Math 106 BC Quiz 05 page 1 10/29/2010 Name
Circle your Section: B (11am) C (noon)
1. Consider the function f(x) = 3
√x=x1/3.
1A. Find the Taylor polynomial approximation P3(x) of f(x) of degree 3 in powers of (x−1) (ie, take x0= 1). Organize
all your work into a table as we have done in class. Write each fraction in your final polynomial in lowest terms.
1B. Set your calculator window to [−2,3.5]×[−2,2.5] and plot both f(x) and P3(x).
Neatly sketch P3(x) on the axes on the right; note that f(x) is already drawn for
you.
1C. To five decimal places, what approximation does P3(x) give for 3
√1.8? How “far off ” from the actual value is this
approximation (again, to 5 decimal places?)
1D. The graphs of P3and fshould seem to be clearly “separated” by more than 0.1 to the right of x= 2.6 on your plot
of the two functions. But use techniques discussed in class to find to 6 decimal places the point pwhere the graphs actually
separate by more than 0.05 somewhere between x0= 1 and x= 2.6. Explain how you found p.
1E. Bonus We’ve discussed how adding more terms to the Taylor polynomial makes it produce better approximations
around x0on an interval outside of which the approximations actually worsen. What does the “interval of convergence”
appear to be in this problem? Explain!