1. Warm up
In terms of letters and variables write the general form of the first 5 polynomials.
For example: the first degree polynomial is represented by: y = ax+b

2. Maclaurin Polynomials
8.7: Taylor and
Maclaurin Polynomials
Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series.
Brook Taylor

3. In mathematics, a Taylor series is a representation of a function as an infinite sum of polynomial terms calculated from the values of the function's derivatives at a single point.
The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use
of this special case of Taylor series in the 18th century.
Colin Maclaurin (February 1698 – June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series centered about x = 0, are named after him.

4. Instead, he created polynomials to approximate functions.
Taylor’s brilliancy improved on the idea of approximating function values with points on the tangent line.
Instead, he created polynomials to approximate functions.
Whereas the tangent line only shares the function value and slope with the function at the point of tangency, Taylor polynomials share many more of the function’s traits.
A second degree Taylor polynomial has the same slope and second derivative value as the function (at the point of tangency) and thus the same concavity or shape as the function.
A third degree Taylor polynomial matches the first three derivatives, a fourth degree the first four derivatives, and so on.
Each iteration produces a polynomial that better matches the points on the original function…often on a much wider interval.

5. For certain functions (categorized as transcendental functions)
if the Taylor Polynomial is infinite, becoming a
Taylor Series, then the polynomial is actually
equivalent to the original function.

6. Suppose we wanted to find a fourth degree polynomial of the form:
that approximates the behavior of at
If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.

9. If we plot both functions, we see that near zero the functions match very well!

10. WOW!!!

11. Our polynomial:
has the form:
This pattern occurs no matter what the original function was!
or:

12. Maclaurin Series:
(generated by f at )
If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series:
Taylor Series:
(generated by f at )

13. example:

14. The more terms we add, the better our approximation.

15. WOW!!!

16. example:
Rather than start from scratch, we can use the function that we already know:

17. example:

18. There are some Maclaurin series that occur often enough that they should be memorized. They are:

19. When referring to Taylor polynomials, we can talk about number of terms, order or degree.
This is a polynomial in 3 terms.
It is a 4th order Taylor polynomial, because it was found using the 4th derivative.
It is also a 4th degree polynomial, because x is raised to the 4th power.
The 3rd order polynomial for is , but it is degree 2.
The x3 term drops out when using the third derivative.
A recent AP exam required the student to know the difference between order and degree.
This is also the 2nd order polynomial.

20. HW: p. 613 # 1-4, 13-29 odd, 41, 43 (refer to example 7)