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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

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\)the annual inflation rate, and \(S=\)the inflated value t years from now. Use this formula to solve. Round answers to the nearest dollar. If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 465,000\) be worth in 10 years?

Use the exponential decay model, \(A=A_{0} e^{k t},\) to solve Exercises \(28-31 .\) Round answers to one decimal place. The half-life of thorium- 229 is 7340 years. How long will it take for a sample of this substance to decay to \(20 \%\) of its original amount?

Solve each exponential equation in Exercises \(1-22\) by expressing each side as a power of the same base and then equating exponents $$6^{\frac{x-3}{4}}=\sqrt{6}$$

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?

The exponential growth models describe the population of the indicated country, \(A\), in millions, \(t\) years after 2006 $$\begin{array{l}\mathrm{Camada}\quadA=33.1e^{0.009\mathrm{t}}\\\\\mathrm{U}_{\mathrm{ganda}}\quad A=28.2 e^{0.034 t}\end{array}$$ In Exercises \(81-84,\) use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. By \(2009,\) the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.
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