MAT137 Calculus! Lecture 48

Transcription

1 official website MAT137 Calculus! Lecture 48 Today: Taylor Series Applications Next: Final Exams

2 Important Taylor Series and their Radii of Convergence 1 1 x = e x = n=0 n=0 x n n! ln(1 + x) = x n (R = 1) n=0 ( 1) n 1 x n cos x = ( 1) n (2n)! n=0 x 2n sin x = ( 1) n x 2n+1 (2n + 1)! tan 1 x = n=0 n=0 n ( 1) n x 2n+1 2n + 1 (R = ) (R = 1) (R = ) (R = ) (R = 1)

3 Applications of Taylor Series Example 1 Find a Taylor polynomial for sin x at 0 whose graph differs by less than from the graph of sin x on the interval [ π, π].

4 Applications of Taylor Series Example 1 Find a Taylor polynomial for sin x at 0 whose graph differs by less than from the graph of sin x on the interval [ π, π].

5 Evaluating Integrals Example Approximate the integral 0 e x2 dx to within of its actual value.

6 Evaluating Integrals Example Explain why he definite the integral I = dx exists x 3 2 Use Taylor series to evaluate I. (Your final answer can be a series)

7 Evaluating Integrals Example Explain why he definite the integral I = dx exists x 3 2 Use Taylor series to evaluate I. (Your final answer can be a series) 3 Can we use the same approach to evaluate the integral 2 1 J = x 3 dx?

8 Finding higher order derivatives Example 4 Let f (x) = e x2. Show that f (2n) (0) = (2n)! n!

9 Second Derivative Test Example 5 ( The second second derivative test ) Let f be an analytic function with domain R. Let a, x R. Using Lagrange formula for the remainder, there is c between x and a such that f (x) = f (a) + f (a)(x a) + f (c) (x a) 2. 2 Suppose that f (a) = 0. Prove that 1 f has a local maximum at a if f 0 throughout an interval whose interior contains a. 2 f has a local minimum at a if f 0 throughout an interval whose interior contains a.

10 A Limit by Taylor Series Example 6 (A limit by Taylor series) Evaluate lim (6x 5 sin 1 x x 6x 4 + x 2 )

11 More Problems

12 Solving Differential Equations Example 7 Consider the initial-value problem y 2xy = x, y(0) = 1. Assume that y may be represented by a power series in x. That is, y = a k x k. k=0 1 Explain why a 0 = 2. 2 Find a series solution of the differential equation.

13 Solving Differential Equations Example 7 Consider the initial-value problem y 2xy = x, y(0) = 1. Assume that y may be represented by a power series in x. That is, y = a k x k. k=0 1 Explain why a 0 = 2. 2 Find a series solution of the differential equation. 3 Show that your series solution is equal to the function 1 2 (3ex2 1)

14 Irrationality of e Theorem e is irrational Proof. Hint: Assume by contradiction that e is rational and use the Taylor series expansion for e x to arrive at a contradiction.

15 Rolling Dice Example 8 In one throw of two dice, the probability of getting a roll of 7 is p = 1/ The sum of two dice If you throw the dice repeatedly, the probability that a 7 will appear for the first time at the nth throw is q n 1 p, where q = 1 p = 5/6. The expected number of throws until a 7 first appears is n=1 nq n 1 p. Find the sum of this series.

16 Finding the Sum of a Series Example 9 (Mystery Series) Consider the series ( 1) k k=1 4 k x 2k 1 Determine the interval of convergence of the power series. 2 Find the function it represents on this interval.

17 Evaluating Integrals Example 10 Consider the definite integral 0 x t 2 sin t dx 1 Evaluate the integral using integration by parts. 2 Evaluate the integral by integrating the Maclaurin series for x 2 sin x.

18 Applications of Taylor Series Example 11 (Approximating ln 2) Consider the following two ways to approximate ln 2. 1 Use the Taylor series for ln(1 + x) centred at 0 and evaluate it at x = 1. Write the resulting series. 2 Use the property ln(a/b) = ln a ln b and the series from the previous part to find the Taylor Series for f (x) = ln ( 1 + x ) centred at 0 1 x 3 At what value of x should the series be evaluated to approximate ln 2? 4 Using four terms of the series, which of the two series gives the best approximation to ln 2? why?

19 Applications of Taylor Series Challenge Question 1 Let f (x) = (x 2 2x + 3) 2. Find f (100) (1)

20 Applications of Taylor Series Example 12 Find the sum of the series (x + 2) n n=0 (n + 3)!

21 Applications of Taylor Series Example 13 Evaluate the limit lim ( 1 x 0 sin x 1 x )

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