Solve ex-3x2 equals 0 by bisection method?

Answer 1

I'm assuming you're trying to write f(x)=exp(x)-3x2=0 as your function. This has 3 solutions where f(x)=0 between (-0.5,-0.4), (0.85, 0.95) and (3.7, 3.8) I won't find all of the roots, only the one between 3.7 and 3.8 in order to show you the procedure/algorithm.

What we do is choose two points x=a, b on the function where it takes on opposite values. We then find the mid point (a+b)/2 between these values and find the value of the functions at f((a+b)/2) we then choose the smallest interval where the points take on opposite values and repeat until we have a small enough value for x.

We will do our iteration until we get within 0.01 of the solution (i.e. the first number to give an answer between -0.01 and 0.01) and i will be using c instead of writing out (a+b)/2 all the time.

iteration 1: a=3.7 b=3.8 c=3.75 f(c)=0.3335 +ve

iteration 2: a=3.7 b=3.75 c=3.725 f(c)=-0.155 -ve

iteration 3: a=3.725 b=3.75 c=3.7375 f(c)=0.086 +ve

iteration 4: a=3.725 b=3.7375 c=3.73125 f(c)=-0.035 -ve

iteration 5: a=3.73125 b=3.7375 c=3.734375 f(c)=0.025 +ve

iteration 6: a=3.73125 b=3.734375 c=3.732813 f(c)=-0.005 between -0.01 and 0.01

ONE solution for f(x)=exp(x)-3x2=0 is x=3.732813 (actual value is x=3.733079029 so we have an absolute error of 0.000266029, a relative error of 0.00007126262207 and a percentage error of 0.007126262207%)