Calculus BC
Worksheet 10.3
Name________________________________________
2/24/14
1) a. Use the Maclaurin series for e x to generate a second order Taylor polynomial for e 2 x centered at x = 0.
b. Use graphs to determine the maximum error between the polynomial and the function e 2 x on the interval
[0.2, 0.2] .
c. For e 2 x , find the Lagrange error bound R2 ( x) on the interval [0.2, 0.2] .
2) Let f ( x)  1  x
a. Generate P3 ( x) centered at x = 0.
b. Find the Lagrange error bound when P3 ( x) is used to estimate f(x) on the interval [0.5, 0.5] .
3) Let f ( x)  ln(1  x)
a. Write P3 ( x) and the general term.
b. Use graphs to determine the order of the polynomial necessary to approximate ln(1.8) with an error of
less than 0.01. (Hint: What value of x makes f(x) = ln(1.8)?)
c. Use the Lagrange error bound formula to determine n such that Pn ( x) approximates ln(1.8) with an error
less than 0.01.
4) Use the Lagrange error bound to estimate the domain for which a P7 ( x) approximation of sin(x) will have an
error less than 0.01.
[2.486, 2.486]
or
[2.116, 2.116]
or
[2.167, 2.167]
AP Calculus BC
Review - Ch 10
Name:_________________________
Wed 3/13/14
The closed interval [0, π] is partitioned into n equal subdivisions each of length
1).
x 

by the numbers x0 , x1, x2,..., xn 1 , xn , with 0  x0  x1  x2  ...  xn 1  x n   .
n
The lim
n 
n
xi cos(xi )x

i
is
1
(A) -2
(B) -1
(C)
1
(D) 2

(E)
y
-2
4
2
-1
1
2
x
-2
-4
2).
The graph above shows a function f with a relative minimum at x = 2. The
approximation of f(x) near x = 2 using a second-degree Taylor polynomial centered about x
= 2 is given by
a  b(x  2)  c (x  2)2 .
Which of the following is true about a, b, and c?
(A) a  0, b  0,c  0
(B) a  0, b  0,c  0
(C) a  0, b  0,c  0
(D) a  0, b  0,c  0
(E) a  0, b  0,c  0
Find the Maclaurin series for the function xe
3).
x 2
Determine if the series converge absolutely, converge conditionally, or diverge. Give
reasons for your answer.
( 5)n
4) 
n 1 n !

n
 n 
5)  

n 1  n  3 

6) What are all the values of x for which the series x 
converges?
(A)
(B)
(C)
(D)
(E)
1  x  1
1  x  1
1  x  1
1  x  1
All real numbers x
x2
2

x3
3

x4
4
n 1
 ...  ( 1)
xn
 ...
n
( 1)n ( )2n
 (2n )! 
n 0

7)
(A) 1
(B) -1
(C)

(D)

(E)
2
8) Find a bound for the truncation error after 99 terms of

( 1)n

n
1
1
(A) 0.0103
(B) 0.0102
8) Find the sum of the telescoping series
(C) 0.102

2 n
n2
(D) 0.101
4

n n (n  3)
1
(A)
14
3
(B)
10
3
(C)
22
9
(D)
14
9
e