Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

Exploration Use a graphing utility to graph \(y_{1}=e^{x}\) and each of the functions \(y_{2}=x^{2}, y_{3}=x^{3}, y_{4}=\sqrt{x}\) and \(y_{5}=|x|\) in the same viewing window. (a) Which function increases at the fastest rate for "large" values of \(x ?\) (b) Use the result of part (a) to make a conjecture about the rates of growth of \(y_{1}=e^{x}\) and \(y=x^{n},\) where \(n\) is a natural number and \(x\) is "large." (c) Use the results of parts (a) and (b) to describe what is implied when it is stated that a quantity is growing exponentially.

Step-by-Step Solution

Answer

For large values of \(x\), \(e^x\) grows faster than \(x^2\), \(x^3\), \(\sqrt{x}\), and \(|x|\). If the rate of growth of a quantity increases at a faster pace as the quantity itself becomes larger, then the quantity is said to be growing exponentially. So, exponential growth means the function grows at a faster rate being increasingly faster the larger the value is.

Step 1: Graph the Functions

Use a graphing utility to graph the following functions in the same viewing window: \(y_{1}=e^{x}\), \(y_{2}=x^{2}\), \(y_{3}=x^{3}\), \(y_{4}=\sqrt{x}\) and \(y_{5}=|x|\).

Step 2: Compare the Rates of Growth

Observe the graphs and identify the function that grows fastest for large values of \(x\). Judging by their general shapes, for \(x > 0\) we expect \(y_{1}=e^{x}\) to grow faster than the others.

Step 3: Make a Conjecture

Due to the observation made in step 2, one can conjecture that an exponential function with base greater than 1 (\(e > 1\)), like \(e^x\), will grow faster than any polynomial function of the form \(y=x^n\), where \(n\) is a natural number, for sufficiently large values of \(x\).

Step 4: Describe Exponential Growth

Based on the previous steps, exponential growth implies a rate of growth that is continually accelerating and will eventually outpace any fixed power of \(x\). For large values of \(x\), an exponential function increases at a faster rate compared to polynomial functions. Thus, if a quantity is growing exponentially, the rate of its growth becomes increasingly faster over time.

Key Concepts

Graphing Utility

A graphing utility is a tool that helps visualize functions by plotting their graphs on a coordinate plane. With modern technology, graphing utilities can be both physical devices and software applications.

They are extremely useful in mathematics, allowing students to better understand and analyze the behavior of different types of functions. Let's look at why they are essential:

• **Visualization**: Graphing utilities provide a visual representation of mathematical functions, which can be especially helpful for understanding complex concepts.
• **Comparison**: By graphing multiple functions in the same view, we can easily compare their behaviors and see how they change relative to each other.
• **Precision**: Graphing tools allow for accurate plotting with detailed scales, making it easier to observe subtle differences.
• **Convenience**: Instead of manually plotting points and drawing curves by hand, a graphing utility can provide instant graphs, saving time and effort.

When exploring exponential functions versus polynomial ones, a graphing utility reveals the stark differences in their rates of growth, particularly for large values of x.

Polynomial Functions

A polynomial function is a mathematical expression consisting of variables raised to whole number powers and coefficients. They can take many forms, including quadratic (\(y = x^2\)), cubic (\(y = x^3\)), and more.

Polynomial functions are fundamental in algebra due to their simple form and wide applicability. Here are key characteristics:

• **Degree**: The largest power of the variable in the polynomial determines its degree. For example, \(x^2\) is a second-degree polynomial, while \(x^3\) is third-degree.
• **Behavior**: The behavior of a polynomial function as \(x\) approaches infinity depends on its degree and leading coefficient.
• **Growth Rate**: As the degree of a polynomial increases, the function grows faster, but polynomial growth is still slower compared to exponential growth for large x.

Understanding polynomial functions helps in analyzing how they eventually fare against exponential functions, providing context for the discussion about growth rates.

Exponential Functions

Exponential functions are expressions where the variable appears in the exponent, like \(y = e^x\). These functions are known for their rapid growth, which far exceeds that of polynomial functions as \(x\) becomes large.

Important aspects of exponential functions include:

• **Base of Exponent**: In the function \(e^x\), "\(e\)" is the base, which is a constant approximately equal to 2.71828.
• **Growth Pattern**: Exponential functions have a growth pattern that accelerates over time. For positive x, \(y = e^x\) tends to rise sharply.
• **Comparison**: Compared to polynomial functions, the exponential function will always grow faster for large values of x.

The significance of exponential growth is seen in various real-world phenomena, such as population growth and compound interest, where quantities increase rapidly as they are proportional to their current amount.

Rates of Growth

Understanding rates of growth helps in determining how quickly a function increases as the variable \(x\) becomes large. Different functions exhibit varying growth rates based on their mathematical structure.

Here's a closer look at how growth rates differ:

• **Constant Rate**: Linear functions increase at a consistent rate, illustrated by functions like \(y = x\).
• **Polynomial Growth**: Polynomial functions grow at increasing rates depending on their degree. However, this rate is still slower than exponential growth.
• **Exponential Growth**: Characterized by accelerating growth, exponential functions can surpass polynomial functions due to their base's power (such as \(e\)).

Realizing the implications of these growth rates is crucial as it allows one to predict long-term behavior in various contexts—whether it's in mathematics or in modeling real-world scenarios where rapid and sustained growth occurs.