The Dirty (half) dozenI thought about calling it the unlucky bakers dozen…but I thought it was

trying to hard.

My third option was …. 6

Practice Test 6…College Board S.A.T. site.

Calc BC Q-6

2012 BC m.c. (ab topic)

2011 #6 (first part of part a for sure…the rest is new(ish), but unsurprising stuff)

2008 practice test some teacher put up somewhere

Yesterday:

We used polynomials to model other elementary (basic) functions.

( ) ( )nP x f x

Yesterday:

We used polynomials to model other elementary (basic) functions.

We were given the basic form to write these ‘polynomial approximations’ (aka: taylor polynomials). It was:

( ) ( )nP x f x

0 1 2 3 4 4( )( ) '( )( ) ''( )( ) '''( )( ) ( )( ) ( )( )( ) ...

0! 1! 2! 3! 4! !

n n

n

f c x c f c x c f c x c f c x c f c x c f c x cP x

n

Yesterday:

We used polynomials to model other elementary (basic) functions.

We were given the basic form to write these ‘polynomial approximations’ (aka: taylor polynomials). It was:

We also found the more terms we used…the better the approximations become (the desmos graph demonstration)

( ) ( )nP x f x

0 1 2 3 4 4( )( ) '( )( ) ''( )( ) '''( )( ) ( )( ) ( )( )( ) ...

0! 1! 2! 3! 4! !

n n

n

f c x c f c x c f c x c f c x c f c x c f c x cP x

n

The polynomials have to be centered somewhere…meaning we have to have a point where we are evaluating all of the derivatives

Yesterday we were all using 0 as the centering point for our ‘taylorpolynomials’…i.e. we set c=0 for our polynomial approx. (taylor poly.)

0 1 2 3 4 4( )( ) '( )( ) ''( )( ) '''( )( ) ( )( ) ( )( )( ) ...

0! 1! 2! 3! 4! !

n n

n

f c x c f c x c f c x c f c x c f c x c f c x cP x

n

0 1 2 3 4 4( )( 0) '(0)( 0) ''(0)( 0) '''(0)( 0) (0)( 0) (0)( 0)( ) ...

0! 1! 2! 3! 4! !

n n

n

f o x f x f x f x f x f xP x

n

The polynomials have to be centered somewhere…meaning we have to have a point where we are evaluating all of the derivatives

Yesterday we were all using 0 as the centering point for our ‘taylorpolynomials’…i.e. we set c=0 for our polynomial approx. (taylor poly.)

0 1 2 3 4 4( )( ) '( )( ) ''( )( ) '''( )( ) ( )( ) ( )( )( ) ...

0! 1! 2! 3! 4! !

n n

n

f c x c f c x c f c x c f c x c f c x c f c x cP x

n

0 1 2 3 4 4( )( 0) '(0)( 0) ''(0)( 0) '''(0)( 0) (0)( 0) (0)( 0)( ) ...

0! 1! 2! 3! 4! !

n n

n

f o x f x f x f x f x f xP x

n

0 1 2 3 4 4( )( ) '(0)( ) ''(0)( ) '''(0)( ) (0)( ) (0)( )( ) ...

0! 1! 2! 3! 4! !

n n

n

f o x f x f x f x f x f xP x

n

When using polynomials to approximate, we are interested in getting the ‘best’ approximate we can.• So far, this means more terms…3,4,5,73, …as many as you can get.

• Another way to better your approximate is to move where you are centering your polynomial approximation … if the center is moved closer to the value you are trying to approximate, your approximation will be better.

What would it look like if we moved where our taylor polynomials were centered?

BTW…This is LT 1 for today!

**The A.P. folks will choose where the polynomial is centered for you!

If we were all using 1 as the centering point for our ‘taylorpolynomials’…i.e. we set c=1 for our polynomial approx. (taylor poly.)

0 1 2 3 4 4( )( ) '( )( ) ''( )( ) '''( )( ) ( )( ) ( )( )( ) ...

0! 1! 2! 3! 4! !

n n

n

f c x c f c x c f c x c f c x c f c x c f c x cP x

n

0 1 2 3 4 4(1)( 1) '(1)( 1) ''(1)( 1) '''(1)( 1) (1)( 1) (1)( 1)( ) ...

0! 1! 2! 3! 4! !

n n

n

f x f x f x f x f x f xP x

n

Let’s do one from yesterday:

Write a 5th degree Taylor Polynomials centered at 0 to apporixmatef(x)= sin(x)

Let’s change it up and center it :

Write a 5th degree Taylor Polynomials centered at 0 to apporixmatef(x)= sin(x)

2

Let’s change it up and center it :

Write a 5th degree Taylor Polynomials centered at 0 to apporixmatef(x)= sin(x)

6

Practice Problems on LT 1:

9.7

Page 644

25-30

LT 2: Using your taylor polynomials to approximate values you are asked to findWe will focus on Maclaurin Polynomials for this LT:

Note to me: Use examples page 645 #41-44

6 more will be provided for home practice (check online for key…after Friday afternoon

LT 3: Interesting stuff…and also likely on A.P.

We used polynomials to model other elementary (basic) functions.

if

then

and

( ) ( )nP x f x

'( ) '( )nP x f x

( ) ( )nP x f x

LT 3 examples--derivatives

2014 #6…part b (don’t know part a yet…won’t for a while.

LT 3 examples--Integration

LT 3: taylor shortcuts cont’d

LT 3: taylor shortcuts cont’d

We used polynomials to model other elementary (basic) functions.

if

then

and

( ) ( )nP x f x

(2 ) (2 )nP x f x

( ) ( )nxP x xf x

Try page 644-645

Page 644 #20, 44