Graph a Function and It's Derivative
It is often difficult to picture the relationship between a function and it's derivative. This tutorial displays the graph of the derivative of a function directly beside the graph of the function, for the sake of comparison. The function is drawn on the left side, the derivative on the right.
Students often expect the derivative of a function to be positive where the function itself is positive and for the derivative of a function to be negative where the function itself is negative. Actually, the derivative is positive where original function is increasing. To emphasize this point, regions where the derivative is positive are colored in blue on both pictures. Also, where the function is increasing, and where the derivative is positive, the graphs are drawn in red.
To help be able to use the visualization skills that you gain from this program, you are expected to guess where a local minimum of the function will be located. You must therefore be able to determine the x coordinate of a point that is at the bottom of a hill. After you enter your function or derivative, a dialogue will appear that explains this. When you click on the dialogue will disappear and the derivative will be graphed on the right hand side. However, nothing will be graphed on the left. You are expected to identify a local minimum by clicking on the derivative graph, on the right. The x coordinate of your graph will be used to draw two green vertical lines of points with the same x coordinate, one on each graph. Hence it does not matter how high or low on the graph that you click, only where you your click is from left and right.
After you click, the function on the left will also be drawn. If your click was correct, the green line on the left should pass through the graph at the bottom of the hill. If your answer was incorrect, note the correct x coordinate on the left hand side. Then find that x coordinate on the right hand side. See if you can determine what property of the derivative graph characterizes the correct x value. I could tell you, but you will remember it if you discover it on your own.
Once you understand how the program works you will not have read the dialogue over and over. There is a button on the dialogue labeled "Dismiss". When you click on that buton, the dialogue will disappear never reappear. If you need to read it again, you will have to resart the applet. It may even be necessary to quit the browser and go back to the page. bookmarking the page before you quit the browser will make it easier to return to the program.
This program is mainly designed so that you can enter a function and see its derivative graphed beside your function. However, you may enter a derivative and see a function drawn, on the left, that has your entry as its derivative. Remember, no matter which you enter, the derivative is always on the right.
The derivative and the function having a particular derivative are drawn via numerical techniques. Computation is thus very rapid. However, this program is unable to display a formula for either of the companion functions drawn.
This tutorial is not as interactive as some of the previous, but addresses a critical issue in the taking of a derivative. I recommend that you graph as many homework problems as can be put into the function input panel. This demonstration can be of considerable assistance. I hope that you find that to be the case.