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AP CALCULUS BC CHAPTER 9 REVIEW
PLEASE REFRAIN FROM USING INSTRUMENTS OF WEAKNESS!
From the 2003 BC Exam: 1. The function f has a Taylor series about x = 2 that converges to )(xf for all x in
the interval of convergence. The nth derivative of f at x = 2 is given by the following
n
n nf
3
)!1()2()( for 1n and .1)2( f
(a) Write the first four terms & the general term of the Taylor Series for f about x = 2.
(b) Find the radius of convergence for the Taylor Series for f about x = 2.
Show the work that leads to your answer!
(c) Let g be a function satisfying 3)2( g and )()( xfxg for all x. Write the first four terms and the general term of the Taylor Series for g about x = 2.
(d) Does the Taylor Series for g as defined in part (c) converge at x = -2?
Give a reason for your answer!
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AP CALCULUS BC CHAPTER 9 REVIEW
From the 1999 BC Exam: 2. Let f be a function that has derivatives of all orders for all real numbers.
Assume ' '' '''(0) 5, (0) 3, (0) 1, (0) 4f f f f
a) Write the third degree Taylor Polynomial for f about 0x and use it to
approximate (0.2)f .
b) Write the fourth degree Taylor Polynomial for g, where 2( ) ( )g x f x ,
about 0x .
c) Write the third degree Taylor Polynomial for h, where
0
( ) ( )x
h x f t dt ,
about 0x .
d) Let h defined as in part c). Given that (1) 3f , either find the exact
value of (1)h or explain why it cannot be determined.
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AP CALCULUS BC CHAPTER 9 REVIEW
3. a. Find a power series for 2
3)(
xxf centered at 0.
b. Find the radius & interval of convergence for the above series.
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AP CALCULUS BC CHAPTER 9 REVIEW
4. a. Build a MacLaurin expansion for xxf sin)( b. For what values of x does this series converge? c. Use your series from problem 3 to determine how many terms are needed to approximate )5.1(f with error less than 10-6
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AP CALCULUS BC CHAPTER 9 REVIEW
5. Determine whether the following converge or diverge – state what test you’re using and find an upper bound if possible!
a.
onn
n
23
2
b.
12
23
5
n nn
n
c.
1
2ln
n n
n
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AP CALCULUS BC CHAPTER 9 REVIEW
6. Consider the sequence
n
n
4
5
11
a. Graph using 10095 n . Indicate the coordinate of a95..a100 rounded to four places
b. Notice your sequence seems to converge? Find the actual value the sequence converges to using analytic means and show work!
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AP CALCULUS BC CHAPTER 9 REVIEW
7. Recall the formal definition of a convergent sequence a.k.a. Lan
For any ,0 )(M if )(Mn then Lan
Find such a )(M for the sequence
2
2
2
13
n
n
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AP CALCULUS BC CHAPTER 9 REVIEW
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