Question 7 Consider the graph of the second derivative of a function f on the interval (a, 6). It is known that the function f has a zero in the interval (a, 6) and f"(b) > 0. f" Give the order of the root of the equation f(a) = 0 in the interval (a, b). Explain your answer. Question 8
Question 7 Consider the graph of the second derivative of a function f on the interval (a, 6). It is known that the function f has a zero in the interval (a, 6) and f"(b) > 0. f" Give the order of the root of the equation f(a) = 0 in the interval (a, b). Explain your answer. Question 8
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![Question 7
Consider the graph of the second derivative of a function f on the interval [a, b]. It is known that the
function f has a zero in the interval (a, 6 and f'(b) > 0.
f"
Give the order of the root of the equation f(r) = 0 in the interval (a, b). Explain your answer.
Question 8
Consider the equation f(x) = e"(1? – a) + br + b = 0, a € R, bER.
8.1 It is known that the equation has a double root, p. Use the theory to explain why the sequence of
iterations generated with the Secant method will converge to the root for suitable choices of the initial
values r, and r1 .
8.2 Find all values of a and b for which p = 0 is a double root of the equation e" (x? - a) + bx +b = 0.
8.3 Without calculating any iterations, determine whether the choices z, = -0.10 and 11 = 0.4 will
guarantee the convergence stated in 8.1 for the double root p = 0 with a and 6 as in 8.2. You may
assume that the choices of xo = -0.10 and r1 = 0.4 are close enough to the root.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2bfae4e2-14d9-473c-9fc8-9308968cb7f2%2Fa3040abf-326a-4276-81a7-44eec6e1d385%2Fg8sw89q_processed.png&w=3840&q=75)
Transcribed Image Text:Question 7
Consider the graph of the second derivative of a function f on the interval [a, b]. It is known that the
function f has a zero in the interval (a, 6 and f'(b) > 0.
f"
Give the order of the root of the equation f(r) = 0 in the interval (a, b). Explain your answer.
Question 8
Consider the equation f(x) = e"(1? – a) + br + b = 0, a € R, bER.
8.1 It is known that the equation has a double root, p. Use the theory to explain why the sequence of
iterations generated with the Secant method will converge to the root for suitable choices of the initial
values r, and r1 .
8.2 Find all values of a and b for which p = 0 is a double root of the equation e" (x? - a) + bx +b = 0.
8.3 Without calculating any iterations, determine whether the choices z, = -0.10 and 11 = 0.4 will
guarantee the convergence stated in 8.1 for the double root p = 0 with a and 6 as in 8.2. You may
assume that the choices of xo = -0.10 and r1 = 0.4 are close enough to the root.
Transcribed Image Text:Question 9
9.1
Consider the following theorem, it's proof and statements to justify the steps of the proof.
Match the bold and underline steps of the proof in Question 9.1.1 - 9.1.3 to all the statement(s) that
justi-
fies (justify ) the steps.
Theorem: A function f is defined on an interval (a, b) with a zero pE (a, b). Assume the sequence
{In} generated with the Newton-Raphson method converges to p.
lentil
If f" is continuous in p and f'(p) + 0, then lim
Proof:
f(z)
then g'(p) = 0 and g" is continuous at p.
f'(x)
It can be shown that if g(z)
= I -
From the Taylor-expansion for g at p we have g(rn) = g(p) + (In - P)g(p) + 0.5(z, - p)*g"(cn) with
C, between rn and p. It follows that
Question Proof
Statements
9.1.1
Xn+1 =p+0.5(z, - p)°g"(cn) (a) g" is continuous.
(b) f'(p) = 0.
(c) p is a fixed point of g.
(d) lim In = p
9.1.2
-en+1 = 0.5 e g"(cn)
len+il
0.5|g"(cn)|
(e) Cn is between In and p.
(f) The sequence of iterations converges to p.
f(z)f"(x)
f(x)
(h) The definition of the error is used.
9.1.3
lim
= 0.5 g"(p)|
(g) g'(x) =
lental - f"(p)
2f'(p)
lim
(i) The iterative formula for Newton-Raphson's method is used.
9.2 The function f with f(x) = (1 - 4)e* satisfies all the conditions of the above mentioned theorem
with p = 4 a simple root of the equation f(r) = (r – 4)e" = 0. The Newton Raphson iterative formula
- 4x, + 4
reduces to In+1 =
In - 3
Use the function f(z) = (r-4)e* to illustrate the theorem in 9.1. (Do not calculate the approximations
In.) You may assume that f'(r) = e"(x – 3) and f"(z) = e"(z – 2).
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