Problem 18 - Algebra 2

Question

Use the graph of \(y=e^{x}\) to evaluate each expression to four decimal places. $$ e^{3} $$

Step-by-Step Solution

Answer

The evaluated expression \(e^{3}\) is approximately 20.0855.

Step 1: Understanding the function

Begin by understanding what the function \(y = e^{x}\) means. This is the exponential function where the base is the number \(e = 2.71828...\) (a mathematical constant), and \(x\) is any real number. The graph of \(y = e^{x}\) passes through the point \((0,1)\) and is always increasing.

Step 2: Mapping the x value to the function

Inspect the expression \(e^{3}\). This corresponds to finding the y-value (output) of the exponential function when \(x=3\). In general terms, to evaluate \(e^{3}\), find the value of \(y\) on the graph of \(y = e^{x}\) when \(x = 3\).

Step 3: Evaluating the expression

The value \(e^{3}\) is approximately 20.0855 when rounded to four decimal places.

Key Concepts

Graph of Exponential Functions

The graph of exponential functions, such as \(y = e^{x}\), is a smooth curve. This curve always rises sharply as \(x\) becomes larger. It's important to note a few characteristics:

• The graph passes through the point \((0,1)\), since \(e^{0} = 1\).
• For any positive value of \(x\), the value of \(e^{x}\) is always greater than 1.
• The function is asymptotic to the x-axis, meaning as \(x\) approaches negative infinity, \(y\) approaches 0 but never actually touches the x-axis.
• The function grows exponentially; thus, small changes in \(x\) can result in large changes in \(y\).

When graphing, these traits help in understanding the steep rise of \(y = e^{x}\) as \(x\) increases.

Evaluating Exponential Expressions

To evaluate an exponential expression like \(e^{3}\), find the y-value on the graph of \(y = e^{x}\) when \(x = 3\). This involves substituting \(3\) into the function, giving evaluated form: \(e^{3}\).

• For small values of \(x\), calculating \(e^{x}\) might seem straightforward.
• However, for values like 3 or larger, it is useful to use tools, like a calculator or logarithm tables, to obtain accurate results.
• Common calculators or computational tools can easily compute \(e^{3} \approx 20.0855\). For four decimal places, this level of precision can help with accurate evaluations.

Mathematical Constants

Mathematical constants are numbers that are fundamentally special to various areas of mathematics. The constant \(e\) is one of the most important constants due to its presence in exponential growth and natural logarithms.

• \(e\) is approximately equal to 2.71828 but is irrational and transcendental, meaning it cannot be expressed as a simple fraction and doesn’t relate algebraically to other numbers.
• It's often called Euler's number, named after the mathematician Leonard Euler who devised many of its properties.
• \(e\) is defining in continuous growth processes; for example, in compound interest, population growth, and radioactive decay.

Understanding \(e\) deepens comprehension of rates of change and connections to the natural world, making it vital in calculus and real-world applications ranging from finance to physics.