Question

Which of the following functions are continuous for all values in their domain? Justify your answers. a. $a(t)=$ altitude of a skydiver $t$ seconds after jumping from a plane b. $n(t)=$ number of quarters needed to park legally in a metered parking space for $t$ minutes c. $T(t)=$ temperature $t$ minutes after midnight in Chicago on January 1 d. $p(t)=$ number of points scored by a basketball player after $t$ minutes of a basketball game

Short answer

Justify your conclusions. a. The skydiver altitude function, $a(t)$, where $t$ is the time in seconds. b. The number of quarters needed to park legally for $t$ minutes, $n(t)$. c. The temperature in Chicago on January 1 at time $t$ minutes after midnight, $T(t)$. d. The number of points scored by a basketball player during a game at time $t$ minutes, $p(t)$. Answer: a. Yes, the skydiver altitude function, $a(t)$, is continuous for all values in its domain because the altitude changes smoothly over time. b. No, the number of quarters needed function, $n(t)$, is not continuous in its domain because the number of quarters required increases in discrete increments. c. Yes, the temperature function in Chicago on January 1, $T(t)$, is continuous for all values in its domain because the temperature changes smoothly over time. d. No, the basketball points function, $p(t)$, is not continuous in its domain because the number of points scored increases in discrete increments.

Step-by-step solution

Consider the context of the function

The function represents the altitude of a skydiver at any given time after jumping from a plane. Since the skydiver's altitude will change smoothly over time, we can anticipate that this function should be continuous.

Analyze the domain of the function

The domain of the function is the set of possible values of $t$. Since $t$ represents time, the domain of this function is $t\ge 0$. Given the context, we can expect the altitude to change smoothly within this domain.

Conclusion

Since the context and domain of $a(t)$ suggest that the altitude of a skydiver should change continuously over time, we can conclude that this function is continuous for all values in its domain. #b. Number of quarters function#

Consider the context of the function

The function represents the number of quarters needed to park legally in a parking meter for $t$ minutes. The number of quarters required can only increase in discrete increments as time increases, which hints that this function may not be continuous.

Analyze the domain of the function

The domain of the function is the set of possible values of $t$. Since $t$ represents time, the domain of this function is $t\ge 0$. However, the number of quarters will always be a discrete (integer) value, meaning the function cannot be continuous in its domain due to the discrete increments.

Conclusion

Since the context and domain of $n(t)$ indicate that the number of quarters needed cannot change continuously over time, we can conclude that this function is not continuous for all values in its domain. #c. Temperature function#

Consider the context of the function

The function represents the temperature at any given time in minutes after midnight in Chicago on January 1. Temperature naturally changes smoothly over time, so we can expect this function to be continuous.

Analyze the domain of the function

The domain of the function is the set of possible values of $t$. Since $t$ represents time, the domain of this function is $0\le t\le 1440$ (as there are 1440 minutes in a day). We can expect temperature to change smoothly within this domain.

Conclusion

Since the context and domain of $T(t)$ suggest that the temperature should change continuously over time, we can conclude that this function is continuous for all values in its domain. #d. Basketball points function#

Consider the context of the function

The function represents the number of points scored by a basketball player at any given time within a basketball game. Points are scored in discrete increments (1, 2, or 3 points at a time), which hints that this function may not be continuous.

Analyze the domain of the function

The domain of the function is the set of possible values of $t$. Since $t$ represents time, and a basketball game has a specific duration, the domain of this function is limited to the time frame of a basketball game, typically $0\le t\le 40$ or $0\le t\le 48$ minutes for college or professional games, respectively. Within this domain, the number of points scored will not change continuously, as they increase in discrete steps.

Conclusion

Since the context and domain of $p(t)$ indicate that the number of points scored cannot change continuously over time, we can conclude that this function is not continuous for all values in its domain.

Key concepts

Function Domains

Function domain refers to the set of input values for which a function is defined. In simpler terms, it is the range of values that you can plug into a function as the independent variable, often denoted by $t$ or $x$. This is crucial as the domain establishes where a function can operate without creating undefined results.
The domain for a time-based function, like a(t) for the altitude of a skydiver, is usually $t\ge 0$ because time cannot be negative. For temperature over a single day, the domain is typically constrained between 0 and 1440 minutes, since there are 1440 minutes in a day. Meanwhile, a basketball game domain might range from 0 to 48 minutes, covering the full length of the game.
Understanding the domain helps in analyzing whether the function will produce meaningful values under given conditions. If a domain is misunderstood, it might lead to incorrect conclusions about the function's behavior.

Discrete Functions

Discrete functions are ones that trek in distinct, separate steps, rather than forming an unbroken curve. Imagine climbing stairs: your feet step on specific stair treads, not in the spaces between them. This distinct separation makes a discrete function jump from one value to another, creating clear intervals.
Take for example the function $n(t)$, which represents the number of quarters needed at a parking meter. Each quarter represents a discrete increase as time progresses. There's no in-between state where you need a 'fraction' of a quarter, thus this function naturally forms a discrete function. Similarly, $p(t)$, counting basketball points, jumps in 1, 2, or 3-point increments. This discrete characteristic stems from the nature of scoring in basketball, where you can't have parts of points.
Discrete functions play a significant role in various applications where entities increase or decrease in distinct states, often reflecting real-world counting or grouping scenarios.

Continuous Functions

Continuous functions flow smoothly throughout their domain without making abrupt jumps or breaks. Think of a gentle river winding through a valley, seamlessly flowing from one point to the next. In mathematical terms, if you can trace the graph of the function without lifting your pencil, it's continuous.
For example, the altitude function $a(t)$ of a skydiver forms a continuous curve as altitude decreases smoothly over time. Similarly, $T(t)$, representing temperature over time, typically evolves smoothly, bridging each moment into the next without skipping values, making it a continuous function.
Understanding continuity is vital in ensuring mathematical models reliably reflect real-world phenomena. It helps in predicting trends and interpreting behaviors in contexts like physics, engineering, and environmental studies, where constant flow and change are essential.

Mathematical Modeling

Mathematical modeling is a powerful tool that uses mathematical expressions to represent real-world situations. It acts as a bridge between abstract math and tangible scenarios, helping us understand or predict how systems behave over time.
For instance, the altitude model $a(t)$ of a skydiver uses mathematics to depict the continuous change in altitude as the skydiver descends. The model for temperature, $T(t)$, simplifies the understanding of how temperature changes throughout a typical day in Chicago. On the other hand, basketball points function $p(t)$ offers a model of scoring, highlighting changes that occur in distinct steps.
Through mathematical modeling, diverse real-world challenges can be translated into manageable mathematical terms, allowing us to simulate, analyze, and solve complex problems systematically. It is foundational in fields like engineering, economics, and environmental science.