Concept explainers
(a) How is the number e defined?
(b) Use a calculator to estimate the values of the limits
correct to two decimal places. What can you conclude about the value of e?
(a)
To define: The number e.
Answer to Problem 1E
The number e is
Explanation of Solution
From the definition, e is the number such that
(b)
To estimate: The values of the limits corrected to two decimal places.
Answer to Problem 1E
The value of the limits
The value of e must be between 2.7 and 2.8.
Explanation of Solution
Result used:
If the exponential function
Calculation:
Obtain the value of the limit
h | |
0.1 | 1.04 |
0.01 | 1.00 |
0.001 | 0.99 |
0.0001 | 0.99 |
0.00001 | 0.99 |
Therefore, the value of limit
Obtain the value of the limit
h | |
0.1 | 1.08 |
0.01 | 1.03 |
0.001 | 1.03 |
0.0001 | 1.03 |
0.00001 | 1.03 |
Therefore, the value of the limit
Obtain the value of e.
Consider the exponential functions
Substitute 2.7 for b in
Substitute 2.8 for b in
Since
From the Result, the simplest differential formula exists when
Since
Therefore, the number e must be between 2.7 and 2.8.
Chapter 3 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
Additional Math Textbook Solutions
Precalculus: A Unit Circle Approach
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Calculus & Its Applications (14th Edition)
Precalculus: Mathematics for Calculus - 6th Edition
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