Taylor Polynomials and Approximations
Taylor Polynomials and Approximations
Polynomial functions can be used to approximate other elementary functions such as sin x, [pic],
and ln x. On your calculator, graph:
[pic]
[pic]
and compare.
Where did the polynomial we entered into y2 come from?
Let [pic]
What happens to this series if we let x = c?
[pic]
Now differentiate [pic] to find [pic].
[pic]
[pic]
Differentiate again, and find [pic]
[pic]
[pic] so [pic]
Now find [pic]
[pic]
[pic]= so [pic]
Do you see a pattern? What is [pic] [pic]
Now substitute your results into
[pic]
|Definition of an nth-degree Taylor polynomial: |
|If f has n derivatives at x = c, then the polynomial |
|[pic] |
|is called the nth-degree Taylor polynomial for f at c, named after Brook Taylor, an English mathematician. |
|If c = 0, then [pic] is called the nth-degree Maclaurin polynomial for f, named after another English mathematician, Colin Maclaurin. |
Ex. (a) Find the Maclaurin polynomial of degree n = 5 for [pic].
(b) Find [pic].
What is the value of [pic]?
What is the error of your approximation?
The error is symbolized [pic].
(c) Find [pic].
What is the value of [pic]?
[pic]
How does the error for [pic] compare to the error for [pic]?
What do you think would happen if we used our polynomial to estimate sin 2.7?
The TI-89 can find a Taylor polynomial. Press F3 and go to down to taylor or go to the Catalog.
The syntax is taylor (function, variable, degree, center). The calculator defaults to a center of zero
if you do not enter a center.
To find the polynomial that we just found, type:
taylor (sin(x), x, 5) or taylor (sin(x), x, 5, 0)
You can evaluate the polynomial for a particular value of x by using the “when” command.
For example, [pic]
Ex. Find the Taylor polynomial of degree n = 6 for [pic] at c = 1.
Ex. Find the Maclaurin polynomial of degree n = 4 for [pic].
Ex. Suppose that g is a function which has continuous derivatives, and that
[pic]
Write the Taylor polynomial of degree 3 for g centered at 2.
Taylor Polynomials and Approximations, Day 2
Yesterday we learned:
|Definition of an nth-degree Taylor polynomial: |
|If f has n derivatives at x = c, then the polynomial |
|[pic] |
|is called the nth-degree Taylor polynomial for f at c, named after Brook Taylor, an English mathematician. |
|If c = 0, then [pic] is called the nth-degree Maclaurin polynomial for f, named after another English mathematician, Colin Maclaurin. |
Ex. (a) Find the third- degree Maclaurin polynomial for [pic].
(b) Use your answer to (a) to find:
[pic]
________________________________________________________________________________
Ex. Suppose that the function [pic] is approximated near x = 0 by a third-degree Taylor
polynomial [pic].
(a) Find the value of [pic].
(b) Does f have a local maximum, a local minimum, or neither at x = 0 ? Justify your answer.
Ex. (a) Find the eighth- degree Maclaurin polynomial for [pic].
(b) Use your answer to (a) to approximate the value of [pic] so that the error in your
approximation is less than [pic]. Justify your answer.
(c) Use your calculator to find the actual value of [pic]. What is the error in the
approximation you found in (b)?
Power Series
If x is a variable, then an infinite series of the form
[pic]
is called a power series centered at c, where c is a constant.
Ex. Find the radius of convergence and the interval of convergence. Be sure to check the endpoints.
(a) [pic]
(b) [pic]
(c) [pic]
Taylor Series
A few days ago we learned to find a Taylor polynomial for a function f. Today we will extend our knowledge of Taylor polynomials to find a Taylor series for a function f.
|Taylor Series centered at x = c: [pic] |
|=[pic] |
If c = 0, the series is called a Maclaurin series.
Ex. Find a Taylor series for [pic] centered at c = 2. Give the first four nonzero terms and
the general term.
__________________________________________________________________________________
There are three special Maclaurin series you must know. These are the series for [pic], sin x, and cos x.
To derive a series for [pic]:
For what values of x does [pic] equal the series that you found? (Hint: Look at problem 2 on last night’s homework.)
| |
|[pic]= for |
__________________________________________________________________________________
To derive a series for sin x:
For what values of x does sin x equal the series that you found? (Hint: Look at problem 6 on last night’s homework.)
| |
|sin x = for |
To derive a series for cos x:
| |
|cos x = for |
We can manipulate these three special series (or any series we are given) to find other series by using the following techniques:
1) Substitute into the series
2) Multiply or divide the series by a constant and/or a variable
3) Add or subtract two series
4) Differentiate or integrate a series
5) Recognizing it as the sum of a geometric power series
Ex. Find a Maclaurin series for [pic] Find the first four nonzero terms and the
general term.
Ex. Find a Maclaurin series for [pic] Find the first four nonzero terms and the
general term.
Ex. Find a Maclaurin series for [pic] Find the first four nonzero terms and the
general term.
Manipulation Techniques, Day 2
Before we try to find a power series by recognizing it as the sum of a geometric power series, let’s do a quick review of geometric series. Geometric series are formed by multiplying by a common ratio r.
Suppose I told you to start with [pic]2 and to let r = 3. What geometric
series would you write?
What if [pic]2 and [pic]?
What if [pic]1 and r = x?
________________________________________________________________________________
Ex. Find a power series for [pic], centered at x = 0. Give the first four nonzero terms
and the general term. For what values of x does your series converge to [pic]?
On your calculator, graph y1 = [pic] and y2 = the first five terms of the series you found.
Trace on each graph to [pic] and x = 2. What do you notice?
________________________________________________________________________________
Ex. Find a power series for [pic], centered at x = 0. Give the first four nonzero terms
and the general term. For what values of x does your series converge to [pic]?
________________________________________________________________________________
Ex. Find a power series for [pic], centered at x = 2. Give the first four nonzero terms
and the general term. For what values of x does your series converge to [pic]?
Differentiation and Integration of Taylor Series and Finding the Sum of a Taylor Series
Ex. Find the sum of [pic]
______________________________________________________________________________
Ex. Find the sum of [pic]
______________________________________________________________________________
Ex. Find the sum of [pic]
|Theorem |
|If the function given by [pic] has a radius of convergence of R > 0, then on the interval [pic], f is differentiable (and therefore |
|continuous). Moreover, the derivative and antiderivative of f are as follows: |
|1) [pic] |
|2) [pic] |
| |
|The radius of convergence of the series obtained by differentiating or integrating a power series is the same as that of the original power |
|series. The interval of convergence, however, may differ as a result of the behavior at the endpoints. |
Ex. The function f is defined by [pic].
(a) Write the Maclaurin series for f. Give the first four nonzero terms and the general term.
For what values of x does the series converge?
(b) Use your answer to (a) to find the Maclaurin series for[pic]. Give the first four nonzero
terms and the general term. For what values of x does the series converge?
(c) Use your answer to (a) to find the sum of the infinite series
[pic]
(d) Use your answer to (a) to find the Maclaurin series for[pic]. Give the first four nonzero
terms and the general term. For what values of x does the series converge?
(e) Use your answer to (d) to find the sum of the infinite series
[pic]
Lagrange Form of the Remainder (also called Lagrange Error Bound or Taylor’s Theorem Remainder
When a Taylor polynomial is used to approximate a function, we need a way to see how accurately the polynomial approximates the function.
|Taylor’s Theorem: If a function f is differentiable through order n + 1 in an interval containing c, |
|then for each x in the interval, there exists a number z between x and c such that |
|[pic]where the remainder [pic] (or error) is given by [pic] The value of [pic] is the maximum value of the [pic] derivative of f for all |
|numbers z between x and c. |
Historically the remainder was not due to Taylor but to a French mathematician, Joseph Louis
Lagrange (1736 – 1813). For this reason, [pic] is called the Lagrange form of the remainder or the Lagrange Error Bound. When applying Taylor’s Formula, we may or may not be able to find the exact value of z. Rather, we would attempt to find a maximum bound for the (n+1)th derivative from which we will be able to tell how large the remainder or error, [pic], is.
________________________________________________________________________________
Ex. 1 The function f has derivatives of all orders for all real numbers x. Assume that
[pic]
(a) Write the third-degree Taylor polynomial for f about x = 2, and use it to
approximate [pic] Give three decimal places.
(b) The fourth derivative of f satisfies the inequality [pic] for all x in the closed interval
[2, 2.3]. Use the Lagrange error bound on the approximation of [pic] found in part (a) to find
an interval [a, b] such that [pic] Give three decimal places.
(c) Could [pic] equal 6.922? Explain why or why not.
(d) Could [pic] equal 6.927? Explain why or why not.
Ex. 2 Let f be the function given by [pic] and let [pic] be the third-degree
Taylor polynomial for f about x = 0.
(a) Find [pic].
(b) Use the Lagrange error bound to show that [pic].
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