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UAE_Math_Grade_12_Vol_1_SE_718383_ch3
33. x + 1 = 0,x = 0 GO01962-Smith-v1.cls July 4, 2016 13:38
2
49. Given the graph of y = f(x), draw in the tangent lines used
0
in Newton’s method to determine x and x after starting at
1
2
2
34. x + 1 = 0,x = 1 x = 2. Which of the zeros will Newton’s method con-
0
0
4x − 8x + 1 verge to? Repeat with x =−2 and x = 0.4.
2
35. = 0,x =−1 0 0
4x − 3x − 7 0
2
( x + 1 ) 1∕3
36. x − 2 = 0,x = 0.5
0
............................................................ y
37. Use Newton’s method with (a) x = 1.2 and (b) x = 2.2 to 2
0
0
2
3
find a zero of f(x) = x − 5x + 8x − 4. Discuss the differ-
ence in the rates of convergence in each case.
38. Use Newton’s method with (a) x = 0.2 and (b) x = 3.0 to
0
0
find a zero of f(x) = x sin x. Discuss the difference in the -2 2 x
rates of convergence in each case.
39. Use Newton’s method with (a) x =−1.1 and (b) x = 2.1
0
0
3
to find a zero of f(x) = x − 3x − 2. Discuss the difference
in the rates of convergence in each case.
40. Factor the polynomials in exercises 37 and 39. Find a rela-
tionship between the factored polynomial and the rate at 50. What would happen to Newton’s method in exercise 49
which Newton’s method converges to a zero. Explain how if you had a starting value of x = 0? Consider the use of
0
the function in exercise 38, which does not factor, fits into Newton’s method with x = 0.2 and x = 10. Obviously,
0
0
this relationship. (Note: The relationship will be explored x = 0.2 is much closer to a zero of the function, but
0
further in exploratory exercise 1.) which initial guess would work better in Newton’s method?
Explain.
In exercises 41–44, find the linear approximation at x = 0 to 51. Show that Newton’s method applied to x − c = 0 (where
2
show that the following commonly used approximations are c > 0 is some constant) produces the iterative scheme x =
n+1
valid for “small” x. Compare the approximate and exact values 1 (x + c∕x ) for approximating √ c. This scheme has been
for x = 0.01, x = 0.1 and x = 1. 2 n n
knownforover2000years.Tounderstandwhyitworks,sup-
√ 1 √
41. tan x ≈ x 42. 1 + x ≈ 1 + x pose that your initial guess (x ) for c is a little too small.
0
2 √
√ 1 How would c∕x compare to c? Explain why the average
43. 4 + x ≈ 2 + x 44. e ≈ 1 + x 0 √
x
4
............................................................ of x and c∕x would give a better approximation to c.
0
0
52. Show that Newton’s method applied to x − c = 0 (where n
n
45. (a) Find the linear approximation at x = 0 to each of f(x) = and c are positive constants) produces the iterative scheme
2x
2
1
(x + 1) ,g(x) = 1 + sin(2x) and h(x) = e . Compare x n+1 = [(n − 1)x + cx 1−n ] for approximating √ c.
n
n
your results. n n
2
(b) Graph each function in part (a) together with its linear 53. Applying Newton’s method to x − x − 1 = 0, show that
approximation derived in part (a). Which function has (a) if x = 3 , x = 13 ; (b) if x = 5 , x = 34 ; (c) if x = 8 ,
1
0
0
1
0
the closest fit with its linear approximation? 89 2 8 3 21 5
x = ; (d) The Fibonacci sequence is defined by F = 1,
1
1
55
46. (a) Find the linear approximation at x = 0 to each of f(x) = F = 1, F = 2, F = 3 and F = F + F for n ≥ 3. Write
x
3
n−1
n−2
n
2
4
sin x, g(x) = tan −1 x and h(x) = sinh x = e − e −x . each number in parts (a)−(c) as a ratio of Fibonacci num-
2
Compare your results. bers. Fill in the subscripts m and k in the following: If x =
0
(b) Graph each function in part (a) together with its linear F n+1 , then x = F m (e) Assuming that Newton’s method
approximation derived in part (a). Which function has F n 1 F k
the closest fit with its linear approximation? converges from x = 3 , determine lim F n+1 .
0
47. For exercise 7, compute the errors (the absolute value of 2 n→∞ F n
the difference between the exact values and the linear 54. Determine the behavior of Newton’s method applied
approximations). Thinking of exercises 7a–7c as numbers to (a) f (x) = 1 (8x − 3); (b) f (x) = 1 (16x − 3), (c) f (x) =
√ 1 2 3
of the form 4 16 +Δx, denote the errors as e(Δx) (where f (x) if 1 5 1 , f(x) = f (x) if 1 1 5 1 < x < 1, f(x) =
1
(32x − 3); (d)f(x) when f(x) = f (x) if
Δx = 0.04, Δx = 0.08 and Δx = 0.16). Based on these three
Copyright © McGraw-Hill Education 48. Useacomputeralgebrasystem(CAS)todeterminetherange with x = < x ≤ 2 3 8 < x ≤ 4
2
5
computations, conjecture a constant c such that e(Δx) ≈
1
and so on,
2
2
c ⋅ (Δx) .
4
3
. Does Newton’s method converge to a zero
0
4
of f?
of x’s in exercise 41 for which the approximation is accurate
to within 0.01. That is, find x such that |tan x − x| < 0.01.
237