Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x=e^{2} $$

Step-by-Step Solution

Answer

The value of \( x \) is approximately 7.39.

Step 1: Recall the formula for the natural exponential function

The natural exponential function is represented as \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. In this problem, we need to find the value of \( e^2 \).

Step 2: Use a calculator to compute the value

Using a scientific calculator, find the value of \( e^2 \) by entering the expression and using the exponential function key. Calculating \( e^2 \) gives approximately 7.389056.

Step 3: Round to the nearest hundredth

To round 7.389056 to the nearest hundredth, look at the third decimal place, which is 9. Since it is 5 or greater, round the second decimal place up from 8 to 9.

Key Concepts

Understanding the Natural Exponential Function

The natural exponential function is a fundamental concept in mathematics. It is denoted as \( e^x \), where \( e \) represents Euler's number. This function grows exceptionally fast compared to polynomial functions and is widely used in modeling growth processes in areas like finance, biology, and physics.

Here's why it's called "natural":

• It occurs naturally in many processes, such as continuously compounded interest.
• The derivative of \( e^x \) is the same as the function itself, making it unique and useful for differential equations.

When you see \( e^x \), understand it as a way to express very rapid growth. Calculating this function requires knowing Euler's number and using it as the base raised to a power.

Exploring Euler's Number

Euler's number, represented by \( e \), is approximately equal to 2.71828 and is a crucial constant in mathematics. It is named after the Swiss mathematician Leonhard Euler, who significantly contributed to its understanding.

Here's what makes \( e \) special:

• It is the base of the natural logarithm, which simplifies various computations.
• You can think of \( e \) as similar to other mathematical constants like \( \pi \) in its importance and appearance in formulas and equations.
• In calculus, \( e \) plays a vital role, particularly in limits and exponential growth problems.

Using calculators to compute powers of \( e \) helps in finding precise results quickly for complex calculations. As seen in our example, determining \( e^2 \) required using a calculator to get an approximate value of 7.389056.

The Importance of Rounding to the Nearest Hundredth

Rounding numbers is crucial for precision and clarity in mathematics. When you "round to the nearest hundredth," you focus on the first two digits after the decimal point, ensuring a balance between accuracy and simplicity.

Let's look at the process:

• Identify the digit in the third decimal place.
• If this digit is 5 or greater, increase the second decimal digit by 1.
• If it is less than 5, leave the second decimal digit unchanged.

For instance, rounding 7.389056 to the nearest hundredth involves looking at the third decimal digit (9), and since it's 5 or greater, we round the second decimal digit up, making it 7.39.

This practice of rounding is essential in providing concise numerical results and is especially valuable in scientific and financial calculations where exact figures are often needed.