AP Calculus BC Monday, 07 April 2014 OBJECTIVE TSW (1) find polynomial approximations of elementary...

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

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)?

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

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

Sec. 9.7: Taylor Polynomials and Approximations

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

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

Sec. 9.7: Taylor Polynomials and Approximations

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

Sec. 9.7: Taylor Polynomials and Approximations

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

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

Sec. 9.7: Taylor Polynomials and Approximations

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

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

Sec. 9.7: Taylor Polynomials and Approximations

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

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.

Sec. 9.7: Taylor Polynomials and Approximations

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

10.2

0.0011

n

n

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