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Chapter 11 Taylor Polynomials and Infinite Series

Chapter 11 Taylor Polynomials and Infinite Series. Chapter Outline. Taylor Polynomials The Newton-Raphson Algorithm Infinite Series Series With Positive Terms Taylor Series. § 11.1. Taylor Polynomials. Section Outline. Determining Taylor Polynomials

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Chapter 11 Taylor Polynomials and Infinite Series

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  1. Chapter 11Taylor Polynomials and Infinite Series

  2. Chapter Outline • Taylor Polynomials • The Newton-Raphson Algorithm • Infinite Series • Series With Positive Terms • Taylor Series

  3. §11.1 Taylor Polynomials

  4. Section Outline • Determining Taylor Polynomials • nth Taylor Polynomial of f (x) at x = a • Using Taylor Polynomials to Approximate Area Under a Curve • nth Taylor Polynomial of f(x) at x = a • Taylor Polynomials to Make Estimates • The Remainder Formula • Determining the Accuracy of an Estimate

  5. First Taylor Polynomial of f(x) at x = 0

  6. nth Taylor Polynomial of f(x) at x = 0

  7. Determining Taylor Polynomials EXAMPLE Sketch the graphs of f(x) = sinx and its first three Taylor polynomials at x = 0. SOLUTION 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.

  8. Determining Taylor Polynomials CONTINUED First Taylor polynomial: Second Taylor polynomial: Third Taylor polynomial:

  9. Determining Taylor Polynomials CONTINUED

  10. Determining Taylor Polynomials EXAMPLE Determine the nth Taylor polynomial for f(x) = ex at x = 0. SOLUTION 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,

  11. Determining Taylor Polynomials CONTINUED

  12. Taylor Polynomials to Approximate Area EXAMPLE Use a second Taylor polynomial at x = 0 to estimate the area under the curve y = ln(1 + x2) from x = 0 to x = ½ . SOLUTION Since the graph of p2(x) is very close to the graph of ln(1 + x2) for x near 0, the areas under the two graphs should be almost the same. The area under the graph of p2(x) is

  13. nth Taylor polynomial of f(x) at x = a

  14. Taylor Polynomials to Make Estimates EXAMPLE Use the second Taylor polynomial of at x = 9 to estimate SOLUTION 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

  15. Taylor Polynomials to Make Estimates CONTINUED Since 9.3 is close to 9, p2(9.3) gives a good approximation to f(9.3), that is, to Therefore,

  16. The Remainder Formula

  17. Determining Accuracy of an Estimate EXAMPLE Determine the accuracy of the estimate in the preceding example. SOLUTION 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 c5/2 ≥ 95/2 = 243, we have c-5/2 ≤ 9. Thus

  18. Determining Accuracy of an Estimate CONTINUED and Thus the error in using p2(9.3) as an approximation of f(9.3) is at most 0.00000694.

  19. §11.2 The Newton-Raphson Algorithm

  20. Section Outline • The Newton-Raphson Algorithm • Iterations of the Newton-Raphson Algorithm • The Newton-Raphson Algorithm to Make Approximations • Amortization of a Loan

  21. The Newton-Raphson Algorithm

  22. Iterations of the Newton-Raphson Algorithm EXAMPLE The polynomial f(x) = x2 + 3x – 11 has a zero between -5 and -6. Let x0 = -5 and find the next three approximations of the zero of f(x) using the Newton-Raphson algorithm SOLUTION Since , formula (1) becomes With x0 = -5, we have

  23. Iterations of the Newton-Raphson Algorithm CONTINUED

  24. Newton-Raphson to Make Approximations EXAMPLE Use three repetitions of the Newton-Raphson algorithm to approximate SOLUTION is a zero of the function f(x) = x3 – 11. Since clearly lies between 2 and 3, let us take our initial approximation as x0 = 2. Since , we have

  25. 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

  26. Amortization of a Loan EXAMPLE 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. SOLUTION 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 i0 = 0.01:

  27. Amortization of a Loan CONTINUED Therefore, the monthly interest rate is approximately 0.75%.

  28. §11.3 Infinite Series

  29. Section Outline • 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

  30. Infinite Series

  31. nth Partial Sum

  32. Convergent & Divergent Series

  33. The Geometric Series

  34. Finding the Sum of a Geometric Series EXAMPLE Determine the sum of the following geometric series. SOLUTION Here a = 1 and r = 1/23. The sum of the series is

  35. Using Geometric Series EXAMPLE What rational number has the decimal expansion SOLUTION This number denotes the infinite series a geometric series with a = 173/1000 and r = 1/1000. The sum of the geometric series is

  36. Geometric Series in Application EXAMPLE 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? SOLUTION 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

  37. Geometric Series in Application CONTINUED This is a geometric series with initial term 20(0.89) and ratio 0.89. Its sum is Thus a $20 billion tax cut creates new spending of about $162 billion.

  38. Geometric Series Using Sigma Notation

  39. Sums of Infinite Series EXAMPLE Determine the sum of the infinite series. SOLUTION

  40. §11.4 Series With Positive Terms

  41. Section Outline • Integral Test for Convergence of a Series • Testing for Convergence of a Series • Comparison Test for Convergence of Series • Comparison Test for Negative Terms

  42. Integral Test for Convergence of Series

  43. Testing for Convergence of a Series EXAMPLE Use the integral test to determine whether the infinite series is convergent or divergent. SOLUTION Here f(x) = 1/(2x + 1)3. We know from Chapter 4 that f(x) is a positive, decreasing, continuous function. Also,

  44. Comparison Test for Convergence of Series

  45. Testing for Convergence of a Series EXAMPLE Use the comparison test to determine whether the infinite series is convergent or divergent. SOLUTION 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.

  46. Comparison Test for Negative Terms

  47. §11.5 Taylor Series

  48. Section Outline • 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

  49. The Power Series

  50. The Taylor Series of f(x) at x = 0

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