# 15 Worksheet on Exponential Function Exponential

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```					Exponential Functions from the Numerical Point of View                            Note: Save this workbook under a different
(Based on Example 2 in the text.)                                                 name to avoid changing the original.

In this sheet, we look at two tables of data and determine which, if either, represents an exponential function.
The data in question are shown under f(x) and g(x).

1. Click on cell C23 and look at the formula: B23/B22 = f(x)/f(x-1).
If f were exponential, these ratios would be the same for all values of x.
Since they are not, we conclude that f is not exponential.

x           f(x)       f(x)/f(x-1)                                      x       g(x)       g(x)/g(x-1)
-2            -7                                                        -2        2/9
-1            -3      0.428571429                                       -1        2/3
0             1      -0.33333333                                        0         2
1             5            5                                            1         6
2             9           1.8                                           2        18

2. Now test the function g by filling in the formula for g(x)/g(x-1).
(Enter the formula in cell H23 and copy it into the rest of the column.)
Press this button to have the computer fill in the cells:

Press this button to revert to empty cells:

Question What can you conclude about the function g?

Go to the sheet "Graphical" for a tutorial on graphing exponential functions.
s workbook under a different
changing the original.

ential function.
Exponential Functions from the Graphical Point of View
(Based on Example 3 in the text.)

In this sheet we graph two exponential functions on the same graph.
Click on cell B14 and you will see the formula 2^X. Click on cell C14 and you
will see the formula 2^(-X). These are the two exponential functions being graphed.
(See the tutorial for Section 1.2 for information about graphing functions.)
Notice the symmetry between these two graphs.
Now try graphing some other exponential functions A(b^x). How do A and b affect the shape of the graph?

X              Y1             Y2                    Xmin             Xmax            NumSteps DeltaX
-3          0.125             8                       -3               3          100      0.06
-2.94    0.13030822     7.67411295
-2.88    0.13584186      7.3615012
-2.82    0.14161049     7.06162397                                             9
-2.76    0.14762408      6.7739625
8
-2.7    0.15389305     6.49801917
-2.64    0.16042824     6.23331664                                             7
-2.58    0.16724094     5.97939699
6
-2.52    0.17434296     5.73582099
-2.46    0.18174656     5.50216727                                             5
-2.4    0.18946457     5.27803164
4
-2.34    0.19751033     5.06302638
-2.28    0.20589775     4.85677954                                             3
-2.22    0.21464136     4.65893435
2
-2.16    0.22375627     4.46914855
-2.1    0.23325825     4.28709385                                             1
-2.04    0.24316374     4.11245531
0
-1.98    0.25348987     3.94493082
-1.92    0.26425451     3.78423059      -4          -3        -2     -1            0    1      2
-1.86    0.27547628     3.63007662
-1.8    0.28717459     3.48220225
-1.74    0.29936968     3.34035168
-1.68    0.31208264     3.20427951
-1.62    0.32533546     3.07375036
-1.56    0.33915108     2.94853843
-1.5    0.35355339     2.82842712
-1.44     0.3685673     2.71320865
-1.38     0.3842188     2.60268371
-1.32    0.40053494      2.4966611
-1.26    0.41754396     2.39495741
-1.2    0.43527528     2.29739671
-1.14    0.45375958     2.20381023
-1.08    0.47302882     2.11403608
-1.02    0.49311635     2.02791896
-0.96    0.51405691     1.94530989
-0.9    0.53588673     1.86606598
-0.84    0.55864357     1.79005014
-0.78    0.58236679     1.71713087
-0.72    0.60709744     1.64718203
-0.66     0.6328783     1.58008262
-0.6    0.65975396     1.51571657
-0.54    0.68777091     1.45397252
-0.48    0.71697762     1.39474367
-0.42    0.74742462     1.33792755
-0.36   0.77916458     1.2834259
-0.3    0.8122524    1.23114441
-0.24   0.84674531    1.18099266
-0.18      0.882703   1.13288389
-0.12   0.92018765    1.08673486
-0.06   0.95926412    1.04246576
2.1649E-15              1              1
0.06   1.04246576    0.95926412
0.12   1.08673486    0.92018765
0.18   1.13288389       0.882703
0.24   1.18099266    0.84674531
0.3   1.23114441     0.8122524
0.36    1.2834259    0.77916458
0.42   1.33792755    0.74742462
0.48   1.39474367    0.71697762
0.54   1.45397252    0.68777091
0.6   1.51571657    0.65975396
0.66   1.58008262     0.6328783
0.72   1.64718203    0.60709744
0.78   1.71713087    0.58236679
0.84   1.79005014    0.55864357
0.9   1.86606598    0.53588673
0.96   1.94530989    0.51405691
1.02   2.02791896    0.49311635
1.08   2.11403608    0.47302882
1.14   2.20381023    0.45375958
1.2   2.29739671    0.43527528
1.26   2.39495741    0.41754396
1.32    2.4966611    0.40053494
1.38   2.60268371     0.3842188
1.44   2.71320865     0.3685673
1.5   2.82842712    0.35355339
1.56   2.94853843    0.33915108
1.62   3.07375036    0.32533546
1.68   3.20427951    0.31208264
1.74   3.34035168    0.29936968
1.8   3.48220225    0.28717459
1.86   3.63007662    0.27547628
1.92   3.78423059    0.26425451
1.98   3.94493082    0.25348987
2.04   4.11245531    0.24316374
2.1   4.28709385    0.23325825
2.16   4.46914855    0.22375627
2.22   4.65893435    0.21464136
2.28   4.85677954    0.20589775
2.34   5.06302638    0.19751033
2.4   5.27803164    0.18946457
2.46   5.50216727    0.18174656
2.52   5.73582099    0.17434296
2.58   5.97939699    0.16724094
2.64   6.23331664    0.16042824
2.7   6.49801917    0.15389305
2.76    6.7739625    0.14762408
2.82   7.06162397    0.14161049
2.88    7.3615012    0.13584186
2.94   7.67411295    0.13030822
3             8          0.125
e of the graph?

3   4
Fitting an Exponential Curve to Data
(Based on Example 5 in the text)

In the text we saw how to find the exponential curve through two given points by substituting in the equation
y = A*b^x and then solving for the coefficients A and b. Here, we will derive and use a formula to do this
automatically.

Start with the two points (x1, y1) and (x2, y2). Substituting them in the above exponential equation gives
y1 = A*b^x1          ...      (1)
y2 = A*b^x2          ...      (2)
Dividing the second equation by the first gives
(y2/y1) = (b^x2)/(b^x1) = b^(x2-x1)
Taking the 1/(x2-x1) power of both sides gives us b:
b = (y2/y1)^(1/(x2-x1))
Once we have b, we can use equation (1) to solve for A:
A = y1/b^x1

Let us use these formulas to solve Exercise 5 in the text:

x1             y1             x2            y2              b             A
1              6             3             24              2             3

Equation: y = 3*(2)^x

Click on the light orange shaded cells to see how we used the formulas. Then change the values of x1, y1, x2, and y2
to see the effect on the result.
1, y1, x2, and y2
The number e

In this worksheet you can explore the convergence of the expression (1+1/m)^m to the number e.

1. Enter the formula =(1+1/A18)^A18 in cell B18 and copy to the cells below it.

2. Compare the values you get to the value of e, which is approximately
2.718281828459
(notice that we calculate this value in the spreadsheet as EXP(1); EXP is the function e^x).

3. Try to extend to larger values of m. You'll notice a problem with computer arithmetic:
The answers become inaccurate because 1/m becomes so small that, eventually,
the computer thinks 1+1/m is equal to 1.

m          (1+1/m)^m
1
10
100
1000
10000
100000
1000000
10000000
100000000
1000000000
10000000000

```
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