Taylor and Maclaurin Series. Approximating functions using Polynomials.

Transcription

1 Taylor and Maclaurin Series Approximating functions using Polynomials.

2 Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear Approximation). f 0 = 1 f x = e x f (0) = e 0 = 1 P x P x 1 = 1(x 0) P x = 1 + x P 1 (x) = 1 + x The 1 st Degree Polynomial Approximation P 1 0 = f 0 = 1 P 1 0 = f 0 = 1

3 A Better Approximation to f x = e x As we move away from x = 0, our line is not a good approximation to e x. How could we get a better approximation? To get a better approximation, we can create a function, P 2 x, that also has the same concavity or 2 nd Derivative as f x = e x. P 2 0 = f 0 = 1 P 2 0 = f 0 = 1 P 2 0 = f 0 = 1 f x = e x P 2 x = 1 + x + x2 2 The 2 nd Degree Polynomial Approximation Graph

4 The next step, in order to get an even better approximation is to allow their third derivatives to equal one another. This can be done with the following equation: P 3 x = 1 + x + x2 2 + x3 6 Graph In general, the function e x can be approximated exactly by the following: P n x = 1 + x + x2 2 + x3 3! + + xn e x = x n

5 For what values of x will e x = x n For all x values in the interval of convergence!!? lim n a n+1 a n < 1 lim n x n+1 n + 1! x n lim n x n+1 n + 1! x n lim n x n + 1 < 1 This holds for all values of x. Therefore, Graph e x = x n everywhere!

6 Use our Polynomial approximation for e x in order to estimate e and e 2. What is the maximum error involved in our estimation? e x = x n e = 1 n = 1 e = 1 e e = e x = x n P n x = 1 + x + x2 2 + x3 6 P 3 1 = *The maximum error will just be the absolute value of the first omitted term. (This is very similar to the alternating series error). *This is a simplified way of calculating the error but will work 90% of the time. We do not have time to actually calculate the error this year.

7 Maclaurin Polynomial Approximation When developing a polynomial to estimate a function centered at c = 0, you can use the following: P n x = f 0 + f 0 x + f 0 2! x 2 + f 0 3! x fn 0 x n P n x = f n 0 x n *A Maclaurin Polynomial is centered at c = 0.

8 Develop a Maclaurin Polynomial Approximation for the graph of f(x) = sin x. P 1 x = x P n x = f n 0 x n P 0 x = sin 0 0! P 1 0 = f(0) P 1 0 = f (0) 1 st Degree Polynomial Approximation P 1 x = 0 + cos 0 1! P 1 x x 0 = = 0 = x (Tangent line at x = 0) P 1 x x 1 = x = x

9 2 nd Degree Polynomial Approximation P n x = P 2 x = 0 + x + f n 0 x n f x = sin x f 0 = sin 0 = 0 P 2 x = x 3 rd Degree Polynomial Approximation f x = cos x f o = cos 0 = 1 P 3 x = x +? P 3 x = x x3 3!

10 P 4 x 4 th Degree Polynomial Approximation P n x = f n 0 = x x3 3! + f4 x = sin x f 0 = sin 0 = 0 P 4 x = x x3 3! 5 th Degree Polynomial Approximation f 5 x = cos x f 5 o = cos 0 = 1 x n P 5 x = x x3 3! + x5 5!

11 Maclaurin Polynomial for f x = sin x P 5 x = x x3 3! + x5 5! P n x = x x3 3! + x5 5! x7 7! + x9 9! + 1 n x 2n+1 2n + 1! sin x = 1 n x 2n+1 2n + 1!

12 For what values of x does sin x = 1 n x 2n+1 2n + 1!? For all x values in the interval of convergence!! lim n a n+1 a n < 1

13 Homework Using the formula for a Maclaurin Polynomial, Write the first 5 nonzero terms of the polynomial P n x = f n 0 approximation for f x = cos x, f x = sin x and f x = e x. x n Try to find the pattern so you can come up with the formula for the n th term of the series in each case. **This is very important for the BC Calculus test.**

14 cos x sin x 1 x2 2! + x4 4! x6 6! + x8 8! + 1 n x 2n 2n! x x3 3! + x5 5! x7 7! + x9 9! + 1 n x 2n+1 2n + 1! e x 1 + x + x2 2 + x3 6 + x x xn

15 Interval of Convergence for Power Series Representation of cos x P n x = 1 x2 2! + x4 4! x6 6! + x8 8! + 1 n x 2n 2n! cos x = 1 n x 2n 2n!

16 Taylor Series Polynomials How can we develop a Polynomial whose center is not c = 0?

17 Taylor and Maclaurin Series We can take this formula and use it to come up with a Power Series that will exactly represent many elementary functions including: sin x, cos x, e x P n x = f c + f c x c + f c 2! x c 2 + f c 3! x c fn c x c n Taylor Series f n c x c n Maclaurin Series f n 0 x n

18 Finding the third degree Taylor Polynomial for f(x) = ln x centered at c = 1. f n c x c n f x = ln x f x = 1/x f x = 1/x 2 f 1 = 0 f 1 = 1 f 1 = 1 f x = 2/x 3 f 1 = 2 P 3 x = 0 0! x ! x ! x ! x 1 3 P 3 x = x 1 1/2 x /3 x 1 3 GRAPH

19 Find the third order Taylor Polynomial for f x = 2x 3 3x 2 + 4x 5 centered at c = 1. f n c x c n f x = 2x 3 3x 2 + 4x 5 f x = 6x 2 6x + 4 f x = 12x 6 f 1 = 2 f 1 = 4 f 1 = 6 f x = 12 f 1 = 12 P 3 x = 2 0! x ! x ! x ! x 1 3 P 3 x = x x x 1 3 P 3 x = 2 x x x 1 2 Now use the expand key on your calculator to verify that P 3 x = f x (Home > F2 > expand).

20 Find the Taylor Series generated by f x c = 2. f n c x c n = e x at f x = f x = f (x) = f x = f 4 x = e x f 2 0! x 2 0 f 2 + x ! e 2 + e 2 (x 2) + f 2 2! e 2 x f 2 3! 2 x x e 2 6 x e2 x 2 n k=0 e 2 k! x 2 k Graph

21 Find the third Taylor polynomial for f x expanded about c = π/6. f n c x c n f x = sin x f x = cos x f x = sin x f x = cos x = sin x, f π/6 = 1/2 f π/6 = 3/2 f π/6 = 1/2 f π/6 = 3/2 P 3 x = f π 6 +f π 6 x π 6 + f π 6 2! x π f π 6 3! x π 6 3 P 3 x = 1 2

22 f x = sin x

23 The General Formula for a Polynomial Approximation What is the equation of the tangent line, P(x), to a function f(x) at the point c? P x f c = f (c)(x c) P x = f c + f (c)(x c)

24 P 1 x = f c + f (c)(x c) P c P x P c x = c = f c = f (c) = f c (f c constant) In order to make P(x) a better approximation for f(x), I am going to add a term so that the second derivatives also equal one another. P 2 x = f c + f c x c + f c 2 x c 2 The 2 nd Degree Polynomial Approximation

25 P 2 x = f c + f c x c + f c x = c P c = f(c) 2 x c 2 First Derivative of P(x) P x = f c + f (c)(x c) P c = f (c) f c constant (f c constant) Second Derivative of P(x) P x = f (c) P c = f (c)

26 Allowing the 3 rd Derivatives to be Equal P 3 x = f c + f c x c + f c 2 x c 2 + f c 6 x c 3 The 3 rd Degree Polynomial Approximation x = c P c = f(c) P c = f (c) P c = f (c) P 3 c = f 3 (c) f c constant (f c constant) P n x = f c + f c x c + f c 2! x c 2 + f c 3! x c fn c x c n The N th Degree Polynomial Approximation

27 Homework (Finney/Demana) Taylor and Maclaurin WS Section 9.2 (13, 14, 16, 20, 21) Try to do 13 two ways: A: Developing a Taylor Polynomial B: Using a Geometric Series

28 Developing a Taylor Series for a Composite Function How do you develop a Taylor Series for a composite function like sin x 2?

29 Taylor and Maclaurin Series We can take this formula and use it to come up with a Power Series that will exactly represent many elementary functions including: sin x, cos x, e x P n x = f c + f c x c + f c 2! x c 2 + f c 3! x c fn c x c n Taylor Series f n c x c n Maclaurin Series f n 0 x n

30 Find the Maclaurin series for f x = sin x 2 f n 0 x n sin x = x x3 3! + x5 5! x7 7! + x9 9! + 1 n x 2n+1 2n + 1! sin(x 2 ) = (x 2 ) (x2 ) 3 3! + (x2 ) 5 5! (x2 ) 7 7! + (x2 ) 9 9! + 1 n (x 2 ) 2n+1 2n + 1! sin x 2 = x 2 x6 3! + x10 5! x14 7! + x18 9! + 1 n x 4n+2 2n + 1! GRAPH

31 Write the first three nonzero terms and then general term of the power series centered at 0 that will represent the following: g x = e x 1 x e x = 1 + x + x2 2 + x3 6 + x x xn g x = 1 + x + x2 2 + x3 6 + x x xn 1 x g x = 1 + x 2 + x xn 1

32 Write the first three nonzero terms and then general term of the power series centered at 0 that will represent the following: g x = 4x sin 3x sin x = x x3 3! + x5 5! x7 7! + x9 9! + 1 n x 2n+1 2n + 1! sin(3x) = (3x) (3x)3 3! + (3x)5 5! (3x)7 7! + (3x)9 9! + 1 n (3x) 2n+1 2n + 1! g x = (4x) 3x (4x) 3x 3 3! + (4x) 3x 5 5! + (4x) 1 n (3x) 2n+1 2n + 1!

33 Homework (Finney/Demana) Taylor and Maclaurin WS Section 9.2 (2, 5, 14, 20, 23a) Section 9.3 (7, 8)