11/17 (late blog)

Sorry I am late, I just hate computers and avoid them at all cost :).

Taylor Polynomials and Approximations:

The form of a convergent power series:

"In this section you will study a general procedure for deriving the power series for a function that has derivatives of all orders. The following theorem gives the form that every convergent power series must take."

If f is represent by a power series f(x) = E an(x-c)^n for all x in an open interval l containing c, then an = f^(n)(c)/n! and
f(x) = f(c) + f'(c)(x-c) + f''(c)/2! * (x-c)^2 +...+ f^n(c)/n! * (x-c)^n +... .

Definition of Taylor and Maclaurin Series:

If a function f has derivatives of all orders at x = c, then the series
E(from n=0 to infinity) f^n(c)/n! * (x-c)^n = f(c) + f'(c)(x-c) + f''(c)/2! * (x-c)^2 +...+ f^n(c)/n! * (x-c)^n +...
is called the Taylor series for f(x) at c. Moreover, if c = 0, then the series is the Maclaurin series for f.

"If you know the pattern for the coefficients of the Taylor polynomials for a function, you can extend the pattern easily to form the corresponding Taylor series."

The convergence of a Tyalor series will always equal f^n(c)/n! * (x-c)^n if lim(as n -> infinity) Rn = 0.

Guidelines for Finding A Taylor Series:

1.) Differentiate f(x) several times and evaluate each derivative at c.
f(c), f'(c), f''(c), f'''(c), ... , f^n(c), ...

2.) Use the sequence developed in the first step to form the Taylor coefficients an = f^n(c)/n!, and determine the interval of convergence for the resulting power series
f(c) + f'(c)(x-c) + f''(c)/2! * (x-c)^2 +...+ f^n(c)/n! * (x-c)^n +... .

3.) Within the interval of convergence, determine whether the series converges to f(x).



Everyone should look at and put to memory the chart of page 684 about power series for elementary functions.

11/14 post

what we did this week in calc class, we basically went over some more stuff with power series and taylor polynomials and maclaurin serires and what not. same old chapter nine stuff that we been doing for the past 2 weeks or whatever.

ok so POWER SERIES:
what you do for this is
1. do ratio test
2. set lim of abs value less than 1
3. solve

i went over taylor polynomials and all that last week. so let's go over derivative rules, since we been using that lately
sin = cos
cox = -sin
tan = sec^2
sec = sectan
1/x^2 = -2/x^3
xsinx = product rule
x/cosx = quotient rule

that is just a few examples of some derivative formulas for ya.

lim rules as n approaches infinity
1. if top degree > bottom degree = +/- infinity
2. if top degree < bottom degree = 0
3. if top degree = bottom degree = divide leading coefficients

yayyyyyyyyyyyyy :)

Post # 11

Okayy, so I'm going to go over some derivative rules and identities because while I was doing the homework I quickly realized that I do NOT remember how to do derivatives because I'm so in integral mode..

So, let's get it.

**The formula for quotient rule is vu^1-uv^1/v^2 or the derivative of the top times the bottom – the derivative of the bottom times the top over the bottom squared.

An example is sin x / x-1. Take the derivative of the top which is (cos x) times the bottom (x-1) – the derivative of the bottom (1) times the top (sin x) over the bottom squared (x-1) ^2.

From there, it is just simple algebra.
The answer comes out to (cos x) (x-1)-sin x/(x-1) ^2.

*You do not use quotient rule when there is only an x on the bottom.
*You just bring the x to the top and make the exponent negative then use the formula U^n.

**Another thing is the product rule..if everyone remembers it.

(first)(derivative of second) + (second)(derivative of first)

**Third, you need to remember when to use chain rule..

*when you have something inside something or something raised to something with a variable..

Also, remember all the trig functions are THE OTHER way around..

Like

Sin = cos

Cos =-sin

Tan = sec^2

Sec = sectan

And so on…

The thing I need most help with is where to go after root test when doing power series…like I get an answer then take the limit then what? And also, what are the quiz orders and when are we taking the HUGE test on everything? THANKSSSS J

Blog

Power Series is something relatively easy, and I found the homework for both this and thee Maclaurin/Taylor Polynomials to be extremely redundant. However, there's like only two basic rules for Power Series.

1.Do the ratio test.
2 set the limit of the absolut value less than 1
3. solve.

Basically this is just a review on the ratio test. HOWEVER. Say I have after the ratio test

limit as n-inf. of abs(x^2/2!)

Now this is the thing. the limit of the abs value is set to less than one right? well, if I plug in infinity, it'll give me inf over a number. which is just infinity. Therefore it diverges. HOWEVER,once again, if I plug in say 1 for x, I'll be left with 1/2 which is less than 1 but not greater than -1 (coming from the absolute value thing where you put -ve < inside of abs< +ve) value . if I plug in 0, I'll be left we something less than 1. therefore at both x=1 and 0, the polynomial converges (aka no infinity)

For what I do not get, and perhaps a question for Brob is exactly how this will be phrased on the AP. Also, I would like to know what to do with the graph ones?? Thanks oh so much.