Root Approximation 1

Calculus Level 1

Consider this equation

$2^x=x^2$

There are 2 trivial solutions, namely $x=2$ and $x=4$

However, the equation is known to also have a negative solution. Find the negative solution.

Give your answer to 3 decimal places.

1 solution

Agnishom Chattopadhyay
Jul 2, 2016

We want to find the root of $2^x = x^2$ which means we want to find the zero of the function $f(x) = 2^x - x^2$

Let's use newton raphson method to get the root near -1

Clearly, $f'(x) = \ln 2 \cdot 2^x - 2 x$

We run the following in Mathematica

1
2
3

In[4]:= T[x_]:= x-(2^x-x^2)/(2^x Log[2]-2x)
NestList[N[T[#]]&,-1,4]
Out[5]= {-1,-0.786923,-0.766843,-0.766665,-0.766665}