1.2 Functions: Modeling Relationships

Introduction

Functions

Comparing models and abstract functions

Use the equation $T = 40 + 0.25N$ that relates the temperature, $T$, to the number of chirps per minute, $N$, to respond to the questions below. The equation is also called “Dolbear's Law”.

If we hear snowy tree crickets chirping at a rate of $92$ chirps per minute, what does Dolbear's Law suggest should be the outside temperature?

What values of $h$ make sense to consider in the context of this function? What values of $V$ make sense in the same context?

What is the domain of the function $f$ in the context of the spherical tank? Why? What is the corresponding codomain? Why?

Determine and interpret (with appropriate units) the values $f(2)$, $f(4)$, and $f(8)$. What is important about the value of $f(8)$?

Consider the claim: “since $f(9) = \frac{\pi}{3} 9^2(12-9) = 81\pi \approx 254.47$, when the water is $9$ meters deep, there is about $254.47$ cubic meters of water in the tank”. Is this claim valid? Why or why not? Further, does it make sense to observe that “$f(13) = -\frac{169\pi}{3}$”? Why or why not?

Can you determine a value of $h$ for which $f(h) = 300$ cubic meters? Why or why not?

Calculus shows that for a tennis ball tossed vertically from a window $48$ feet above the ground at an initial vertical velocity of $32$ feet per second, the ball's height above the ground at time $t$ (where $t = 0$ is the instant the ball is tossed) can be modeled by the function $h = g(t) = -16t^2 + 32t + 48$. Discuss the differences between the model $g$ and the abstract function $f$ determined by $y = f(x) = -16x^2 + 32x + 48$.

What is the height of the water when $t = 0$? When $t = 1$? When $t = 2$? How long will it take the tank to completely drain? Why?

What is the domain of the model $h = q(t)$? What is the domain of the model $V = p(t)$?

How much water is in the tank when the tank is full? What is the range of the model $h = q(t)$? What is the range of the model $V = p(t)$?

We will frequently use a graphing utility to help us understand function behavior, and strongly recommend https://www.desmos.com/ because it is intuitive, online, and free. To learn more about Desmos, see their http://learn.desmos.com/.

In https://www.desmos.com/calculator/btlyyxsdr4, you can see how we enter the (abstract) function $V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t)$, as well as the corresponding graph the program generates. Make as many observations as you can about the model $V = p(t)$. You should discuss its shape and overall behavior, its domain, its range, and more.

How does the model $V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t)$ differ from the abstract function $y = r(x) = \frac{256\pi}{3} - \frac{\pi}{24} x^2(24-x)$? In particular, how do the domain and range of the model differ from those of the abstract function, if at all?

How should the graph of the height function $h = q(t)$ appear? Can you determine a formula for $q$? Explain your thinking.