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Chapter 3: Problem 78

What is the natural exponential function?

Short Answer

Expert verified
The natural exponential function, denoted by \( e^x \), is a mathematical function with 'e' (approximately 2.71828) as its base and 'x' as its exponent. It is always positive, increasing for all real 'x' values, and both its derivative and integral are itself. Its applications are numerous, notably in exponential growth and decay problems.

Step by step solution

01

Definition of Natural Exponential Function

The natural exponential function is denoted by \( e^x \), where 'e' is the base, while 'x' is the exponent. Here, 'e' is a mathematical constant approximately equal to 2.71828.
02

Properties of Natural Exponential Function

1) For any real number x, the value of \( e^x \) is always positive. \n 2) \( e^0 \) is equal to 1. \n 3) The function \( e^x \) is an increasing function for all real numbers x. \n 4) The derivative and integral of \( e^x \) is \( e^x \) itself, a remarkable characteristic of the natural exponential function.
03

Role of Natural Exponential Function

The natural exponential function is widely applied, particularly, in calculations involving exponential growth or decay - such as population growth, radioactive decay. Additionally, it appears often in applications involving calculus due to its special property that the derivative of \( e^x \) is \( e^x \) itself.

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Most popular questions from this chapter

$$\text { Solve for } y: 7 x+3 y=18$$

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In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation.. $$3^{x}=2 x+3$$
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