Power, Taylor, & Maclaurin Series Page 1

Transcription

1 Power, Taylor, & Maclaurin Series Page 1 Directions: Solve the following problems using the available space for scratchwork. Indicate your answers on the front page. Do not spend too much time on any one problem. Note: Let ln(x) denote the natural logarithm of x with base e If, what is? Page 1 of 22

2 Power, Taylor, & Maclaurin Series Page 2 02 If, then must be None of the above 03 If, then is Page 2 of 22

3 Power, Taylor, & Maclaurin Series Page 3 04 What are the first four nonzero terms in the power series expansion of about? 05 The first three terms of the Maclaurin series of are None of the above Page 3 of 22

4 Power, Taylor, & Maclaurin Series Page 4 06 The coefficient of in the Taylor series for about is 07 The coefficient of in the Taylor series for about is Page 4 of 22

5 Power, Taylor, & Maclaurin Series Page 5 08 The coefficient of in the Taylor series for about is 2 09 Which is the approximate value of Taylor polynomial for centered at? obtained by using a fourth-degree Page 5 of 22

6 Power, Taylor, & Maclaurin Series Page 6 10 The Taylor series for centered at is Page 6 of 22

7 Power, Taylor, & Maclaurin Series Page 7 11 The figure above shows the graph of and where is the Taylor polynomial for centered at zero. Which of the following statements is true? I. is a good approximation for II. is a good approximation for III. I only II only III only I and III only I, II, and III Page 7 of 22

8 Power, Taylor, & Maclaurin Series Page 8 12 The graph above shows a function f with a relative minimum at The approximation of near by the second-degree Taylor polynomial centered about is given by Which of the following is true about and Page 8 of 22

9 Power, Taylor, & Maclaurin Series Page 9 13 Let be a function that is everywhere differentiable. The table below provides information about and its first, second, and third derivatives for selected values of x Which of the following best approximates Page 9 of 22

10 Power, Taylor, & Maclaurin Series Page The first three nonzero terms in the Maclaurin series about of are 15 The Maclaurin series for a function is given by What is the value of the fourth derivative of at Page 10 of 22

11 Power, Taylor, & Maclaurin Series Page The graph of the third-degree Maclaurin polynomial that approximates intersects the graph of at how many points. None One Two Three Four 17 What is the approximation of the value of Taylor polynomial about for? obtained by using a fourth-degree Page 11 of 22

12 Power, Taylor, & Maclaurin Series Page The Maclaurin series for is Which of the following is a power series expansion for? 19 The graph of the function represented by the Taylor series, centered at intersects the graph of at Page 12 of 22

13 Power, Taylor, & Maclaurin Series Page Let be the first four nonzero terms of the Maclaurin polynomial used to approximate the value of Determine the area bounded by the graph and the x-axis for 21 For if then Page 13 of 22

14 Power, Taylor, & Maclaurin Series Page For all if then Page 14 of 22

15 Power, Taylor, & Maclaurin Series Page If then Page 15 of 22

16 Power, Taylor, & Maclaurin Series Page If for all real then 25 The Taylor series of a function about is given by What is the value of 0 7 Page 16 of 22

17 Power, Taylor, & Maclaurin Series Page Let be a function that is continuous and differentiable for all The derivative of this function is given by the power series If then Page 17 of 22

18 Power, Taylor, & Maclaurin Series Page Let value of Be the Taylor series for a function What is the the tenth derivative of at? 28 converges for in [ ] [ ) ( ] Cannot be determined Page 18 of 22

19 Power, Taylor, & Maclaurin Series Page What are all the values of for which the series converges? ( ] [ ) [ ] [ ] 30 What are all the values of converges? for which the series All values of x Page 19 of 22

20 Power, Taylor, & Maclaurin Series Page What are all the values of converges? [ ] for which the series * + ( + * ) 32 The power series for what values of? converges only only only All real numbers Page 20 of 22

21 Power, Taylor, & Maclaurin Series Page The power series converges for all real numbers. For values in the interval * +, what is the minimum number of terms of the power series necessary to approximate the value of with an error whose absolute value is less than ? Let E be the error when the Taylor polynomial is used to approximate at Which of the following is true? Page 21 of 22

22 Power, Taylor, & Maclaurin Series Page ( Let be a function whose seventh derivative is ) If is the interval of convergence of the power series for this function, then the Taylor polynomial of degree six centered at will approximate with an error of not more than 36 Let be a function whose Taylor series converges for all If where is the n th derivative of what is the minimum number of terms of the Taylor series centered at necessary to approximate with an error less than ? Three Four Five Six Ten Page 22 of 22