going to be a point where all of a sudden the elevation changes precipitously, even if I come to a cliff technically the cliff is going to have some mhm, gradual change in elevation. So that would be continuous than part D. We're being asked to consider the cost of a taxi ride as a function of distance traveled. This one may be a little different because taxi fares tend to go up in increments. So if I consider distance traveled in the taxi D. For distance and see for cost when I first get in the taxi I am going to be charged a certain amount for just hailing the taxi. And then as soon as I have passed a certain distance they're going to add an increment. As soon as I passed a certain distance they'll add an increment. As soon as I've gone a certain distance past that they're adding an increment. So this would be a discontinuous function because um the cost goes up for every um 8th of a mile it's traveled rather than just going up consistently. Yeah. And then finally we are being asked to consider situation the current in the circuit for the lights in a room as a function of time. I think that a lot of these are dependent on specifically how you think about addressing them. This one there's probably two different ways you can think about it, my input, it's time, my output is the amount of current that's in the circuits. Current is only working if the lights are on. So when I flip a light switch on, there's no current until I flip the switch on. As soon as I flip the switch on, current is flowing. As soon as I flip the switch off, current stops flowing again. If I switch on that light and then switch on another light, the current will jump even higher, but it's very much an on off situation. The current goes from zero two current that's running in an instant, so that is a discontinuous function.

Michael C.

University of Southern California

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