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Chapter 9: Problem 34

Use a calculator to find each of the following to four decimal places. $$ e^{-2.64} $$

Short Answer

Expert verified
0.0703

Step by step solution

01

Identify the Expression

The expression we need to evaluate is \( e^{-2.64} \).
02

Enter the Expression into the Calculator

Turn on your calculator and enter the value exactly as shown: \( e^{-2.64} \). Ensure you use the exponent function correctly, which is typically represented by a button labeled \( e^x \).
03

Execute the Calculation

Press the 'equals' button (often labeled as '=' or 'ENTER') to compute the value.
04

Round the Result

After obtaining the result from the calculator, round it to four decimal places. The display should show approximately 0.0703.

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Key Concepts

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

Understanding e^x
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The notation \(e^x\) is an example of such a function. Here, 'e' represents Euler's number, a constant approximately equal to 2.71828. Exponential functions grow rapidly and are used in many fields such as biology, economics, and physics. Understanding \(e^x\) is crucial as it forms the basis for more complex exponential calculations, including those involving negative exponents.
Using a Calculator for Exponential Functions
Calculators make it easy to compute exponential functions like \(e^{-2.64}\). To use a calculator efficiently:
  • Turn on the calculator and locate the exponent button, often labeled as \(e^x\).
  • Enter the exponent value. In this case, it will be -2.64.
  • Check the input to ensure it's entered correctly.
  • Press the 'equals' button (labeled as '=' or 'ENTER') to compute the result.
By following these steps, you ensure accurate calculations and save time.
Rounding Decimals Properly
When you obtain a result from your calculator, it might not always land exactly on four decimal places. Rounding is important to make the result more readable and consistent. Here's how to round to four decimal places effectively:
  • Identify the fifth decimal place in the number.
  • If the fifth decimal place is 5 or greater, round the fourth decimal place up by one.
  • If the fifth decimal place is less than 5, keep the fourth decimal place as it is.
In our example, the calculator might show 0.070321. Hence, we round it to 0.0703 for four decimal places.
Working with Negative Exponents
Negative exponents indicate reciprocal values in exponential functions. For instance, \(e^{-2.64}\) can be interpreted as \(\frac{1}{e^{2.64}}\). This means instead of growing exponentially, the value decreases. Understanding this helps visualize why the result is less than one. Remember, negative exponents are very common in real-world applications such as decay processes in chemistry and depreciating assets in finance.

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Most popular questions from this chapter

Atmospheric pressure \(P\) at an elevation \(a\) feet above sea level can be estimated by $$ P=P_{0} e^{-0.00004 a} $$ where \(P_{0}\) is the pressure at sea level, which is approximately 29.9 in. of mercury (Hg). Explain how a barometer, or some other device for measuring atmospheric pressure, can be used to find the height of a skyscraper.

Solve for \(x\). Give an approximation to four decimal places. $$ \frac{3.01}{\ln x}=\frac{28}{4.31} $$

The number of bacteria in a culture grows at an exponential growth rate of \(139 \%\) per hour. What is the doubling time for these bacteria?

Growth of Bacteria. The bacteria Escherichia coli (E. coli) are commonly found in the human bladder. Suppose that 3000 of the bacteria are present at time \(t=0 .\) Then \(t\) minutes later, the number of bacteria present is $$ N(t)=3000(2)^{t / 20} $$ If \(100,000,000\) bacteria accumulate, a bladder infection can occur. If, at 11: 00 A.M., a patient's bladder contains \(25,000 E\) coli bacteria, at what time can infection occur?

The Richter scale, developed in \(1935,\) has been used for years to measure earthquake magnitude. The Richter magnitude \(m\) of an earthquake is given by $$ m=\log \frac{A}{A_{0}} $$ where \(A\) is the maximum amplitude of the earthquake and \(A_{0}\) is a constant. What is the magnitude on the Richter scale of an earthquake with an amplitude that is a million times \(A_{0} ?\)
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