Question

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 1

a) Continuous

b) Continuous

c) Continuous

d) Discontinuous

e) Continuous

Step-by-step explanation:

In examining whether a function is continuous or discontinuous, we consider if there are any interruptions in the value of the function as the input changes. Here's an analysis for each given scenario:

1. The temperature at a specific location as a function of time tends to be continuous because temperature normally changes gradually over time without abrupt jumps.
2. The temperature at a specific time as a function of the distance due west from New York City could be continuous, assuming that there are no abrupt changes in the geographical factors that influence temperature.
3. The altitude above sea level as a function of the distance due west from New York City is typically continuous, generally changing smoothly as one moves across the landscape.
4. The cost of a taxi ride as a function of the distance traveled is often a piecewise function, with portions being continuous within certain distance intervals, but possibly discontinuous at specific points where the rate changes (e.g., base fare to metered fare).
5. The current in the circuit for the lights in a room as a function of time is generally continuous when the lights are on, but if the light switch is flipped, this creates a discontinuity at the moment the lights turn on or off.

A continuous probability function has been defined such that probability equals area under the curve of the function over an interval. For real-world phenomena like temperature and altitude, these functions tend to be continuous as they reflect gradual changes over time or distance.

Answer 2

(a) The temperature at a specific location as a function of time.

This is a continuous function as the temperature cannot increase in an instant like time.

(b) The temperature at a specific time as a function of the distance due west from New York City.

This is a continuous function as the temperature in one location is affected by its neighboring places.

(c) The altitude above sea level as a function of the distance due west from New York City.

The altitude above sea level can be discontinuous at a cliff, or continuous at very deep hole.

(d) The cost of a taxi ride as a function of the distance traveled.

This is a discontinuous function as the cost still raises if you make a stop.

(e) The current in the circuit for the lights in a room as a function of time.

This is a discontinuous function as the function takes the value of 0 when the switch is off and 1 when the switch is on.

The electron traveling speed makes this discontinuous.