Question

4. (a) How is the number \(e\) defined?

(b) What is an approximate value for \(e\)?

(c) What is the natural exponential function?

Short answer

(a) The slope of the tangent line to \(y = {b^x}\) at the point \(\left( {0,1} \right)\) is exactly 1, then the number is called e.

(b) The approximated value of e is 2.71828.

(c) The natural exponential function is \(f\left( x \right) = {e^x}\).

Step-by-step solution

(a) Define the number e

Recall that the exponential function \(y = {b^x}\). For any value of \(b\), the graph of the exponential function always cross through the \(y\)-axis at \(\left( {0,1} \right)\).

For the base\(b\), if the slope of the tangent line to\(y = {b^x}\)at the point\(\left( {0,1} \right)\)is exactly 1, then the number is called e.

(b) The approximate value of e

The value of the number e lies in between the numbers 2 and 3.

In between the graphs of\(y = {2^x}\), and\(y = {3^x}\)the graph of\(y = {e^x}\)lies.

The approximated value of e is shown below:

\(e \approx 2.71828\)

Thus, the approximated value of e is 2.71828.

(c) The natural exponential function

The function \(f\left( x \right) = {b^x}\) is the exponential function, if \(b \approx 2.71828\), then \(f\left( x \right) = {e^x}\) is called the natural exponential function.

Thus, the natural exponential function is \(f\left( x \right) = {e^x}\).