© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 1 of 57 Chapter 11 Taylor Polynomials and Infinite Series

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 2 of 57  Taylor Polynomials  The Newton-Raphson Algorithm  Infinite Series  Series With Positive Terms  Taylor Series Chapter Outline

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 3 of 57 § 11.1 Taylor Polynomials

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 4 of 57  Determining Taylor Polynomials  nth Taylor Polynomial of f (x) at x = a  Using Taylor Polynomials to Approximate Area Under a Curve  n th Taylor Polynomial of f (x) at x = a  Taylor Polynomials to Make Estimates  The Remainder Formula  Determining the Accuracy of an Estimate Section Outline

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 5 of 57 First Taylor Polynomial of f ( x ) at x = 0 DefinitionExample First Taylor Polynomial of f (x) at x = 0:

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 6 of 57 n th Taylor Polynomial of f ( x ) at x = 0

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 7 of 57 Determining Taylor PolynomialsEXAMPLE SOLUTION Sketch the graphs of f (x) = sin x and its first three Taylor polynomials at x = 0. To determine these polynomials, we must first determine the first, second, and third derivatives of the function and then evaluate the function and its three derivatives at x = 0. Now we can find the first three Taylor polynomials.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 8 of 57 Determining Taylor Polynomials First Taylor polynomial: CONTINUED Second Taylor polynomial: Third Taylor polynomial:

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 9 of 57 Determining Taylor PolynomialsCONTINUED

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 10 of 57 Determining Taylor PolynomialsEXAMPLE SOLUTION Determine the n th Taylor polynomial for f (x) = e x at x = 0. To determine these polynomials, we must first determine the first, second, and third derivatives of the function and then evaluate the function and its three derivatives at x = 0. From the pattern of calculations, it is clear that f (k) (0) = 1 for each k. Therefore,

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 11 of 57 Determining Taylor PolynomialsCONTINUED

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 12 of 57 Taylor Polynomials to Approximate AreaEXAMPLE SOLUTION Use a second Taylor polynomial at x = 0 to estimate the area under the curve y = ln(1 + x 2 ) from x = 0 to x = ½. Since the graph of p 2 (x) is very close to the graph of ln(1 + x 2 ) for x near 0, the areas under the two graphs should be almost the same. The area under the graph of p 2 (x) is

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 13 of 57 n th Taylor polynomial of f ( x ) at x = a

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 14 of 57 Taylor Polynomials to Make EstimatesEXAMPLE SOLUTION Use the second Taylor polynomial of at x = 9 to estimate Here a = 9. Since we want the second Taylor polynomial, we must calculate the values of f (x) and of its first two derivatives at x = 9. Therefore, the desired Taylor polynomial is

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 15 of 57 Taylor Polynomials to Make Estimates Since 9.3 is close to 9, p 2 (9.3) gives a good approximation to f (9.3), that is, to CONTINUED Therefore,

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 16 of 57 The Remainder Formula

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 17 of 57 Determining Accuracy of an EstimateEXAMPLE SOLUTION Determine the accuracy of the estimate in the preceding example. The second remainder for a function f (x) at x = 9 is where c is between 9 and x (and where c depends on x). Here, and therefore We are interested in x = 9.3, and so 9 ≤ c ≤ 9.3. We observe that since c 5/2 ≥ 9 5/2 = 243, we have c -5/2 ≤ 9. Thus

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 18 of 57 Determining Accuracy of an Estimate and CONTINUED Thus the error in using p 2 (9.3) as an approximation of f (9.3) is at most

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 19 of 57 § 11.2 The Newton-Raphson Algorithm

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 20 of 57  The Newton-Raphson Algorithm  Iterations of the Newton-Raphson Algorithm  The Newton-Raphson Algorithm to Make Approximations  Amortization of a Loan Section Outline

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 21 of 57 The Newton-Raphson Algorithm

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 22 of 57 Iterations of the Newton-Raphson AlgorithmEXAMPLE SOLUTION The polynomial f (x) = x 2 + 3x – 11 has a zero between -5 and -6. Let x 0 = -5 and find the next three approximations of the zero of f (x) using the Newton- Raphson algorithm Since, formula (1) becomes With x 0 = -5, we have

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 23 of 57 Iterations of the Newton-Raphson AlgorithmCONTINUED

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 24 of 57 Newton-Raphson to Make ApproximationsEXAMPLE SOLUTION Use three repetitions of the Newton-Raphson algorithm to approximate is a zero of the function f (x) = x 3 – 11. Since clearly lies between 2 and 3, let us take our initial approximation as x 0 = 2. Since, we have

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 25 of 57 Loans When a loan of P dollars is paid back with N equal periodic payments of R dollars at interest rate i per period, the equation relating P, N, R and i is becomes

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 26 of 57 Amortization of a LoanEXAMPLE SOLUTION A mortgage of $100,050 is repaid in 240 monthly payments of $900. Use two iterations of the Newton-Raphson method to determine the monthly rate of interest. Here P = 100,050, R = 900, and N = 240. Therefore, we must solve the equation Let Then Apply the Newton-Raphson algorithm to f (i) with i 0 = 0.01:

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 27 of 57 Amortization of a LoanCONTINUED Therefore, the monthly interest rate is approximately 0.75%.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 28 of 57 § 11.3 Infinite Series

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 29 of 57  Infinite Series  nth Partial Sum  Convergent and Divergent Series  The Geometric Series  Finding the Sum of a Geometric Series  Using Geometric Series  Geometric Series in Application  Geometric Series Using Sigma Notation  Sums of Infinite Series Section Outline

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 30 of 57 Infinite Series DefinitionExample Infinite Series: An infinite addition of numbers

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 31 of 57 n th Partial Sum DefinitionExample nth Partial Sum: The sum of the first n terms of an infinite series, denoted S n

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 32 of 57 Convergent & Divergent Series DefinitionExample Convergent Series: An infinite series whose partial sums approach a limit Divergent Series: An infinite series whose partial sums do not approach a limit

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 33 of 57 The Geometric Series

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 34 of 57 Finding the Sum of a Geometric SeriesEXAMPLE SOLUTION Determine the sum of the following geometric series. Here a = 1 and r = 1/2 3. The sum of the series is

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 35 of 57 Using Geometric SeriesEXAMPLE SOLUTION What rational number has the decimal expansion This number denotes the infinite series a geometric series with a = 173/1000 and r = 1/1000. The sum of the geometric series is

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 36 of 57 Geometric Series in ApplicationEXAMPLE SOLUTION Compute the effect of a $20 billion federal income tax cut when the population’s marginal propensity to consume is 89%. What is the “multiplier” in this case? Express all amounts of money in billions of dollars. Of the increase in income created by the tax cut, (0.89)(20) billion dollars will be spent. These dollars become extra income to someone and hence 89% will be spent again and 11% saved, so additional spending of (0.89)(0.89)(20) billion dollars is created. The recipients of those dollars will spend 89% of them, creating yet additional spending of billion dollars, and so on. The total amount of new spending created by the tax cut is thus given by the infinite series

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 37 of 57 Geometric Series in Application This is a geometric series with initial term 20(0.89) and ratio Its sum is CONTINUED Thus a $20 billion tax cut creates new spending of about $162 billion.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 38 of 57 Geometric Series Using Sigma Notation

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 39 of 57 Sums of Infinite SeriesEXAMPLE SOLUTION Determine the sum of the infinite series.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 40 of 57 § 11.4 Series With Positive Terms

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 41 of 57  Integral Test for Convergence of a Series  Testing for Convergence of a Series  Comparison Test for Convergence of Series  Comparison Test for Negative Terms Section Outline

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 42 of 57 Integral Test for Convergence of Series

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 43 of 57 Testing for Convergence of a SeriesEXAMPLE SOLUTION Use the integral test to determine whether the infinite series is convergent or divergent. Here f (x) = 1/(2x + 1) 3. We know from Chapter 4 that f (x) is a positive, decreasing, continuous function. Also,

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 44 of 57 Comparison Test for Convergence of Series

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 45 of 57 Testing for Convergence of a SeriesEXAMPLE SOLUTION Use the comparison test to determine whether the infinite series is convergent or divergent. Compare the series with the divergent series This series diverges since the k has an exponent less than or equal to one, namely 1. The kth terms of these two series satisfy because By the comparison test, the series diverges.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 46 of 57 Comparison Test for Negative Terms

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 47 of 57 § 11.5 Taylor Series

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 48 of 57  The Power Series  The Taylor Series of f (x) at x = 0  Taylor Series Expansions  Using Taylor Series Expansions  Taylor Series to Estimate Integrals  Types of Power Series Section Outline

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 49 of 57 The Power Series DefinitionExample Power Series: where a 0, a 1, a 2,... are suitable constants, and where x ranges over values that make the series converge to f (x)

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 50 of 57 The Taylor Series of f ( x ) at x = 0 DefinitionExample Taylor Series of f (x) at x = 0:Examples will follow.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 51 of 57 Taylor Series ExpansionsEXAMPLE SOLUTION Find the Taylor series expansion of Therefore,

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 52 of 57 Using Taylor Series ExpansionsEXAMPLE SOLUTION Use the result in the last example to compute ln(4). Take x = -1 in the Taylor series expansion of ln(1 – 3x). Then This infinite series may be used to compute ln(4) to any degree of accuracy required.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 53 of 57 Taylor Series ExpansionsEXAMPLE SOLUTION Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of e x. We begin with the series expansion To obtain the Taylor series for e x 2, we replace every occurrence of x with x 2 in the Taylor series for e x.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 54 of 57 Taylor Series Expansions To obtain the Taylor series for x 3 e x 2, we multiply the last series by x 3, term by term. CONTINUED

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 55 of 57 Taylor Series to Estimate IntegralsEXAMPLE SOLUTION Find an infinite series that converges to the value of the given definite integral. We begin with the series expansion To obtain the Taylor series for e -x 2, we replace every occurrence of x with -x 2 in the Taylor series for e x.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 56 of 57 Taylor Series to Estimate Integrals Integrating, we obtain CONTINUED This infinite series converges to the value of the definite integral. Summing up the four terms displayed gives the approximation , which is accurate to three decimal places. This approximation can be made arbitrarily accurate by summing additional terms.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 57 of 57 Types of Power Series (i)There is a positive constant R such that the series converges for |x| R. (ii)The series converges for all x. (iii)The series converges only for x = 0. Given any power series (sum from k to infinity of a k x k ), one of three possibilities must occur.