From early in the course, we have associated the derivative of a function with velocity: indeed, if a given function represents the position of a moving object along an axis, then its derivative measures precisely the instantaneous velocity of the object at a given time. Further, we know that regardless of the function under consideration, the derivative measures not only the slope of the tangent line at a given point, but also the function’s instantaneous rate of change with respect to the input variable. But what is the meaning of this instantaneous rate of change in contexts other than velocity? For instance, what if a function measures the size of a population at a given time? or the total revenue being generated by a sales of a product? or the rate at which a car consumes gasoline at a given speed? In what follows, we take a closer look at the meaning of the derivative in applied contexts.
These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.
Explore a GeoGebra applet that returns the value of a forward difference: Forward Difference Applet. For the function \(f(x) = \ln(x+1)\text{,}\) use the applet to estimate the value of \(f'(-0.75)\) using the forward difference and \(\Delta x=0.25\text{.}\) The difference quotient and secant line boxes may be of help. Which value displayed is the estimate? Submit a screen capture of the Forward Difference Applet, in which you found the forward difference for the function \(f(x) = \ln (x+1)\text{.}\) Comment on which value shown in your screen capture is the (forward difference) estimate of \(f'(-0.75)\text{.}\)
Ask Copilot to illustrate the difference between using forward, central, and backward differences to estimate the value of a derivative. You will want an image that illustrates these. How does Copilot perform on this request?
📝 [Submit] Do Using the data from Activity 1.5.3, use a spreadsheet to determine backwards, forward, and central differences as done in the video above. Microsoft Excel or Google Sheets are both examples of spreadsheets. Submit a screen capture of the spreadsheet finding backwards, forward, and central differences using the data in Activity 1.5.3.
Ask Copilot “If I have a function representing how much hair \(H(t)\) my dog sheds in month \(t\) after the start of January 2020, what are the units of the derivative of \(H\text{?}\)”
Explore an applet that illustrates a forward difference graphically and much, much more. The applet, Derivative Function: Without Words, is very cool. In a sentence or two, describe what the applet is attempting to illustrate. Feel free to vary the input function in your investigation.
If \(q = f(p)\) gives the number of pounds of sugar produced when the price per pound is \(p\) dollars, then what are the units and the meaning of the statement \(f'(3) = 50\text{?}\)
Note: You might think at first that the statement has something to do with the velocity of the water, but in fact a flow rate of 10 cubic feet per second could be achieved either with very slowly moving water through a large pipe, or with very rapidly moving water through a narrow pipe. If we look at the units - cubic feet per second - we realize that we are being given the rate of change of a quantity measured in cubic feet (i.e. a rate of change of a volume).
\(f'(3)=50\) means that when the price of sugar is $3 per pound, the rate at which the pounds produced changes is 50 pounds of sugar per dollars. That is, a one dollar increase in the price of sugar increases production by 50 pounds.
Visualize all the water flowing through the pipe ending up in a tank somewhere. If \(V(t)\) is the volume of water in the tank at time \(t\text{,}\) then we are being told the rate of change of \(V(t)\) is 10, or \(V'(t)=10\) cubic feet per second.
