Problem 39 - Precalculus with Limits

Question

Graphing a Natural Exponential Function In Exercises \(39-44\) , use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. $$f(x)=e^{x}$$

Step-by-Step Solution

Answer

A table of selected values for the function \(f(x)=e^{x}\) is: (-2, 0.14), (-1, 0.37), (0, 1), (1, 2.72), (2, 7.39). The graph of the function starts slightly above the x-axis for negative x-values, crosses the y-axis at \(f(x)=1\), and then increases rapidly for positive x-values.

Step 1: Use a Graphing Utility

Access a graphing utility either online or download one. Enter the equation \(f(x)=e^{x}\) and generate a table of values. Normally, one would select x-values ranging from -2 to 2 to get a rudimentary understanding of the function's behavior across the x-axis.

Step 2: Generate Table of Values

Using the graphing utility, create a table of values for the function at the selected x-values. This might appear as follows; for x-values: -2, -1, 0, 1, 2, you get f(x) values of 0.14, 0.37, 1, 2.72, 7.39 respectively.

Step 3: Sketch the Graph

Begin by sketching out your x and y axes on a piece of graph paper. Label the x-axis with the selected x-values and the y-axis with the corresponding f(x) values (it may be necessary to use a reasonable scale to accommodate all the values). Plot the points and join them. The function being a natural exponential, would start slightly above the x-axis for negative x-values, crosses the y-axis at \(f(x) = 1\), and then increases rapidly for positive x-values.

Key Concepts

Natural Exponential Function

A natural exponential function is a type of exponential function in which the base of the exponent is Euler's number, denoted by the symbol \(e\). Euler's number \(e\) is approximately equal to 2.71828 and is a fundamental constant in mathematics.

The natural exponential function is represented as \(f(x) = e^x\). This function is unique because it's the only continuous growth rate that is equal to its own derivative. This property makes it incredibly useful in natural processes such as population growth, radioactive decay, and compound interest calculations.

Learning about the natural exponential function is important for understanding the behavior of exponential growth over time. It helps in predicting trends and solving real-world problems where growth or decay processes are involved. The graph of this function showcases exponential growth, where the curve gets steeper as \(x\) increases.

Exponential Growth

Exponential growth refers to an increase that occurs at a consistent rate over time. With the natural exponential function, you can observe exponential growth when the variable \(x\) is positive.

One of the hallmarks of exponential growth is that, at every equal time increment, the quantity grows by a fixed percentage. This results in a rapid increase as the value of \(x\) grows larger, which is distinct from linear growth where increases happen at a constant amount.

In the case of \(f(x) = e^x\), as \(x\) takes on larger positive values, \(f(x)\) increases drastically. This is why exponential growth is represented by a steep curve. Understanding this concept is key for fields like biology, finance, and environmental science where predicting future outcomes based on current trends is critical.

Graphing Utility

A graphing utility is a tool that allows you to visualize mathematical functions and generate related tables and graphs. For functions like \(f(x) = e^x\), using a graphing utility makes it much easier to interpret the behavior of the graph across different x-values.

Most graphing utilities permit you to:

• Input functions in symbolic form.
• Select a range of x-values to analyze.
• Automatically generate tables of values for these x-values.
• Visualize functions in the form of graphs which can be manipulated for closer inspection.

By leveraging a graphing utility, you can efficiently plot complex functions and uncover their characteristics, such as intercepts, intervals of increase, and points of inflection, all of which can be immediately visualized.

Table of Values

A table of values is a mathematical tool used to represent specific outputs for given inputs in a function. This table is especially useful when graphing a function by providing a clear summary of how the function behaves at certain points. It helps in plotting accurate graphs.

When constructing a table of values for \(f(x) = e^x\), you typically choose a range of \(x\) values. Commonly chosen points, like ranging from \(-2\) to \(2\), help capture the key characteristics of the graph, showing the behaviors from exponential decay, crossing the y-axis, and the rapid growth.

• For \(x = -2\), \(f(x) = 0.14\).
• For \(x = -1\), \(f(x) = 0.37\).
• For \(x = 0\), \(f(x) = 1\).
• For \(x = 1\), \(f(x) = 2.72\).
• For \(x = 2\), \(f(x) = 7.39\).

Using this table, you can plot these points on a graph to effectively sketch the function's curve. This method assists in gaining an intuitive sense of how the function behaves across different intervals of \(x\).