Problem 10 - College Algebra

Question

In Exercises \(1-10,\) approximate each number using a calculator. Round your answer to three decimal places. $$ e^{-0.75} $$

Step-by-Step Solution

Answer

The expression \( e^{-0.75} \) can be approximated to be equal to \( 0.472 \) when rounded to three decimal places.

Step 1: Understanding the Natural Exponent

The natural exponential function, denoted as \( e^x \), is a mathematical function indicating the amount of growth resulting from continuously compounded interest over \( x \) units of time, where \( e \) is Euler's number, approximately equal to \( 2.71828 \). For the function \( e^{-0.75} \), a negative exponent means you're looking at exponential decay, rather than growth. This implies division, or fraction.

Step 2: Using Calculator for Exponential Calculation

ICalculate the value of \( e^{-0.75} \) using a scientific calculator. Make sure your calculator is set to use the exponential base \( e \). If necessary, consult your calculator's user manual or perform an online search to learn how to perform this operation on your specific model of calculator.

Step 3: Rounding the Result

After you get result, remember to round it to three decimal places. This involves looking at the digit in the fourth decimal place and determining whether the digit in the third decimal place should stay the same or increase by one.

Key Concepts

Exponential Decay

Exponential decay describes the process by which an original amount decreases at a rate proportional to its current value. Imagine you have a substance that loses half its mass every hour; this is a classic example of exponential decay. In the context of the provided exercise, the negative exponent in the natural exponential function, represented as \( e^{x} \), suggests that the value is diminishing over time rather than increasing.

So, when calculating \( e^{-0.75} \), we are essentially determining to what extent a quantity has reduced after 0.75 units of time, assuming the process follows an exponential decay model. This concept is crucial in various fields, including physics, chemistry, and finance.

Euler's Number

Euler's number, denoted as \( e \), is a mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and is pivotal in the natural exponential function \( e^{x} \). Euler's number arises in many areas of mathematics, especially in calculations involving continuous growth or decay.

It is also widely used in calculus to simplify derivatives and integrals involving exponential functions. When we use \( e \) in an equation like \( e^{-0.75} \), we're tapping into a profound mathematical truth that helps us model complex, real-world situations like population growth, radioactive decay, and even compound interest in finance.

Scientific Calculator Usage

A scientific calculator is an invaluable tool for students and professionals in scientific disciplines. It helps perform complex calculations, including those involving exponential functions like \( e^{x} \). To calculate \( e^{-0.75} \), you'll need to locate the function for Euler's number, often labeled as 'e^', 'exp', or a similar variant.

After inputting the exponent (-0.75 in this case), your calculator will display the result. If you are unfamiliar with the process or the specific functions of your calculator, remember to refer to the user manual or search for a tutorial online. Knowing how to effectively use your calculator can greatly enhance your ability to solve mathematical problems efficiently.

Rounding Decimals

Rounding decimals is a way of simplifying a number while retaining its value as closely as possible to the original. For example, if you need to round a number to three decimal places, you look at the fourth decimal point. If this number is 5 or higher, you increase the third decimal by one.

If it's less than 5, the third decimal remains unchanged. This rule allows for consistency and precision across various fields of study and applications. Correct rounding is essential for accurate data representation and is particularly important when reporting financial figures, scientific measurements, and statistical data.