TAYLOR SERIES

TAYLOR POLYNOMIALS: Use an nth degree Taylor polynomial to approximate f(x) near x=c.
f ' ' (c) f ' ' ' (c) f ( n) (c)
Pn ( x)  f (c)  f ' (c)( x  c)  ( x  c) 
2
( x  c)  ... 
3
( x  c) n
2! 3! n!
*A Maclaurin polynomial is centered at x = 0.
Remember: Don’t go beyond the degree asked for on the test!

TAYLOR’S THEOREM:
f ' ' (c) f ' ' ' (c) f ( n) (c)
f ( x)  f (c)  f ' (c)( x  c)  ( x  c) 2  ( x  c) 3  ...  ( x  c) n  Rn ( x) , where
2! 3! n!
f ( n 1) ( z )
Rn ( x)  ( x  c) n1 .  (Lagrange error bound) You don’t need to find z. Just look for the
(n  1)!
max value of f ( n1) ( z ) on the interval between x and c.

TAYLOR & MACLAURIN SERIES:


f ( n ) (c ) f ' ' (c ) f ' ' ' (c ) f ( n ) (c )

n 0 n!
( x  c)  f (c)  f ' (c)( x  c) 
n

2!
( x  c) 
2

3!
( x  c)  ... 
3

n!
( x  c) n  ....

Taylor series converge to a function f(x) if Rn ( x)  0 as n   .

SAVE TIME ON THE EXAM BY MEMORIZING THESE COMMON TAYLOR SERIES!

( x  1) 2 ( x  1) 3 ( x  1) 4 (1) n1 ( x  1) n 
(1) n1 ( x  1) n
ln x  ( x  1)     ....   ...  
2 3 4 n n 1 n

x2 x3 xn xn
e  1 x 
x
  ....   ...  
2! 3! n! n 0 n!
n 2 n 1
x 3
x 5
x 7
(1) x 
(1) n x 2 n1
sin x  x     ...   ....  
3! 5! 7! (2n  1)! n 0 ( 2n  1)!

x 2
x 4
x 6
(1) xn 2n 
(1) n x 2 n
cos x  1     ...   ...  
2! 4! 6! (2n)! n 0 (2n)!

CONVERGENCE: Use ratio test to determine convergence of Taylor series.

If a n is the general term of a Taylor series,
a n1
1. lim  1  the series converges
n  a
n

a n1
2. lim  1  the series diverges
n  a
n

INTEGRALS & DERIVATIVES OF TAYLOR SERIES can be calculated term by term from
a known Taylor series. Integrals and derivatives have
 the same radius of convergence
 the same interval of convergence, except maybe at the endpoints of the interval.
(You must test the endpoints in the series to determine convergence there.)
AP CALCULUS BC NAME
Unit 10 - Review

#’s 1 – 11: Multiple Choice & #’s 12 – 13: Free Response * means no calculator!

1 x2
*1. The Maclaurin series for is  x n . Which is a power series expansion for ?
1 x n 0 1 x2
A. 1  x 2  x 4  x 6  x 8  ... B. x 2  x 3  x 4  x 5  ... C. x 2  2 x 3  3x 4  4 x 5  ...
D. x 2  x 4  x 6  x 8  ... E. x 2  x 4  x 6  x 8  ...

x4 x5 x6 x n 3
*2. A function f has Maclaurin series given by    ...   ... . Which of the
2! 3! 4! (n  1)!
following is an expression for f(x)?
A.  3x sin x  3x 2 B.  cos(x 2 )  1 C.  x 2 cos x  x 2
D. x 2 e x  x 3  x 2 E. e x  x 2  1
2

1
*3. What is the coefficient of x 2 in the Taylor series for about x = 0?
(1  x) 2
1 1
A. B. C. 1 D. 3 E. 6
6 3

*4. What is the approximation of the value of sin 1 obtained by using the fifth-degree Taylor
Polynomial about x = 0 for sin x?
1 1 1 1 1 1
A. 1   B. 1   C. 1  
2 24 2 4 3 5
1 1 1 1
D. 1   E. 1  
4 8 6 120


*5. If a
n 0
n x n is a Taylor series that converges to f(x) for all real x, then f ' (1) 
  
A. 0 B. a1 C.  an
n 0
D.  na
n 1
n E.  na
n 1
n
n 1

6. Let P( x)  3x 2  5x 3  7 x 4  3x 5 be the fifth-degree Taylor polynomial for the function f

about x = 0. What is the value of f ' ' ' (0) ?
5 1
A. -30 B. -15 C. -5 D.  E. 
6 6


 1n  x  3  n
*7. What are all values of x for which the series 

n n 1
 converges?
2
5 1 5 1 5 1
A.  x B.   x   C.   x  
2 2 2 2 2 2
1 1 1
D.   x   E. x  
2 2 2
*8. Which of the following is the Maclaurin series for e 3 x ?
x 2 x3 x 4 27 x 2 81x 3 243x 4
A. 1  x        B. 3  9 x     
2 3! 4! 2 3! 4!
9 x 2 27 x 3 81x 4 3x 2 3x 3 3x 4
C. 1  3x      D. 1  3x     
2 3! 4! 2 3! 4!
9 x 2 27 x 3 81x 4
E. 1  3x     
2 3! 4!
*9. What is the interval of convergence of the power series 

x  3n ?
n 1 n  2
n

A. 1  x  5 B. 1  x  5 C. 1  x  5 D. 2  x  4 E. 2  x  4

*10. The third-degree Taylor polynomial for a function f about x = 4 is

x  43  x  42  x  4  2.
512 64 4
What is the value of f ' ' ' (4) ?
1 1 1 3 81
A.  B.  C. D. E.
64 32 512 256 256

1
*11. Which of the following is the Maclaurin series for ?
1  x 2
A. 1  x  x 2  x 3     B. 1  2 x  3 x 2  4 x 3     C. 1  2 x  3 x 2  4 x 3    
x2 x3 x4
D. 1  x 2  x 4  x 6     E. x       
2 3 4

 
*12. Let f be the function given by f ( x)  sin 5 x   , and let P(x) be the third-degree Taylor
 4
polynomial for f about x = 0.
a. Find P(x).
b. Find the coefficient of x 22 in the Taylor series for f about x = 0.
1 1 1
c. Use the Lagrange error bound to show that f    P   .
 10   10  100
x
d. Let G be the function given by G( x)   f (t )dt . Write the third-degree Taylor polynomial for G
0
about x = 0.

*13. The function f is defined by the power series


(1) n x 2 n x2 x4 x6 (1) n x 2 n
f ( x)    1    ...   ... for all real numbers x.
n 0 (2n  1)! 3! 5! 7! (2n  1)!
a. Find f ' (0) and f ' ' (0) . Determine whether f has a local maximum, a local minimum, or
neither at x = 0.
1 1
b. Show that 1  approximates f(1) with error less than .
3! 100
c. Show that y = f(x) is a solution to the differential equation xy' y  cos x