1 carry out three steps c1 c2 c3 of the bisection method to

1) carry out three steps (c1, c2, c3) of the bisection method to find the roots of the following equations in the givien intervals

A) e^x - 5sinx= 0 [0, Pi/2]

B) xe^x -2 = 0 [0, 1]

Solution

A)

f(x)=e^{x}-5sinx

f(0)=1>0,f(pi/2)=e^{pi/2)-5=-0.1895<0

So by intermediate value theorem there is a root in this interval

x1=(0+Pi/2)/2=Pi/4

f(x1)=e^{Pi/4}-5sin(Pi/4)=-1.34<0

So there is root in interval: (0,x1)

x2=(0+x1)/2=Pi/8

f(x2)=e^{Pi/8}-5sin(Pi/8)=-0.432<0

So there is a root in the interval: (0,x2)

x3=(0+x2)/2=Pi/16

f(x3)=e^{Pi/16}-5sin(Pi/16)=0.241

The root hence must be in the interval: (x3,x2)

The estimate for the root using three iterations is:x=Pi/16

B)

g(x)=xe^x-2

g(0)=-2<0,g(1)=e-2>0

So by intermediate value theorem ,g(x) has a root in (0,1)

x1=(0+1)/2=0.5

g(0.5)~-1.176<0

So there must be a root in the interval: (x1,1)

x2=(x1+1)/2=0.75

g(x2)~-0.41<0

So there must be a root in the interval:(x2,1)

x3=(x2+1)/2=0.875

g(x3)=0.099>0

This is the estimate for the root using bisection method with three iterations.

Based on the sign there must be a root in the interval: (x2,x3)