4.10E: Exercises for Section 4.9

In exercises 1 - 5, write Newton’s formula as for solving .

1)

2)

Answer

3)

4)

Answer

5)

In exercises 6 - 8, solve using the iteration , which differs slightly from Newton’s method. Find a that works and a that fails to converge, with the exception of

6) with

Answer

fails, works

7) with

8) What is the value of for Newton’s method?

Answer

In exercises 9 - 16, compute and using the specified iterative method.

Start at

a. and

b.

9)

10)

Answer

a.
b.

11)

12)

Answer

a.
b.

13)

14)

Answer

a.
b.

15)

16)

Answer

a.
b.

In exercises 17 - 26, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess that is not the exact root.

17)

18)

Answer

or

19)

20)

Answer

or

21)

22) choose

Answer

23)

24)

Answer

or

25)

26)

Answer

In exercises 27 - 30, use Newton’s method to find the fixed points of the function where ; round to three decimals.

27)

28) on

Answer

29)

30)

Answer

Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton’s method to the derivative function to find its roots, instead of the original function. In exercises 31 - 32, consider the formulation of the method.

31) To find candidates for maxima and minima, we need to find the critical points Show that to solve for the critical points of a function , Newton’s method is given by .

32) What additional restrictions are necessary on the function ?

Answer

We need to be twice continuously differentiable.

In exercises 33 - 40, use Newton’s method to find the location of the local minima and/or maxima of the following functions; round to three decimals.

33) Minimum of

34) Minimum of

Answer

35) Minimum of

36) Maximum of

Answer

37) Maximum of

38) Maximum of

Answer

39) Minimum of closest non-zero minimum to

40) Minimum of

Answer

In exercises 41 - 44, use the specified method to solve the equation. If it does not work, explain why it does not work.

41) Newton’s method,

42) Newton’s method,

Answer

There is no solution to the equation.

43) Newton’s method, starting at

44) Solving starting at

Answer

It enters a cycle.

In exercises 45 - 48, use the secant method, an alternative iterative method to Newton’s method. The formula is given by

45) a root to accurate to three decimal places.

46) Find a root to accurate to four decimal places.

Answer

47) Find a root to accurate to four decimal places.

48) Find a root to accurate to four decimal places.

Answer

49) Why would you use the secant method over Newton’s method? What are the necessary restrictions on ?

In exercises 50 - 54, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton’s method.

50)

Answer

Newton: iterations, secant: iterations

51)

52)

Answer

Newton: three iterations, secant: six iterations

53)

54)

Answer

Newton: five iterations, secant: eight iterations

In exercises 55 - 56, consider Kepler’s equation regarding planetary orbits, , where is the mean anomaly, is eccentric anomaly, and measures eccentricity.

55) Use Newton’s method to solve for the eccentric anomaly when the mean anomaly and the eccentricity of the orbit round to three decimals.

56) Use Newton’s method to solve for the eccentric anomaly when the mean anomaly and the eccentricity of the orbit round to three decimals.

Answer

In exercises 57 - 58, consider a bank investment. The initial investment is . After years, the investment has tripled to

57) Use Newton’s method to determine the interest rate if the interest was compounded annually.

58) Use Newton’s method to determine the interest rate if the interest was compounded continuously.

Answer

59) The cost for printing a book can be given by the equation . Use Newton’s method to find the break-even point if the printer sells each book for