Problem 9 - College Algebra

Question

Approximate each number using a calculator. Round your answer to three decimal places. \(e^{-0.95}\)

Step-by-Step Solution

Answer

After computation and rounding, the result of \( e^{-0.95} \) approximated to three decimal places is 0.386.

Step 1: Calculate the Value

First, compute the value of \( e^{-0.95} \) by using a scientific calculator. This can be done by entering -0.95, pressing the 'EXP' or 'e^x' button which represents the Euler's number (e) raised to the power of the entered value.

Step 2: Round to Three Decimal Places

After you find the value, you round the obtained number to three decimal places. Rounding is done to simplify numbers and to keep the answer within a reasonable amount of error.

Key Concepts

Euler's number

Euler's number, often denoted as \( e \), is a fundamental mathematical constant approximately equal to 2.71828. Euler's number is the base of the natural logarithm and is widely used in calculus and complex analysis. It arises naturally in various growth and decay processes, such as population growth and radioactive decay.

When you perform calculations involving \( e \), you're dealing with exponential functions, which are used extensively in real-world applications. An exponential function is one where the function has \( e \) as its base, like \( e^x \). Exponential functions have the unique property of growing or decaying at a rate proportional to their current value.

In the given exercise, you are required to compute \( e^{-0.95} \). This represents exponential decay, and using Euler's number helps us understand and predict rapid changes in systems. Calculating it is best handled with a scientific calculator, where the power key is used to determine the effect of raising \( e \) to a given exponent.

scientific calculator

A scientific calculator is a vital tool for performing complex calculations in various fields of science and engineering. It can compute functions that go beyond basic arithmetic, such as trigonometric functions, logarithms, and exponentials, which are handy for working with Euler's number.

To calculate expressions like \( e^{-0.95} \) on a scientific calculator, you'll usually follow these steps:

• Enter the exponent, here -0.95.
• Press the 'EXP' or 'e^x' button. This instructs the calculator to calculate \( e \) raised to the given power.

Some calculators might also have direct \( e^x \) functionalities, making them particularly convenient for those working with exponential functions regularly, such as in physics or finance.

Always refer to your specific model's instructions since calculator buttons and functionality can differ. Using a scientific calculator ensures precise and rapid computations that are essential in higher-level mathematics.

rounding numbers

Rounding numbers is a simple yet crucial mathematical skill, especially when dealing with approximations and real-world measurements. It is the process of reducing the digits in a number while preserving its value as close as possible to the original number. This makes it easier to read and work with the numbers.

To round a number to three decimal places, like in the exercise, you need to look at the fourth decimal place. For example, if you calculate \( e^{-0.95} \) using a scientific calculator and get a value of, say, 0.386568, you would:

• Identify the digit in the fourth decimal place (the number 5 in this instance).
• Since 5 is greater than or equal to 5, you increase the third decimal place by 1.

This turns the rounded number into 0.387.

This process helps present data in a straightforward manner when unnecessary precision might complicate understanding. Rounding ensures that your calculations communicate effectively while maintaining accuracy.