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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
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Practice Test 6…College Board S.A.T. site.
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Calc BC Q-6
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2012 BC m.c. (ab topic)
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2011 #6 (first part of part a for sure…the rest is new(ish), but unsurprising stuff)
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2008 practice test some teacher put up somewhere
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Yesterday:
We used polynomials to model other elementary (basic) functions.
( ) ( )nP x f x
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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
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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
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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
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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
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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.
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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
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Let’s do one from yesterday:
Write a 5th degree Taylor Polynomials centered at 0 to apporixmatef(x)= sin(x)
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Let’s change it up and center it :
Write a 5th degree Taylor Polynomials centered at 0 to apporixmatef(x)= sin(x)
2
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Let’s change it up and center it :
Write a 5th degree Taylor Polynomials centered at 0 to apporixmatef(x)= sin(x)
6
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Practice Problems on LT 1:
9.7
Page 644
25-30
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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
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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
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LT 3 examples--derivatives
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2014 #6…part b (don’t know part a yet…won’t for a while.
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LT 3 examples--Integration
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LT 3: taylor shortcuts cont’d
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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
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Try page 644-645
Page 644 #20, 44