Problem 7 - College Algebra

Question

In Exercises \(1-10,\) approximate each number using a calculator. Round your answer to three decimal places. $$ e^{23} $$

Step-by-Step Solution

Answer

The rounded value of \(e^{23}\) to three decimal places would be approximately 9744803446.117.

Step 1: Calculate \(e^{23}\)

Use a scientific calculator to calculate \(e^{23}\). Most calculators have a button for \(e\) (usually labelled as 'EXP' or 'ex'). If not, can use the approximation \(e \approx 2.71828\) for the base.

Step 2: Round to Three Decimal Places

Once the approximate value is found, it needs to be rounded to three decimal places. If the digit in the fourth decimal place is 5 or greater, increase the digit in the third decimal place by one. If it's less than 5, leave the third decimal place digit as it is.

Key Concepts

Scientific Calculator

A scientific calculator is an essential tool for solving mathematical problems that involve complex calculations. It differs from a basic calculator in its ability to perform functions like exponentiation, trigonometry, and more. When finding the value of exponential expressions like \(e^{23}\), a scientific calculator is invaluable because it simplifies the process.

• Look for the button labelled 'EXP' or sometimes 'ex'. This represents the exponential function \(e\), where \(e\) is a constant approximately equal to 2.71828.
• Ensure your calculator is in the correct mode for computing exponentials—often this is the standard (not degree) mode.

By familiarizing yourself with these functions, solving exponential expressions can be done quickly and accurately.

Rounding Numbers

Rounding numbers is crucial when presenting results in a manageable and easy-to-read format. When you compute a number like \(e^{23}\) using a calculator, you'll get several decimal places, which may be impractical for most applications. Follow these steps to round a number to three decimal places:

• Identify the digit in the fourth decimal place. This is the deciding factor.
• If this digit is 5 or greater, increase the third decimal digit by one.
• If it's less than 5, keep the third decimal digit the same.

Rounding makes it easier to communicate and work with the results both in mathematics and real-world applications.

Approximation Methods

Approximation methods are techniques used to find near values of complex expressions. These methods are used to give a reasonable estimate of a number when exact data is either unavailable or unnecessary. For instance, approximating the base \(e\) as 2.71828 enables manual computations when a calculator isn't handy.

• Often, approximations simplify complex formulas and make them accessible without extensive computation.
• They are also a learning tool to understand the behavior of functions, such as exponential growth.

Approximating \(e^{23}\) without a calculator might involve using repeated multiplication of the approximation for \(e\), though this isn't typically practical for large exponents.

Natural Exponential Function

The natural exponential function is one of the most significant mathematical functions. Denoted as \(e^x\), where \(e\) is the natural base, it constantly appears in various fields ranging from mathematics, physics, to economics.

• This function has properties that make it unique, such as its derivative and integral both being \(e^x\), making it very useful in calculus.
• It describes growth processes, like populations or radioactive decay, which occur continuously.

Understanding \(e^x\) deepens comprehension of exponential growth and decay, equipping you with tools to tackle problems involving continuous rates of change.