Q. 24 - Calculus

Question

In this exercise you will use a calculator to investigate the number e.

1. Make a table of values that describes the behavior of the quantity $(1+h{)}^{1/h}$ as $h\to 0$.
2. Make a table of values that describes the behavior of the quantity $\frac{e^h-1}{h}$ as $h\to 0$.
3. What do your tables of values have to do with Definition 1.25 and Theorem 1.26?

Step-by-Step Solution

Answer

Part(a) The table is,

0.1	2.537425
0.01	2.7028138
0.001	2.7169239
0.0001	2.7181459
0.00001	2.7182682
0.000001	2.7182805

Part(b) The table is, 

h	f(h)
0.1	1.0517092
0.01	1.0050167
0.001	1.0005002
0.0001	1.00005
0.00001	1.000005
0.000001	1.0000005

Part(c) Both the tables satisfies the Definition 1.25 and Theorem 1.26 .

Part(a) Step 1. Given Information.

We are given a number e,

Part(a) Step 2. Table of values.

The table of values for $(h,(1+h{)}^{1/h})$ is as follows,

h	$(1+h{)}^{1/h}$
0.1	2.537425
0.01	2.7028138
0.001	2.7169239
0.0001	2.7181459
0.00001	2.7182682
0.000001	2.7182805

Part(b) Step 1. Table of values.

The table of values for $\left(h,\frac{e^h-1}{h}\right)$ is as follows,

h	f(h)
0.1	1.0517092
0.01	1.0050167
0.001	1.0005002
0.0001	1.00005
0.00001	1.000005
0.000001	1.0000005

Part(c) Step 1. Tables of values related with Definition 1.25 and Theorem 1.26

The table of values satisfies the Definition 1.25 and Theorem 1.26 .

The value conveys to e and the value is converging to $1$.