Numerical Analysis
Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Chapter 0.5, Problem 1E

a.

To determine

To prove: f(c)=0 for some 0<c<1 : f(x)=x34x+1 .

a.

Expert Solution
Check Mark

Explanation of Solution

Given information: f(x)=x34x+1

Theorem used:

Intermediate Value Theorem: If f is a continuous function on [a,b] and if y is a number between f(a) and f(b), then there exists a number c with acb such that f(c)=y .

Calculation:

Consider the equation, f(x)=x34x+1

Substituting x=1 in f(x)=x34x+1 , we get

  f(1)=134(1)+1=14+1=2

Substituting x=0 in f(x)=x34x+1 , we get

  f(0)=034(0)+1=00+1=1

Therefore, we have

  2<0<1f(1)<0<f(0)

Using Intermediate value Theorem, there exists c(0,1) such that f(c)=0 .

Hence, it is proved.

b.

To determine

To prove: f(c)=0 for some 0<c<1 : f(x)=5cosπx4 .

b.

Expert Solution
Check Mark

Explanation of Solution

Given information: f(x)=5cosπx4

Theorem used:

Intermediate Value Theorem: If f is a continuous function on [a,b] and if y is a number between f(a) and f(b), then there exists a number c with acb such that f(c)=y .

Calculation:

Consider the equation, f(x)=5cosπx4

Substituting x=1 in f(x)=5cosπx4 , we get

  f(1)=5cosπ4=5(1)4=54=9

Substituting x=0 in f(x)=5cosπx4 , we get

  f(0)=5cosπ(0)4=54=1

Therefore, we have

  9<0<1f(1)<0<f(0)

Using Intermediate value Theorem, there exists c(0,1) such that f(c)=0

Hence, it is proved.

c.

To determine

To prove: f(c)=0 for some 0<c<1 : f(x)=8x48x2+1 .

c.

Expert Solution
Check Mark

Explanation of Solution

Given information: f(x)=8x48x2+1

Theorem used:

Intermediate Value Theorem: If f is a continuous function on [a,b] and if y is a number between f(a) and f(b), then there exists a number c with acb such that f(c)=y .

Calculation:

Consider the equation, f(x)=8x48x2+1

Substituting x=1 in f(x)=8x48x2+1 , we get

  f(1)=8(1)48(1)2+1=88+1=1

Substituting x=0 in f(x)=8x48x2+1 , we get

  f(0)=8(0)48(0)2+1=00+1=1

Substituting x = 12 in f(x)=8x48x2+1, we get

  f(12)=8( 1 2)48( 1 2)2+1=8(1 16)8(14)+1=122+1=12

Therefore, we have

  12<0<1f(12)<0<f(0)

Using Intermediate value Theorem, there exists c(0,12) such that f(c)=0 .

Hence, it is proved.

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Chapter 0 Solutions

Numerical Analysis

Ch. 0.2 - Find the binary representation of the base 10...Ch. 0.2 - Convert the following base 10 numbers to binary....Ch. 0.2 - Convert the following base 10 numbers to binary....Ch. 0.2 - Find the first bits in the binary representation...Ch. 0.2 - Find the first 15 bits in the binary...Ch. 0.2 - Convert the following binary numbers to base :...Ch. 0.2 - Convert the following binary numbers to base...Ch. 0.3 - Convert the following base 10 numbers to binary...Ch. 0.3 - Convert the following base 10 numbers to binary...Ch. 0.3 - For which positive integers k can the number 5+2k...Ch. 0.3 - Find the largest integer k for which in double...Ch. 0.3 - Do the following sums by hand in IEEE double...Ch. 0.3 - Do the following sums by hand in IEEE double...Ch. 0.3 - Prob. 7ECh. 0.3 - Is 1/3+2/3 exactly equal to I in double precision...Ch. 0.3 - Prob. 9ECh. 0.3 - Prob. 10ECh. 0.3 - Does the associative law hold for IEEE computer...Ch. 0.3 - Prob. 12ECh. 0.3 - Prob. 13ECh. 0.3 - Prob. 14ECh. 0.3 - Do the following operations by hand in IEEE double...Ch. 0.3 - Prob. 16ECh. 0.4 - Identify for which values of x there is...Ch. 0.4 - Find the roots of the equation x2+3x814=0 with...Ch. 0.4 - Explain how to most accurately compute the two...Ch. 0.4 - Evaluate the quantity xx2+17x2 where x=910 ,...Ch. 0.4 - Evaluate the quantity 16x4x24x2 where x=812 ,...Ch. 0.4 - Prove formula (0.14).Ch. 0.4 - Calculate the expressions that follow in double...Ch. 0.4 - Prob. 2CPCh. 0.4 - Prob. 3CPCh. 0.4 - Prob. 4CPCh. 0.4 - Prob. 5CPCh. 0.5 - Prob. 1ECh. 0.5 - Find c satisfying the Mean Value Theorem for f(x)...Ch. 0.5 - Find c satisfying the Mean Value Theorem for...Ch. 0.5 - Find the Taylor polynomial of degree 2 about the...Ch. 0.5 - Find the Taylor polynomial of degree 5 about the...Ch. 0.5 - a. Find the Taylor polynomial of degree 4 for ...Ch. 0.5 - Carry out Exercise 6 (a)-(d) for f(x)=lnx .Ch. 0.5 - (a) Find the degree 5 Taylor polynomial centered...Ch. 0.5 - Prob. 9E
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