Name____________________________
Calculus
Period______
Lesson 9.3 Taylor’s Theorem
We will learn three methods to calculate error. The method chosen depends upon the type of series; is the series
geometric, alternating, or neither?
I. Geometric Series – Create a new formula.
1. Find a formula for the truncation error if we use 1  x 2  x 4  x 6 to approximate
1
over (-1 , 1).
1  x2
Calculate the maximum error for x  0.5 .
2. Find a formula for the truncation error if we use P6 ( x) to approximate
1
on
1  3x
 1 1
  ,  . Calculate the
 3 3
maximum error for x  0.3 .
II. Alternating Series – Use the next term.
x2
3. If cos x is replaced by 1  and x  0.5 , what estimate can be made of the error?
2
Alternating Series (continued) – Use the next term.
x3
4. For what values of x can we replace sin x by x 
with error magnitude no greater than 3 104 ?
3!
III. Remainder Estimation Theorem – Develop the ‘next’ term and substitute the maximum x-value into the
formula.
f  n  ( x)  x n
error 
n!
x2
is used when x is small. Use the Lagrange form of the remainder to get
2
a bound for the maximum error when x  0.1.
5. The approximation ln(1  x)  x 
III. Remainder Estimation Theorem (continued) – Develop the ‘next’ term and substitute the maximum xvalue into the formula.
f  n  ( x)  x n
error 
n!
x2
6. The approximation e  1  x 
is used when x is small. Use the Remainder Estimation Theorem to
2!
estimate the error when x  0.2 .
x
7. The approximation
x4  2
estimate the error when x  0.1.
1
x is used when x is small. Use the Remainder Estimation Theorem to
4