answersLogoWhite

0

Bisection Method:

Begin with the interval [0, pi/2]. The midpoint of the interval is x1 = pi/4. Calculate the value of the function at x1: f(x1) = pi/4 - cos(pi/4). Since f(x1) > 0, the solution must be in the interval [0, pi/4]. Now consider the midpoint of this interval, x2 = pi/8. Calculate the value of the function at x2: f(x2) = pi/8 - cos(pi/8). Since f(x2) < 0, the solution must be in the interval [pi/8, pi/4]. Now consider the midpoint of this interval, x3 = 3pi/16. Calculate the value of the function at x3: f(x3) = 3pi/16 - cos(3pi/16). Since f(x3) > 0, the solution must be in the interval [pi/8, 3pi/16]. Continue this process, calculating the midpoint of the interval and the value of the function at the midpoint, until the difference between the lower and upper bounds of the interval is less than or equal to the error of 0.005.

Newton’s Method:

Try an initial guess of x0 = 1. Calculate the value of the function at x0: f(x0) = 1 - cos(1). Calculate the derivative of the function at x0: f'(x0) = 1 + sin(1). Calculate the next x-value using the Newton’s method formula: x1 = x0 - f(x0)/f'(x0) = 1 - (1 - cos(1))/(1 + sin(1)) = 0.6247. Calculate the value of the function at x1: f(x1) = 0.6247 - cos(0.6247). Calculate the derivative of the function at x1: f'(x1) = 1 + sin(0.6247). Calculate the next x-value using the Newton’s method formula: x2 = x1 - f(x1)/f'(x1) = 0.6247 - (0.6247 - cos(0.6247))/(1 + sin(0.6247)) = 0.739. Continue this process until the difference between two successive x-values is less than or equal to the error of 0.005.

Secant Method:

Start with two initial x-values, x0 = 0 and x1 = 1. Calculate the value of the function at x0 and x1: f(x0) = 0 - cos(0) = 0, f(x1) = 1 - cos(1). Calculate the next x-value using the Secant method formula: x2 = x1 - f(x1)(x1 - x0)/(f(x1) - f(x0)) = 1 - (1 - cos(1))(1 - 0)/(1 - cos(1) - 0) = 0.6247. Calculate the value of the function at x2: f(x2) = 0.6247 - cos(0.6247). Calculate the next x-value using the Secant method formula: x3 = x2 - f(x2)(x2 - x1)/(f(x2) - f(x1)) = 0.6247 - (0.6247 - cos(0.6247))(0.6247 - 1)/(0.6247 - cos(0.6247) - 1) = 0.7396. Continue this process until the difference between two successive x-values is less than or equal to the error of 0.005.

User Avatar

David Denton

Lvl 10
2y ago

Still curious? Ask our experts.

Chat with our AI personalities

DevinDevin
I've poured enough drinks to know that people don't always want advice—they just want to talk.
Chat with Devin
BeauBeau
You're doing better than you think!
Chat with Beau
FranFran
I've made my fair share of mistakes, and if I can help you avoid a few, I'd sure like to try.
Chat with Fran
More answers

We start with the interval

[

,

]

=

[

0

,

/

2

]

[a,b]=[0,π/2] since

(

0

)

=

1

f(0)=−1 and

(

/

2

)

=

/

2

1

>

0

f(π/2)=π/2−1>0, indicating a root between

0

0 and

/

2

π/2.

Iteration 1:

=

(

0

/

2

)

/

2

=

/

4

c=(0+π/2)/2=π/4,

(

/

4

)

=

/

4

cos

(

/

4

)

0.553

f(π/4)=π/4−cos(π/4)≈0.553, which is positive. So, we set

[

,

]

=

[

0

,

/

4

]

[a,b]=[0,π/4].

Iteration 2:

=

(

/

4

0

)

/

2

=

/

8

c=(π/4+0)/2=π/8,

(

/

8

)

=

/

8

cos

(

/

8

)

0.132

f(π/8)=π/8−cos(π/8)≈0.132, still positive. So,

[

,

]

=

[

0

,

/

8

]

[a,b]=[0,π/8].

Iteration 3:

=

(

/

8

0

)

/

2

=

/

16

c=(π/8+0)/2=π/16,

(

/

16

)

=

/

16

cos

(

/

16

)

0.191

f(π/16)=π/16−cos(π/16)≈−0.191, which is negative. So,

[

,

]

=

[

/

16

,

/

8

]

[a,b]=[π/16,π/8].

We continue the bisection iterations until the width of the interval is less than

0.005

0.005. The final result is approximately

0.209

x≈0.209.

User Avatar

Suresh Yadav

Lvl 2
1y ago
User Avatar

Add your answer:

Earn +20 pts
Q: Use the Bisection, Newton’s, and Secant Methods to find the solution of the equation x - cos x = 0 over the interval[0,pi/2] accurate to within error = 0.005, wherex is in radian. For Newton’s method, try initial guesses including x0 = 1?
Write your answer...
Submit
Still have questions?
magnify glass
imp