Application: Modeling Traffic In Johnson, Vermont Pt. 2
I've copied your work from the previous activity, and also hidden the function builder. As you can see, the model is a sine function and goes back and forth between low traffic in the middle of the night, and higher traffic in the afternoon.
Let's use what we've learned in the previous activities, and see what the model predicts are the busiest and quietest minutes of traffic throughout the day. To get started, type the following code snippets to calculate the first and second derivatives:
derivative(g) and derivative(g,2). It's ok that you can barely see the derivatives. They are very small and are located near the x-axis. Also, just like earlier, don't be concerned if your derivatives have funky fractions in them. This is due to the code Geogebra uses to calculate derivatives.
As we learned in the previous activities, in order to identify the busiest and quietest minutes of the day on Route 15, first we must find the critical points of the model, g. To find the critical points, we need to find where the derivative, g', is 0. We'll do this just like before with the snippet intersect(g',y=0,0,1000). The reason I set the right bound to 1000 is that we can see that the maximum and minimum traffic occurs in this range just by looking at the graph. This should create two points (116.854,0) and (849.15998,0). The x-coordinates of these points are the critical points of the model g. Now we need to find out if the critical points are maximums or minimums. We'll use the Second Derivative Test for that.
Now, because we can see the graph of g, we already know that 116.854 will be the time of the least traffic, and 849.15998 will be the time of the most traffic, but let's use the Second Derivative to check this anyway.
Type g''(116.854), and note that the result is a (small) positive number. The Second Derivative Test says when the second derivative is positive, the critical point is a minimum. Indeed we can see this is true: the 116.854th minute of the day is indeed the quietest time of the day on Route 15. Plot it by typing (116.854,g(116.854)).
Now wrap up our analysis by also calculating g''(849.15998) and observe that the result is a (small) negative number, confirming that 849.15998 is the time of the most traffic according to the model. Plot it by typing (849.15998,g(849.15998)).
Nice analysis! Let's move on to one more application before we get to some new content.