8.7 Taylor Polynomials and
Approximations
objective:
Find Taylor and Maclaurin polynomial approximations of elementary functions
Title: Apr 2 ­ 11:50 AM (1 of 17)
The goal of this section is to show how polynomial functions can be used as approximations for other elementary functions
For example, we want to approximate the function
with a polynomial. The idea is we will choose a value of c in which the function and the polynomial have the same value.
In this case, we will use x = 0 as a starting point. And at x = 0, . Therefore, .
Title: Apr 2 ­ 11:55 AM (2 of 17)
This line, , is the zero degree polynomial approximation to . Title: Apr 2 ­ 12:08 PM (3 of 17)
Since that is not a very close approximation, we are going to add the requirement that the value and slope must agree at x =0. Therefore we are making a linear approximation.
Once again, we will use x = 0 as a starting point. At x = 0, and So then, Now we have Title: Apr 2 ­ 12:05 PM (4 of 17)
And, We see that is a close approximation at x = 0, and near x = 0. The approximating polynomial is said to be centered at x = 0.
Title: Apr 2 ­ 12:33 PM (5 of 17)
To make a better approximation, we can also require the polynomial to agree with the function at higher order derivatives. For example, we would create by requiring the second derivatives to be equal.
solve for the a's:
Title: Apr 2 ­ 12:32 PM (6 of 17)
As you can see, this is an even better approximation to the function.
Title: Apr 2 ­ 12:49 PM (7 of 17)
Now for the third degree polynomial, we have the following:
Title: Apr 2 ­ 12:53 PM (8 of 17)
So... What do you think the next polynomial will be?
Title: Apr 2 ­ 12:59 PM (9 of 17)
This is the nth­degree polyomial approximation to centered at x=0. Title: Apr 2 ­ 1:01 PM (10 of 17)
When you center an approximation about x=0, this is called a Maclaurin polynomial.
If you choose to center about x=c and c does not equal 0, then this is called a Taylor polynomial.
The Maclaurin polynomial is a special type of Taylor polynomial.
Title: Apr 2 ­ 1:04 PM (11 of 17)
Definition of a Taylor Polynomial
Let's try
again!! Title: Apr 5­9:05 AM (12 of 17)
Title: Apr 2 ­ 1:17 PM (13 of 17)
Your answer should be:
Use this to approximate ln(1.1).
Answer is 0.0953083
Answer here!!
Title: Apr 2 ­ 1:34 PM (14 of 17)
Title: Apr 2 ­ 1:17 PM (15 of 17)
• Homework: Problems on Handout
Title: Apr 2 ­ 1:50 PM (16 of 17)
Title: Apr 3­12:54 PM (17 of 17)