Honors AP Calculus / Mr. Hansen	Name: _______________________________________
4/26/2007
	Bonus points (for Mr. Hansen’s use only): __________

Test on Chapter 12 (100 pts.)

Instructions

• Calculator is permitted.
• Point values will be written on the board. Also check the board for any possible hints or typo corrections.
• Remember that in most cases, the problems are separable, just as they usually are on the AP exam. That means, for example, that simply because you cannot get part (a), you should still work all the other parts, because you will usually be able to do them (if you know what you’re doing, that is).

1.	Let P5(x) = c0 + c1(x – p/4) + c2(x – p/4)^2 + c3(x – p/4)^3 + c4(x – p/4)^4 + c5(x – p/4)^5 denote the Taylor polynomial of degree 5 that approximates the cosine function in a neighborhood centered about p/4.
(a)	Write out P5(x) so that the exact numeric values of the ci constants are clearly shown. Simplification is not required.
(b)	On a sketch showing both P5(x) and cos x over the interval [–4, 8], label each function.
(c)	State the value of P5(p/4) and explain why it is no surprise that this value precisely matches cos(p/4).
(d)	Find the value closest to x = 7 for which cos x = P5(x). No work is expected.
(e)	For x values in a neighborhood of the answer to part (d), is the Taylor polynomial P5(x) useful for estimating cos x? Explain your reasoning.
(f)	Use the Lagrange form of the remainder to find a bound for the absolute value of the error when P5(p) is computed. Show your work.
(g)	Compute the actual absolute value of the error when P5(p) is used to estimate cos p.
(h)	Write the standard Maclaurin series for cos x and state its interval of convergence.
(i)	Use 6 nonzero terms from (h), i.e., a Maclaurin polynomial of degree 10, to estimate cos p.
(j)	Explain how it is that P5(p) gives a better estimate than the Maclaurin series used in (h) and (i), even though the Maclaurin series is a much higher-degree polynomial and therefore presumably of better quality.
2.	Consider the standard Taylor series for ln x, centered about 1.
(a)	Write this Taylor series and state its interval of convergence.
(b)	Prove rigorously that the series converges.
(c)	To what value does the series in (b) converge? You may use part (a) as a hint. Show enough work to justify your answer.
(d)	Determine the number of terms of (b) that are required to compute a value that is within .0005 of the true limit. Show your work on the reverse side.