Problem 70 - Introductory and Intermediate Algebra for College Students 4th

Question

a. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. c. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. d. Describe what you observe in parts (a)-(c). Try generalizing this observation.

Step-by-Step Solution

Answer

The series expansions are getting closer to the graph of \(e^{x}\) as more terms are added. Therefore, one can say that the series expansion is an approximation of \(e^{x}\), and this approximation improves as we consider more terms in the series.

Step 1: Graph the functions in (a): \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\)

Using a graphing tool, plot both functions within a viewing rectangle that allows for comparisons. Identify the points where the two functions intersect. Observe the overall shape and nature of the curves.

Step 2: Graph the functions in (b): \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\)

Again, use the graphing tool to plot both functions within a similar viewing rectangle as in step 1. Note the differences and similarities from the functions in part (a). Recall that the additional term in the function, \(\frac{x^{3}}{6}\), is the next term in the series expansion of \(e^{x}\).

Step 3: Graph the functions in (c): \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\)

Plot these functions in the same manner as the previous steps. Remember to use the same viewing rectangle for consistency. The addition of another term, \(\frac{x^{4}}{24}\), is just extending the series expansion of \(e^{x}\). Compare this graph with the previous ones.

Step 4: Observation and Generalization

Now that we have created all the necessary graphs, the next step is to carefully observe the trajectories of the curves. Compare them and note down the findings. The generalization step will involve determining the patterns and trends from the observations and drawing a conclusion about the series expansion of \(e^{x}\) and its impact on the resulting graphs.

Key Concepts

Series Expansion

Understanding the concept of series expansion is crucial when exploring the behavior of functions such as the exponential function. In essence, the series expansion for an exponential function like \( e^x \) is expressed as an infinite sum of terms, where each term is a fraction of a power of \( x \) divided by the factorial of the index of that power.

For example, the series expansion of \( e^x \) can be written as \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \). Each new term added provides a more accurate approximation of the exponential function over a wider range of \( x \) values. In the exercise provided, parts (a)-(c) represent truncated series expansions of \( e^x \), meaning that the series is cut off after a finite number of terms.

When graphing these truncated series against the actual \( e^x \) function, one can observe how each additional term results in the approximation curve getting closer to the true curve of the exponential function. This concept is fundamental when studying mathematics because it offers insights into how functions can be approximated and thus utilized for various practical applications, such as calculus and solving differential equations.

Graph Comparison

Comparing graphs is a visual approach to understanding the relationships between different mathematical expressions. In the exercise, we compare the graph of the actual exponential function \( y = e^x \) with graphs of its truncated series expansions.

Initially, with fewer terms in the series, the exact and approximate functions swiftly diverge from each other as \( x \) increases or decreases from zero. As more terms are included, the series expansion becomes a closer representation of \( e^x \), and the two functions remain closer to each other over a larger domain. This visual comparison helps students grasp the power of a series expansion in approximating functions.

By observing the graphs, students can note how the curves intersect at different points and how adding more terms of the series changes the graph’s shape and improves the accuracy of the approximation. A key observation is that, even with a few terms, the series expansion serves as a good approximation of the exponential function near \( x = 0 \). This property is particularly useful in numerous fields, such as physics and engineering, where exponential behavior occurs frequently and precise calculations are essential.

Exponential Function Properties

The exponential function, denoted by \( y = e^x \), is a mathematical expression that defines one of the most important functions in mathematics due to its unique properties. Key properties of the exponential function include:

• It is always positive, \( e^x > 0 \), for all real numbers \( x \).
• The rate of increase of the exponential function is proportional to its value, which is why it is commonly found in growth and decay processes, such as population growth or radioactive decay.
• Its graph has a characteristic curved shape, steeply increasing as \( x \) becomes large and rapidly approaching zero as \( x \) becomes large and negative.
• The function has a horizontal asymptote at \( y = 0 \), meaning the function approaches zero but never actually reaches zero for negative values of \( x \).
• The area under the curve of the exponential function, from \( -\infty \) to any number \( a \), equals the definite integral which results in \( e^a \), illustrating the connection between exponential functions and the concept of the integral in calculus.

Exponential functions form the basis for natural logarithms, another critical concept in mathematics. They are used in solving certain types of differential equations and appear in the compound interest formula, underscoring their relevance not only in pure mathematics but also in financial mathematics and various scientific disciplines.

By studying and graphing exponential functions, students can become familiar with these properties, which are foundational concepts in higher mathematics and essential in the real-world application of mathematical equations.