Another method for introducing e.

James Taylor, Newcastle Grammar

This paper was written after I attended Mary Coupland's conference presentation last year (see Reflections 22(1)).

The students do not need the theory behind the following presentation, but here it is:

If  then

So, if we want a k so that  then we want:

Hence the required base for the exponent is

Hence

I use a graphics calculator (the Sharp EL9300) to investigate  for  The result then follows very smoothly and  pops out to the satisfaction of the students.

The worksheet that we use (see opposite page) gives the teaching process. I have removed spacing lines and I have put the expected values for some answers in italics so that you can see what is going on. Write them out, obviously, if you want to use the sheet.

(Note: The EL9300 has a bug and gives an incorrect derivative at x = 0 for all exponential functions except base e.)

The exponential function worksheet

1.   (a)  Clear all functions on the EL9300, and enter

       (b) Using MENU, turn on .

       (c)  Press GRAPH.

       (d) Press LEFT ARROW and record and  for 3 different points (record all decimals).

              In the third row, use your calculator to work out

Y

      (e)  Complete: For

2.   (a)  Edit Y1 so it reads

       (b) Press GRAPH.

       (c)  Press LEFT ARROW and record and  for 3 different points (record all decimals).

              In the third row, use your calculator to work out

Y

       (d) Complete :

              For

       (e)  Do you notice anything about your answer to 2(d) and 1(e)?

3.   (a)  Edit Y1 so it reads

       (b) Press GRAPH.

       (c)  Press LEFT ARROW and record and  for 3 different points (record all decimals).

              In the third row, use your calculator to work out

Y

       (d) Complete :

              For

       (e)  Do you notice anything about your answer to 3(d) and 1(e)?

4.   Complete: It would seem that:

5.   What will be the value of k such that

6.   For this value of k, what is the value of

       Call this number a.

7.   Edit your equation at Y1 so that it reads  (Enter the value of a.)

8.   Graph  and see if