Problem 67 - College Algebra

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 observation is that as more terms are added to the polynomial expression, it looks like the polynomial graph is starting to envelop the graph of the function \(y=e^{x}\). This suggests that the more terms added to the polynomial approximation, the more closely it resembles the actual function.

Step 1: Graphing first set of functions

For the first part, graph the two functions \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) using graphing tools. The x-limits and y-limits should be set appropriately so that both the functions are visible in the same viewing rectangle.

Step 2: Graphing second set of functions

In this step, graph the two functions \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\). Again ensure both the functions are visible within the same viewing rectangle.

Step 3: Graphing third set of functions

Similarly, for the third part, 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. Make sure all the functions are clearly noticeable.

Step 4: Making observations

After graphing all the functions, make some observations for parts (a), (b) and (c). In general, the graphs may appear to be getting closer as more terms are added to the polynomial. This can be generalized as, as the polynomial expression increases in terms, it becomes a better approximation of the exponential function \(y=e^{x}\).