Problem 82 - College Algebra Essentials

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 graph of the function \(y=1+x+\frac{x^{2}}{2}\) starts to resemble the exponential function \(y=e^{x}\) in a certain viewing rectangle. As we add more terms to the function in the form \(\frac{x^{n}}{n!}\), the function approximates \(y=e^{x}\) even more closely. This exercise demonstrates that the infinite series \(1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}+... \) is equivalent to the exponential function \(y=e^{x}\).

Step 1: Graphing Part a

Graph the two functions \(y = e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) within the same viewing rectangle. This can be done using graphing software, a graphing calculator or by plotting points manually.

Step 2: Graphing Part b

Graph the functions \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. Again, this can be accomplished using a graphing tool.

Step 3: Graphing Part c

Graph the functions \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. With each addition of the new term in the second function, we note changes in the graph.

Step 4: Making Observations

Observe the graphs plotted in the previous steps. Look for patterns and changes that occur as the function evolves from \(y=1+x+\frac{x^{2}}{2}\) to \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) while comparing it to the \(y = e^{x}\) function.

Step 5: Generalizing Observations

Based on the graphs and observations, lay out a general rule or pattern identified from the functions and their graphs. This should summarize similarities or differences in the shape, points and behavior of the functions.

Key Concepts

Exponential Function Properties

Exponential functions, like the natural exponential function denoted as extstyleonly{\(y = e^{x}\)}, have unique characteristics that set them apart from other functions.

• Their rate of growth is proportional to their current value, which results in a rapid increase as the value of x increases.
• The graph of an exponential function always lies above the x-axis, as the function is always positive for real number inputs.
• Exponential functions have a constant base, in this case, Euler's number extstyleonly{\(e\)}, which is approximately equal to 2.71828.
• They possess an horizontal asymptote, typically at y=0, indicating that the graph approaches but never reaches the x-axis.
• The derivative of an exponential function remains proportional to the function itself, signaling constant relative growth.

These properties lead to distinct graph behaviour, making them easy to recognize when graphing against polynomial functions like extstyleonly{\(y = 1 + x + \frac{x^2}{2}\)}.

Taylor Series Approximation

The Taylor series is a powerful tool for approximating complicated functions with polynomials. To approximate extstyleonly{\(e^{x}\)}, the Taylor series around x = 0 (also known as the Maclaurin series) is used, which produces terms like extstyleonly{\(\frac{x^n}{n!}\)} for n=0,1,2,3,...,
By adding increasingly more terms of the series, extstyleonly{\(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...\)}, the approximation becomes more accurate around the point of expansion, in this case, x=0. As observed in the exercise, including more terms from the series gives a graph that moves closer to resembling the true function extstyleonly{\(y = e^{x}\)}.
This process demonstrates a fundamental aspect of Taylor series: given enough terms, it can approximate functions incredibly well over a specified range.

Comparing Functions on a Graph

When comparing the graphs of functions, in particular an exponential function alongside its Taylor series approximation, several observations are indicative of their similarities and differences:

• Initially, for small values of x, the approximation and the exponential function are nearly indistinguishable, which shows the effectiveness of the Taylor series in local approximation.
• As x moves away from 0, the approximation starts to deviate from the true exponential function, highlighting the limitation of polynomial approximations for large values of x.
• The degree of the polynomial in the approximation correlates with its accuracy—higher-degree polynomials typically offer a better fit over a wider range.

The exercise illuminates this by illustrating the convergence of the Taylor polynomial towards the exponential function as more terms are added. Understanding these differences is crucial not only for graphing but also for interpreting the behavior of functions in various scientific and engineering contexts.