Download A function f(x) can be approximated using a polynomial Pn(x) (order n)

Calculus AP BC Taylor Series and Error Analysis
BC: Q402.CH9B – Taylor Series: Error Analysis (LESSON 3)
THM E1: If a series for f (a ) is strictly alternating and decreasing in absolute value to zero, the
error in using Pn (a) to approximate f (a ) is less than or equal to the first omitted non-zero term:
Pn (a)  f (a) with error  Rn (a)  an1
THM E2: Taylor’s Formula with Remainder
The error in using Pn (x) to approximate f (x) is equal to R n ( x) 
f ( n1) (c)( x  a) n1
for some
(n  1)!
value c for a  c  x0 .
Pn ( x)  f ( x) with error  R n ( x) 
f ( n1) (c)( x  a) n1
for a  c  x0
(n  1)!
Random Theorem: If x is any real number, then lim n
| x |n
 0.
n!
Calculus AP BC Taylor Series and Error Analysis
1. Use the first two nonzero terms of the Maclaurin series to approximate sin(0.1). Estimate the
error.
1
2. Approximate  sin( x 2 )dx to four decimal places. Estimate the error.
0
Calculus AP BC Taylor Series and Error Analysis
1
.
x 1
(a) Write the first four terms and the general term of the Taylor series expansion of
f(x) about x  2 .
(b) Use the result from part (a) to find the first four terms and the general term of the
series expansion about x  2 for ln | x  1 | .
3
(c) Use the series in part (b) to compute a number that differs from ln by less than
2
0.05. Justify your answer.
3. Let f be the function defined by f ( x) 
Calculus AP BC Taylor Series and Error Analysis
4. The Taylor series about x  5 for a certain function f converges to f(x) for all x in the interval
1
(1) n n!
of convergence. The nth derivative of f at x = 5 is given by f ( n ) (5)  n
, and f (5)  .
2
2 (n  2)
(a) Write the third-degree Taylor polynomial for f about x = 5.
(b) Find the radius of convergence of the Taylor series for f about x = 5.
(c) Show that the sixth-degree Taylor polynomial for f about x = 5 approximates f(6) with
1
.
error less than
1000
Calculus AP BC Taylor Series and Error Analysis
 1n x 2 n
n  0 2n  1!

5. The function f is defined by the power series f ( x)  
for all real numbers.
(a). Find f / (0) and f // (0) . Determine whether f has a local maximum, a local minimum, or
neither at x  0 . Give a reason for your answer.
1
1
(b). Show that 1  approximates f (1) with error less than
.
100
3!
(c) Show that y  f (x) is a solution to the differential equation xy /  y  cos x .
Calculus AP BC Taylor Series and Error Analysis
CALCULATOR ACTIVE
6. Use the first two nonzero terms of the Maclaurin series to approximate sin(x). Estimate the
maximum error if |x|<1.
x2
7. If cos x is replaced by 1 
and | x | 0.5 , what estimate can be made of the error?
2
Calculus AP BC Taylor Series and Error Analysis
CALCULATOR ACTIVE
x
8A. The approximation 1  x  1  is used when x is small. Estimate the maximum error
2
when | x | 0.01
8B. Use the linearization (the first order polynomial) above to approximate 1.5 . Use the
Lagrange error bound analysis to determine the maximum error.
9. Use a Taylor polynomial of order 3 to find the cubic approximation of f ( x)  1 /(1  x) at
x  0 . Give the upper bound for the magnitude of the error for | x | 0.1 .
Calculus AP BC Taylor Series and Error Analysis
CALCULATOR ACTIVE
10. Let h be a function having derivatives of all orders for x  0 . Select values of h and its first
four derivatives are indicated in the table above. The function h and these four derivatives are
increasing on the interval 1  x  3 .
A. Write the first-degree Taylor polynomial for h about x  2 and use it to approximate h(1.9) .
Is this approximation greater than or less than h(1.9) ? Explain your reasoning.
B. Write the third-degree Taylor polynomial for h about x  2 and use it to approximate h(1.9) .
C. Use the Lagrange error bound to show that the third-degree Taylor polynomial for h about
x  2 approximates h(1.9) with error less than 3  10 4 .