Problem 18 Use the graph of $y=e^{x}$ to ... [FREE SOLUTION]

Chapter 8: Problem 18

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

Short Answer

Expert verified

The evaluated expression $e^{3}$ is approximately 20.0855.

Step by step solution

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.

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 $x = 3$.

Evaluating the expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檛 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.

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91影视

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

Short Answer

Step by step solution

Understanding the function

Mapping the x value to the function

Evaluating the expression

Key Concepts

Graph of Exponential Functions

Evaluating Exponential Expressions

Mathematical Constants

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