AP Questions Tay/Mac #1
Name:__________________________________
1) Use a 6th degree Maclaurin polynomial to approximate cos(.4)
(1) n n
x converges?
2) What are all values of x for which the series 
n  2 ln n

(A) -e < x ≤ e
(B) -1 ≤ x < 1
(C) -e ≤ x < e
(D) -1 < x ≤ 1
(E) -1 ≤ x ≤ 1
3) The Taylor series for
sin( x 2 )
centered at x = 0 is
x2
(A)
(1)k x 2 k 1

k  0 (2k  1)!
(B)
(1)k x 2 k

k  0 (2k  1)!
(D)
1  (1) k x 2 k 1

x k 1 (2k  1)!
(E)
(1)k x 4 k

k  0 (2k  1)!

(1)k x 2 k 1

(2k )!
k 0


(C)

4) The figure to the right shows the graph of y = f(x) and y = T(x) where T(x) is a Taylor polynomial for f(x) centered at
zero. Which of the following statements are true?
I. T(0.5) is a good approximation for f(0.5)
II. T(1.5) is a good approximation for f(1.5)
III. T(0) = f(0)
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
5) The first three nonzero terms in the Taylor series about x = 0 of xe-x are
(A) x  x 2 
x3
2!
(B) x  x 2 
x3
2!
(C)  x  x 2 
x3
2!
(D) x  x 2 
x3
2!
(E) 1  x 
x2
2!
(1)n 1 x 2 n 1
, then f’(x) =

(2n  1)!
n 0

6) For all x if f(x) =
(A)
(1)n 1 x 2 n

n  0 (2n  1)!
(B)
(1)n x 2 n

(2n)!
n 0
(D)
(1)n 1 x 2 n

(2n)!
n 0
(E)
(1)n x 2 n

n  0 (2n  1)!


(1)n 1 x 2 n

n  0 (2n  2)!


(C)

x3
7) Let E be the error when the Taylor polynomial T(x) = x  is used to approximate f(x) = sinx at x = 0.5. Which of the
3!
following is true?
(A) |E| < 0.0001
(B) 0.0001 < |E| < 0.0003
(D) 0.005 < |E| < 0.007
(E) 0.07 < |E|
(C) 0.0003 < |E| < 0.005
(8) The Taylor Series of a function f(x) about x = 3 is given by
f ( x)  1 
3( x  3) 5( x  3)2 7( x  3)3
(2n  1)( x  3) n


 ... 
 ...
1!
2!
3!
n!
What is the value of f’’’(3)?
(A) 0
(B) 1.167
(C) 2.5
(D) 5
(E) 7
AP Questions Tay/Mac #2
Name:__________________________________
2012 AP Test
2010 AP Test
2011 AP Test
AP Questions Tay/Mac #3
Name:__________________________________
The Maclaurin series for ex is ex = 1  x 
x 2 x3
xn
  ...   ... The continuous function f is defined by
2 6
n!
2009 AP Test
e( x 1)  1
for x≠1 and f(1) = 1. The function f has derivatives of all orders at x = 1.
f ( x) 
( x  1)2
2
(a) Write the first four nonzero terms and the general term of the Taylor series for e( x 1) about x = 1.
(b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor
series for f about x = 1.
(c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b).
(d) Use the Taylor series for f about x = 1 to determine whether the graph of f has any points of inflection.
2
2007 AP Test
Let f be the function given by f ( x)  e
 x2
.
(a) Write the first four nonzero terms and the general term of the Taylor series for f about x = 0.
1  x 2  f ( x)
.
x 0
x4
(b) Use your answer to part (a) to find lim
x
(c) Write the first four nonzero terms of the Taylor series for
e
t 2
dt about x = 0. Use the first two terms of your
0
1/ 2
answer to estimate
e
t 2
dt .
0
1/ 2
(d) Explain why the estimate found in part (c) differs from the actual value of
e
0
t 2
dt by less than
1
.
200
The function f is defined by the power series
x 2 x 2 3x3
(1)n nx n
f(x) =  

 ... 
 ...
2
3
4
n 1
2006 AP Test
for all real numbers x for which the series converges. The function g is defined by the power series
x x 2 x3
(1)n x n
g(x) = 1  
  ... 
 ...
2! 4! 6!
(2n)!
for all real numbers x for which the series converges.
(a) Find the interval of convergence of the power series for f. Justify your answer.
(b) The graph of y = f(x) – g(x) passes through the point (0,-1). Find y’(0) and y’’(0). Determine whether y has a
relative minimum, a relative maximum, or neither at x = 0. Give a reason for you answer.