# The number e . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.1, Problem 1E

(a)

To determine

## To define: The number e.

Expert Solution

The number e is limx0ex1x=1.

### Explanation of Solution

From the definition, e is the number such that limx0ex1x=1.

(b)

To determine

### To estimate: The values of the limits corrected to two decimal places.

Expert Solution

The value of the limits limh02.7h1h and limh02.8h1h are approximately 0.99 and 1.03.

The value of e must be between 2.7 and 2.8.

### Explanation of Solution

Result used:

If the exponential function f(x)=bx is differential at 0, then it is differentiable everywhere and f(x)=f(0)bx, where f(0)=limh0bh1h.

Calculation:

Obtain the value of the limit limh02.7h1h.

 h 2.7h−1h 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 limh02.7h1h is approximately 0.99.

Obtain the value of the limit limh02.8h1h.

 h 2.8h−1h 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 limh02.8h1h is approximately 1.03.

Obtain the value of e.

Consider the exponential functions 2.7x and 2.8x.

Substitute 2.7 for b in f(x)=bx and apply the Result,

f(x)=f(0)(2.7)x,  f(0)=limh02.7h1h

Substitute 2.8 for b in f(x)=bx and apply the Result,

f(x)=f(0)(2.8)x,  f(0)=limh02.8h1h

Since limh02.7h1h0.99 and limh02.8h1h1.03, the respective derivative function is f(x)=0.99(2.7)x and f(x)=1.03(2.8)x.

From the Result, the simplest differential formula exists when f(0)=1 for all possible values of b in f(x)=bx.

Since f(0)b=2.7<1<f(0)b=2.8, there is a number b between 2.7 and 2.8 for which f(0)=1 and that number b is called as e.

Therefore, the number e must be between 2.7 and 2.8.

### Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers! 