Estimating the Graph of the Derivative Function
Estimating Points on the Derivative Function
In the app: Enter the formula for a function, f, in the input box. It is graphed in blue.
Note that there are 10 free points A-J. What we want to do is to place these points so that the graph of the derivative function f ' (x) will go through them. For example, take point A and let it have any x-value you desire. The y-value then needs to be the value of f'(x(A)) where x(A) is the x-coordinate of the point A. Recall that this is the slope of the tangent line to the graph of the function f at the point with this x-value on the curve. One good way to estimate this is to compute a symmetric difference quotient at that point by taking two points on the curve on either side of the point of interest and computing the slope between these two points. Whatever this x-value is the adjust point a up or down so that it has the same y-value as this slope. Repeat for the other 9 points. You might start by placing points at all the locations where the derivative touches the x-axis. Next pick some points with positive derivative and some with a negative derivative. Of course, not all functions have derivatives that take on positive, negative, and zero values.
Check your work by checking the Derivative Function checkbox. How close does the derivative function come to going through your 10 points?
There is also one aide that you can use. Check on the Tangent Line Segments checkbox. If this is activated, then there will appear little tangent line segments. The centers of these line segments are the same x-values as your 10 points A-J. Their slopes are the values of the derivatives at these x-values. The change in x from endpoint to endpoint on these line segments is always 1, so the change in y is the value of the derivative. Showing these can help you place the points correctly.
Try some with and without the tangent line segments turned on. Experiment with different functions.