In the previous section, I combined many techniques from the whole course together in order to sketch curves. In this section, I will also be combining many areas of the course. The goal here is not visualization but interpretation: if there is a function which is a model of some real world phenomenon, I can use all the tools of calculus to understand what that model is saying. All of the information in curve sketching applies here, but I can also ask a number of questions that relate to the model.
In addition to the mathematical domain, are there domain restrictions due to the model interpretation. (Very commonly, for example, the model measures only positive input values, so I restrict the domain to positive numbers.)
Questions of intercepts are often questions of the starting value (y-intercept) or places where the model reaches zero values (x-intercepts).
Questions about vertical asymptotes tell me where the model reaches unreasonable values; perhaps this is where the model breaks down.
Questions about limits at \(\infty\) and horizontal asymptotes give me information about the long term behaviour of the model.
The derivative gives me the growth rate of the model.
I can ask for an interpretation of the constants in the model.
I can ask for a narrative: globally, qualitatively, what is the model saying? How can I holistically describe the behaviour?
Finally, I can critique the model. Are there mathematical reasons (such as vertical asymptotes) where I expect the model may no longer fit the real world phenomenon?
It’s best to see this through examples. Unlike curve sketching, where I went through the same list deliberatively one item at a time, here I’ll use which ever tools seem germaine to the model at hand.
Subsection11.2.2Examples
Example11.2.1.
Conisder a model of radioactivity on a contaminated site, where radioactivity is measured as \(r\) in grays and \(t\) is time in years. This is the model.
The domain of the model is \(t \geq 0\text{,}\) showing that observations start at year 0. The starting value at year 0 is \(r(0) = 3\) grays.
The function is piecewise, but I can check that it is continuous at its cross-over point.
There are no vertical asymptotes. The limit at \(\infty\) is 0, so \(y=0\) is a horizontal asymptote. That means the long term behaviour is a decay to negligable radiation.
The derivative is positive on the first section \((0
\leq t \leq 100)\) and negative on the second section \((t > 100)\text{.}\) Therefore, I expect a maximum at \(t=100\text{.}\) The radioactivity of the site is increasing for 100 years and decreasing at all times afterward.
If safe level of radioactivity are under \(5\) grays, I can ask when the site is safe. To do so, I simply solve \(r(t) = 5\text{.}\) In approximate values, the site become unsafe at \(t = 63.2\) years and becomes safe again after \(t = 167.8\) years.
As a narrative, qualitative summary, it looks like there is contamination that slowly increases the radioactivity over the first 100 years. At 100 years, something suddenly changes: either the contamination source is removed or some kind of cleaning process exists that removes contamination faster than it is added. From that point on, contamination decays, eventually dropping to near-zero levels.
Figure11.2.2.A Model of Radioactivity
Example11.2.3.
Conisder a population model, where \(p\) is population in thousands and \(t\) is time in years.
This is exponential decay which starts at \(100\) when \(t=0\text{.}\)
When I include the sine term, the coefficient infront of the exponential function varies between \(90\) and \(110\text{.}\) This will effect the trajectory of the exponential decay. However, since \(10\) is smaller than \(100\text{,}\) the effect is minimal. I would expect an exponential decay curve with small sinusoidal osscilation along its trajectory.
The starting value is \(p=100\text{.}\) The longer term behaviour, in the limit, is \(p=0\text{.}\)
The population is globally decreasing and decaying. However, due to the sinusoidal behaviour, there may be local small time frames where the population is briefly increasing. Since \(t\) is in years, perhaps this sinusoidal term measure seasonal variation in growth.
Figure11.2.4.A Population Model
Example11.2.5.
Consider a model of temperature in a chemical reaction where \(T\) is in degree celcius and \(t\) is in minutes. This model is given by a differential equation. This is a relatively common situation: often in observations of the world, I can observe a differential equation instead of directly observing the function.
For this differential equation, I can just integrate to solve. That gives the function \(T = -3t^2 + 6t + c\text{.}\) To specify the function completely, I would need an initial temperature. Let’s say that initial temperature is 12 degrees: \(T = -3t^2 + 6t + 12\)
This is a downward quadratic. If I complete the square, I get the vertex form: \(T= -3(t^2 - 2t + 1) +
15 = -3 (t-1)^2 + 15\text{.}\)
Therefore, there is a maximum at \(t=1\) minutes. That maximum is \(T(1) = 15\) degrees. The function rises in temperature for a minute, then starts to drop in a parabolic shape.
The starting temperature, as given, is \(T(0) = 12\) degrees.
The long term behaviour is \(T \rightarrow -\infty\text{.}\) However, this is obviously unreasonable. First, mostly likely there is an ambient temperature which the system will eventually reduce to. Second, even if this is happening in the vacuum of space, temperature is still bounded below at \(-273\) degrees. I conclude that this model is only meant to operate for relatively small \(t\text{.}\)
Figure11.2.6.A Temperature Model
The description of models is, in many ways, the culmulation of this course. For the majority of students in this calculus class, you will only use the techniques of calculus sparingly throughout your degree and in any future work. However, when you use these techniques, you are likely to be presented with functions describing some phenomena. I hope you now have some skills to look at those function and analyze their expected behaviour.
