PLEASE REFRAIN FROM USING INSTRUMENTS OF ......AP CALCULUS BC CHAPTER 9 REVIEW PLEASE REFRAIN FROM...
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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 ) ( x f for all x in the interval of convergence. The nth derivative of f at x = 2 is given by the following n n n f 3 )! 1 ( ) 2 ( ) ( for 1 n 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 ) ( ) ( x f x g 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!
Transcript of PLEASE REFRAIN FROM USING INSTRUMENTS OF ......AP CALCULUS BC CHAPTER 9 REVIEW PLEASE REFRAIN FROM...
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!
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.
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.
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
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
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!
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
AP CALCULUS BC CHAPTER 9 REVIEW
