In Exercise 45 of Section 1.3, we presented the following infinite power series formu-las...

In Exercise 45 of Section 1.3, we presented the following infinite power series formu-las for the cosine and sine functions:

a. Graph y = cos x using the following settings:

b. On the same screen, graph the polynomial

which represents the first four terms of the infinite series for cos x. Recall that 2! = 2 • 1 = 2 and 4! = 4 • 3 • 2 • 1 = 24. (Rather than determining the numer-ical values of the denominators, it would be much easier to use the "!" factorial key on your calculator,)

c. On what interval along the x-axis do the cosine function and this polynomial ap-pear to be converging to the same graph?

d. In the window settings, change Xmin to -2π and Xmax to 2 π and explain what happens to these functions when they are graphed on this larger interval, away from the interval in which they seem to converge.

e. Using the same window settings given in part (d), input the next two terms in the formula for the infinite series:

Graph this along with y = cos x. What is happening to the interval of convergence

f. Change the window settings for Xmin to -4- and Xmax to 4ir and experiment to see how many additional terms of the infinite series you need to include in order to have the graphs appear to converge on [-4ir, 4ir].