The Derivative by Zooming

Prerequisites:

  1. The definition of the derivative as the limit of slopes of secant lines.
  2. The concept of the derivative as the slope of a tangent line, which represents an instantaeous rate of change.
Goals:
  1. To use graphical zooming to visualize the limiting process when finding a derivative.
  2. To see both the increasing agreement of the secant lines with the function near the point of interest and the approach of the secant lines to a tangent line at that point.
  3. To guess derivatives by estimating the limiting value of the secant slope approximations.
  4. To use patterns to predict derivatives before zooming.
  5. To see patterns that foreshadow the power rule.

The derivative of a function at a point can be thought of in two ways. It is the instantaneous rate of change of the function at that point. It is also the slope of the tangent line at the point.

In this laboratory we zoom in on the graph of a function and measure the slope of a secant line through the point of interest at each stage of zooming. We continue to zoom in until the function looks more and more like the straight secant line. At each stage we measure the slope of the secant line. We will see that the secant slopes get closer to a final value or limit which we take to be the slope of the tangent at the point.

When you start start Maple, be sure Maple is in Worksheet mode, not Document mode. Then copy and paste the following code into your Maple worksheet. The color is only for your own reference (Maple does not use purple). You may want to edit the purple characters, but probably not the red.



restart: with(plots):

f:= x -> x^2:                          #the function
a:= 1;                                 #x-coord of point where tangent is desired
h:= .9;                                #x-displacement for computing slope of secant

rise:=f(a+h)-f(a);
run:=h;
SecantSlope:=rise/run;
secant:=x->f(a)+SecantSlope*(x-a):

xView := 3/2:                         #Expansion factor for window around point of interest
xMin := a-(xView)*h:
xMax := a+(xView)*h:

Function:=plot(f(x),x=xMin..xMax,color=red,scaling=unconstrained, title="Zoom View",  titlefont=[TIMES, BOLD, 14],axes=boxed):
Secant:=plot(secant(x),x=xMin..xMax,linestyle=DOT,thickness=2,color=blue):
Triangle:=pointplot({[a,f(a)],[a+h,f(a)],[a+h,f(a+h)]},symbol=circle,color=black):
RiseLine:=pointplot({[a+h,f(a)],[a+h,f(a+h)]}, connect=true):
RunLine:=pointplot({[a,f(a)],[a+h,f(a)]}, connect=true):

FunctionGlobal:=plot(f(x),x=a-(xView)..a+xView,color=red,scaling=unconstrained, title="Global View", titlefont=[TIMES, BOLD, 14]):
SecantGlobal:=plot(secant(x),x=a-(xView)..a+xView,linestyle=SOLID,color=blue):
Intersection:=pointplot({[a,f(a)],[a+h,f(a+h)]},symbol=circle,color=black):

Zoom:=display(Function,Secant,Triangle,RiseLine,RunLine):
Global:=display(FunctionGlobal,SecantGlobal,Intersection):

A := array(1..2):
A[1] := display(Zoom):
A[2] := display(Global):
display(A);


Do all the parts below. However, only turn in answers to questions 4 through 10. Turn these in as handwritten answers on a separate piece of paper.

  1. After pasting this code into Maple, press <enter> (or <return>). When Maple executes the commands, you should see two graphs: Zoom View and Global View.

  2. In Zoom View you should see the graph of a function, a secant line through (a, f(a)) and (a+h, f(a+h)), and a triangle showing the rise and run from which the slope of the secant line is found. Find and identify all these features. We will be using the Zoom View to zoom in on the graph near the point (a, f(a)).

    The Global View graph shows a view of the same function and secant line and their intersection points, but on a fixed scale that does not change as we zoom.

  3. The parts of the Maple code in purple are parts that we will be editing. First, try changing h to 0.3 instead of 0.9. What changed? Similarly, change the value of a from 1 to 0.5. Experiment until you understand the roles of h and a. When you are done experimenting, change the values of a and h back to their original values: a:=1; h:=.9;.

  4. Turn in answers to the following questions on a separate sheet of paper.

  5. Zoom in on the point (1,1) by shrinking the values of h until you're sure of the value the slopes are approaching. What is that value? Be sure to try some very small window sizes such as .001.

  6. Describe the relation between the graphs of the function and secant line suggested by the Zoom View when h is very small.

  7. Describe the relation between the graphs of the function and secant line suggested by the Global View when h is very small.

  8. In question 4 you found the slope of f(x)=x^2 at x=1. Now find it at x=2, 3, 4 by changing the Maple line a:=1; to a:=2; and repeating the procedure, then a:=3;, etc. What pattern do you notice in the slopes of the tangent lines?

  9. Use the pattern you found in the previous step to predict the slope for some value of a that is not an integer. Set a to that value and test your prediction. Describe what you predicted and found.

  10. Set the value of a back to 1. Change the line of Maple that defines the function from f:=x->x^2; to f:=x->x^3; and find the slope of the new function at a:=1;. Repeat for f:=x->x^4; and f:=x->x^5;. What pattern do you notice in these slopes or instantaneous rates? Keep a fixed at 1 for each of these functions.

  11. Use the pattern you found in the previous step to predict the slope at a:=1 for some x^n where the value of n is not an integer. Check your prediction. Describe your prediction and what you found.

Last Update: 2007/08/09