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Chapter 5: Problem 7

Answer True or False. You do not need a calculator for these exercises. Rather, use the fact that e is approximately 2.7 $$e^{-1}<0$$

Short Answer

Expert verified
False

Step by step solution

01

Identify the Expression

The expression given is \(e^{-1}\). This represents the reciprocal of \(e\), since \(e^{-1} = \frac{1}{e}\).
02

Consider the Approximate Value of e

We are given that \(e \approx 2.7\). Therefore, \(e^{-1} = \frac{1}{2.7}\).
03

Evaluate the Inequality

We need to check if \(e^{-1} < 0\). Since \(e^{-1} = \frac{1}{2.7}\) is a positive number (as both the numerator and the denominator are positive), \(e^{-1}\) is greater than 0.
04

Conclude the Truth Statement

Since \(e^{-1}\) is actually a positive number, it follows that \(e^{-1} < 0\) is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are fundamental in describing growth and decay processes, such as population growth, radioactive decay, and interest calculations.
The most common base used in natural exponential functions is the mathematical constant e, approximately 2.71828. This constant is called Euler's number, and it naturally arises in various mathematical contexts, including calculus, complex numbers, and financial calculations.
  • An example of an exponential function is \(f(x) = e^x\), where e is the base and x is the exponent.
  • Exponential growth occurs when the exponent is positive, leading to a rapid increase in the function's value.
  • Exponential decay occurs when the exponent is negative, resulting in a decrease.

Understanding how exponential functions work is crucial for working with equations and modeling real-world scenarios that involve rapid change.
Inequalities
Inequalities are mathematical statements used to compare two values or expressions. They are essential for solving equations, optimizing problems, and describing ranges.
Here are the symbols commonly used in inequalities:
  • \< (less than)
  • \> (greater than)
  • \leq (less than or equal to)
  • \geq (greater than or equal to)

Trying to determine the truth of statements involving inequalities is crucial in mathematics. In the exercise given, we evaluated an inequality, \(e^{-1} < 0\), to determine whether the reciprocal of e was less than zero. However, as shown, \(e^{-1}\) is positive, illustrating that understanding signs and values is key to solving inequalities effectively.
Reciprocals
The reciprocal of a number is one divided by that number. Reciprocals are fundamental in arithmetic and algebra, helping simplify fractions and solve equations with division.
For a given number x, its reciprocal is represented as \( \frac{1}{x} \). For positive numbers, reciprocals are positive, while for negative numbers, they are negative.
  • The reciprocal of e, known as e鈦宦�, is \( \frac{1}{e} \).
  • When e is approximately 2.7, its reciprocal becomes \( \frac{1}{2.7} \,\approx 0.37\).
  • Reciprocals reflect the idea of "flipping" a number's position in a fraction, changing denominators into numerators and vice versa.

Reciprocals play a vital role in the exercise, as recognizing them allows us to deduce whether the derived expression is positive or negative and enhances our understanding of complex equations.

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Most popular questions from this chapter

Radiocarbon Dating: Because rubidium-87 decays so slowly, the technique of rubidium-strontium dating is generally considered effective only for objects older than 10 million years. In contrast, archeologists and geologists rely on the radiocarbon dating method in assigning ages ranging from 500 to 50,000 years. Two types of carbon occur naturally in our environment: carbon-12, which is nonradioactive, and carbon-14, which has a half-life of 5730 years. All living plant and animal tissue contains both types of carbon, always in the same ratio. (The ratio is one part carbon- 14 to \(10^{12}\) parts carbon-12.) As long as the plant or animal is living, this ratio is maintained. When the organism dies, however, no new carbon-14 is absorbed, and the amount of carbon-14 begins to decrease exponentially. since the amount of carbon-14 decreases exponentially, it follows that the level of radioactivity also must decrease exponentially. The formula describing this situation is $$\mathcal{N}=\mathcal{N}_{0} e^{k T}$$ where \(T\) is the age of the sample, \(\mathcal{N}\) is the present level of radioactivity (in units of disintegrations per hour per gram of carbon), and \(\mathcal{N}_{0}\) is the level of radioactivity \(T\) years ago, when the organism was alive. Given that the half-life of carbon-14 is 5730 years and that \(\mathcal{N}_{0}=920\) disintegrations per hour per gram, show that the age \(T\) of a sample is given by $$T=\frac{5730 \ln (\mathcal{N} / 920)}{\ln (1 / 2)}$$

(a) Use a graphing utility to estimate the root(s) of the equation to the nearest one-tenth (as in Example 6). (b) Solve the given equation algebraically by first rewriting it in logarithmic form. Give two forms for each answer: an exact expression and a calculator approximation rounded to three decimal places. Check to see that each result is consistent with the graphical estimate obtained in part (a). $$e^{t-1}=16$$

Solve each equation and solve for \(x\) in terms of the other letters. $$y=A e^{k x}$$

Solve the inequalities. \(\frac{1}{\log _{2} x}+\frac{1}{\log _{3} x}+\frac{1}{\log _{4} x}>2 \quad(\text { for } x>1)\)

Use the following information on \(p H\) Chemists define pH by the formula pH \(=-\log _{10}\left[\mathrm{H}^{+}\right],\) where [H \(^{+}\) ] is the hydrogen ion concentration measured in moles per liter. For example, if \(\left[\mathrm{H}^{+}\right]=10^{-5},\) then \(p H=5 .\) Solutions with \(a\) pH of 7 are said to be neutral; a p \(H\) below 7 indicates an acid: and a pH above 7 indicates a base. (A calculator is helpful for Exercises 49 and 50.1 What is the hydrogen ion concentration for black coffee if the pH is \(5.9 ?\)
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