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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

Use the exponential decay model for carbon- \(14, A=A_{0} e^{-0.000121 t}\) to solve Exercises \(19-20\). Skeletons were found at a construction site in San Francisco in \(1989 .\) The skeletons contained \(88 \%\) of the expected amount of carbon-14 found in a living person. In \(1989,\) how old were the skeletons?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After 100 years, a population whose growth rate is \(3 \%\) will have three times as many people as a population whose growth rate is \(1 \%\)

Exercises \(51-56\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(76,\) you will use your graphing utility to obtain these functions.) Teenage Drug Use $$\begin{array}{|lc|} \hline & \text { Percentage Who Have Used } \\ \hline \text { Country } & \text { Marijuana } & \text { Other IIlegal Drugs } \\\ \hline \text { Czech Republic } & 22 & 4 \\ \text { Denmark } & 17 & 3 \\ \text { England } & 40 & 21 \\ \text { Finland } & 5 & 1 \\ \text { Ireland } & 37 & 16 \\ \text { Italy } & 19 & 8 \\ \text { Northern Ireland } & 23 & 14 \\ \text { Norway } & 6 & 3 \\ \text { Portugal } & 7 & 3 \\ \text { Scotland } & 53 & 31 \\ \text { United States } & 34 & 24 \end{array}$$

One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.

Write each equation in its equivalent exponential form. $$\log _{6} 216=y$$
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