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For each function f (x) and interval [a, b] in Exercises 81–86, use the Intermediate Value Theorem to argue that the function must have at least one real root on [a, b]. Then apply Newton’s method to approximate that root.

f(x)=x2-5,[a,b]=[1,3]

Short Answer

Expert verified

For the function, f(1)<0&f(3)>0and the function is continuous, so the function must have at least one root betweenx=1&x=3.

The approximate root of the function isx=4721

Step by step solution

01

Step 1. Given information

The given function is f(x)=x2-5.

Given interval is[a,b]=[1,3].

02

Step 2. Use Intermediate Value Theorem.

Determine function value at x=1.

f(x)=x2-5f(1)=12-5f(1)=-4

Determine the function valuer at x=3.

f(x)=x2-5f(3)=32-5f(3)=4

Here localid="1648506233549" f(1)<0&f(3)>0and function is continuous.

so the function must have at least one root in the interval[1,3].

03

Step 3. approximation of root of function.

Differentiate the function.

limzxf(z)-f(x)z-x=limzxz2-5-x2-5z-x=limzxz2-x2z-x=limzxz+x=2x

Take x1=1&f(1)=4for the approximation of the root.

Find derivative of the function at x=1

f'(x)=2xf'(1)=2

Determine the equation of a tangent to f(x)by using the points1,f(1)=(1,-4).

role="math" localid="1648504538068" y-(-4)=2(x-1)y=2x-6

Determine the roots of the tangent to f(x).

0=2x-6x=3

Find the value of f at x=3.

f(3)=32-5f(3)=4

So x=3is not a root of function.

04

Step 4. approximation of root of function. 

Take x2=3&f(3)=4for the next approximation of the root.

Find derivative of the function at x=3

f'(3)=2(3)f'(3)=6

Determine the equation of a tangent to function by using the points (3,f(3))=(3,4)

y-4=6(x-3)y=6x-14

Determine the roots of the tangent to the function

0=6x-14x=146=73

Find the value of the function at x=73

f73=732-5f73=49

So x=73is not a root of function.

05

Step 5. approximation of root of function. 

Take x=73&f73=49to approximate the root.

Find derivative of the function atx=73.

f'73=273f'73=143

Determine the equation of a tangent to function by using the points73,f73=73,49.

y-49=143x-73y=143x-949

Determine the roots of the tangent to the function

role="math" localid="1648505995938" 0=143x-949x=9442=4721

Find the value of the function atx=4721.

f4721=47212-5f4721=0.009

So is x=4721not a root of function but the function value is very close to zero for this.

so approximate root isx=4721.

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