AP Calculus BC Monday, 07 April 2014 OBJECTIVE TSW (1) find polynomial approximations of elementary functions and compare them with the elementary functions;

AP Calculus BCMonday, 07 April 2014

• OBJECTIVE TSW (1) find polynomial approximations of elementary functions and compare them with the elementary functions; (2) find Taylor and Maclaurin polynomial approximations of elementary functions; and (3) use the remainder of a Taylor polynomial.

• ASSIGNMENTS DUE TOMORROW (TEST DAY)– Sec. 9.4– Sec. 9.5– Sec. 9.6

• TEST TOPICS PowerPoint is already on my website.

Sec. 9.7: Taylor Polynomials and Approximations


What does the graph of look like? xf x e

Just by looking at the graph, what is an approximation for f (1.5)?


We could get a better approximation ifwere represented as a polynomial P.

A first degree approximation at x = 1.5 would look like this:

xf x e


Now suppose we wanted to approximateat a different x-value, say x = 0.

If we use a 1st-degree polynomial, we would want P(0) = f(0) and

xf x e

0 0 .P f 1 0 1P x a a x

1 1 1P x x

1st degree

0 1f

1 11P x a x

1 0 10 0P a a

1 0 0 1P f

0 1a 1 0 0P f

xf x e

0 1f

1 1P x a

1

1 1a

1 1P x x


A 2nd-degree polynomial approximation forwould have P(0) = f(0),

xf x e 0 0 , andP f

22 0 1 2P x a a x a x 2

2

11

2P x x x

0 0 .P f

2 1, and0 1 d

f P Pdx

1 2Since , must be .1 ..P xP x x


A 3rd-degree polynomial?

4th-degree?

2 33 0 1 2 3P x a a x a x a x

2 33

1 11

2 3!P x x x x

2 3 44 0 1 2 3 4P x a a x a x a x a x

2 3 44

1 1 11

2 3! 4!P x x x x x


nth-degree?

2 30 1 2 3

nn nP x a a x a x a x a x

2 31 1 11

2 3! !n

nP x x x x xn


These examples are centered at c = 0. In general, c could be any value, so the polynomial would be written as

2

0 1 2

n

n nP x a a x c a x c a x c

2 1

1 2 32 3n

n nP x a a x c a x c na x c

2

2 32 2 3 1n

n nP x a a x c n n a x c

1 2 2 1nn nP x n n n a


From this, we get the following:

42 3 4 2

3 4

4

1 1 11 1 1 1 1 1

1

2! 3! 41 1

2

14

!1 1

13

f f fx f f x x x

x

x

x

P x x


Ex: Find the Taylor polynomials P0, P1, P2, P3, and P4 for centered atc = 1.

lnf x x

0 01P x f

1 1 1 11x f f xP x

2

2

211 1 1 1

2!1

1 12

Pf

x f f x x x x

2

3

3 2 31 11 1 1 1 1

2!1 1

1 123!

13

xf f

x f f x xP x x x


Ex: Find the Maclaurin polynomials P0, P2, P4, and P6 for . Useto approximate the value of .

cosf x x 6P x cos 0.1

0 1P x

22

11

2P x x

2 44

1 11

2 4!P x x x

2 4 66

1 1 11

2 4! 6!P x x x x

6 0.1 0.995004165278P

Actual value: 0.995004165278


Ex: Find the third Taylor polynomial for

centered at sin ,f x x

.6

c

sinf x x

1

6 2f

cosf x x

36 2

f

sinf x x

1

6 2f

cosf x x

36 2

f


Ex: Find the third Taylor polynomial for

centered at sin ,f x x

.6

c

2 3

36 6

6 6 6 2! 6 3! 6

f fP x f f x x x

2 3

31 3 1 32 2 6 2 2! 6 2 3! 6

P x x x x


Ex: Use a fourth Maclaurin polynomial to approximate the value of

Consider this:

1.1 is closer to 1 than 0, so an approximation using

and an x-value of 0.1 would be more accurate than using and

an x-value of 1.1.

ln 1.1 .

ln 1f x x

lnf x x


Ex: Use a fourth Maclaurin polynomial to approximate the value of ln 1.1 .

ln 1f x x 0 0f

11f x x

21f x x

32 1f x x

44 6 1f x x

0 1f

0 1f

0 2f

4 0 6f


Ex: Use a fourth Maclaurin polynomial to approximate the value of ln 1.1 .

42 3 4

40 0 0

0 02! 3! 4!

f f fP x f f x x x x

2 3 44

1 1 12 3 4

P x x x x x

ln 1.1 ln 1 0.1 4 0.1P 0.0953083333

Actual value: 0.0953101798


An approximation technique is of little value without some idea of its accuracy.

To measure the accuracy of approximating a functional value by the Taylor polynomialyou can use the concept of a remainder, defined as follows.

f x ,nP x ,nR x

n nf x P x R x

Exact Value Approximate Value

Remainder


Another way to look at this is that

The absolute value of is called the error associated with the approximation. In other words,

This is summarized in Taylor’s Theorem, which gives the Lagrange form of the remainder.

.n nR x f x P x

nR x

Error .n nR x f x P x



When applying Taylor’s Theorem,

You will probably not find the exact value of z!!!

(If you could, an approximation would not be necessary.)

You try to find bounds for from which you are able to tell how large the remainderis.

1nf z

nR x


Ex: The third Maclaurin polynomial for sin x isgiven by

Use Taylor’s Theorem to approximate by and determine the accuracy of the approximation.

sin 0.1

3

3 .3!

xP x x

3 0.1P

3sin 0.1 0.1P

3

3

0.10.1 0.1 0.099833

3!P

sin 0.1 0.099833




sin 0.1

3

3 .3!

xP x x

3 0.1P

Using Taylor's Theorem,

3

3

sin3!

xx Rx x

3 13

3 1

3 13 !!

f zx

xx

where 0 < z < 0.1.

4

3 4

,3! 4!

f zx

xx




sin 0.1

3

3 .3!

xP x x

3 0.1P

43sin , so the error 0.1 is given byf z z R

30 0.1R

4

3

sin0.1 0.1

4!

zR

Since 0 sin 1,z 4 4sin 10.1 0.1

4! 4!

z

40.0

10.1

4!00004


Ex: Determine the smallest degree of the Taylor polynomial expanded about c = 1 that should be used to approximate so that the error is less than 0.001.

ln 1.2

nP x

(We want 1.2 0.001.)nR

1.2 1.2 1.2n nR f P

11

1.21 !

nn

n

f zR x c

n

11

so 1.2 1 0.0011 !

nnf z

n



ln 1.2

nP x

(We want 1.2 0.001.)nR

1.2 1.2 1.2n nR f P

lnf x x 1

f xx

2

1f x

x

3

2f x

x

4

4

3!f x

x 5

5

4!f x

x

1 1 !1

nn

n

nf x

x. . .

1

1or equivalently,

!1

nn

n

nf x

x



ln 1.2

nP x

(We want 1.2 0.001.)nR

11

1.2 1 0.0011 !

nnf z

n

1

1

!1

nn

n

nf x

x

1 11.2

!

1 0.0011 !

1n

n

n

n

nz

1

1

0.2 0.001

1

n

nz n

where c < z < x, or 1 < z < 1.2



ln 1.2

nP x

(We want 1.2 0.001.)nR

1

1

0.20.001

1

n

nz n

"What is the smallest value of n that will make the inequality true?"

"What value of z will give us the safest estimate?"

When z is smallest (so the denominator will be smallest and the fraction will be largest.)

Use a -value of 1.z

Use a calculator to determine this.



ln 1.2

nP x

(We want 1.2 0.001.)nR

10.2

0.0011

n

n

n = 3: You should use a 3rd degree Taylor Polynomial

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AP Calculus BC Monday, 07 April 2014 OBJECTIVE TSW (1) find polynomial approximations of elementary functions and compare them with the elementary functions;