How Do You Solve 4x^2 - e^x = 0 Using Numerical Methods?

[stunner5000pt]

Find the intervals containing solutions to the following equation

$$4x^2 - e^x = 0$$
I haven o clue on where to start really?
WOuldi expand e^x using taylor series? i mean one could do this
$$x - 2ln x = ln 4$$
so then would i do log expansion by taylor series? Or would i use bisection method? But how owuld i pick the two points for bisection method? Please assist!

And as always, your help is greatly appreciated!

 

[saltydog]

Find the intervals containing solutions to the following equation

$$4x^2 - e^x = 0$$
I haven o clue on where to start really?
WOuldi expand e^x using taylor series? i mean one could do this
$$x - 2ln x = ln 4$$
so then would i do log expansion by taylor series? Or would i use bisection method? But how owuld i pick the two points for bisection method? Please assist!

And as always, your help is greatly appreciated!

Why not just plot:

$$y1(x)=4x^2$$

and

$$y2(x)=e^x$$

interesections, bingo-bango.

 

[quicknote]

I would suggest using a numerical method such as the bisection method or the Newton-Raphson method to find the intervals containing solutions to the equation 4x^2 - e^x = 0. These methods involve iteratively narrowing down the intervals until a solution is found within a certain tolerance.

To use the bisection method, you would first need to pick two points within the interval of interest and then evaluate the function at those points. If the function values have opposite signs, then there is a root within that interval. You can then divide the interval in half and repeat the process until the interval is small enough to contain the solution.

Alternatively, you could use the Newton-Raphson method which involves choosing an initial guess for the solution and then using the derivative of the function to refine the guess until it converges to the actual solution. This method may be more efficient than the bisection method, but it requires knowledge of the derivative of the function.

In this case, both methods could potentially work, but it may be more straightforward to use the bisection method since it does not require knowledge of the derivative. Whichever method you choose, it is important to carefully select the initial interval or guess to ensure that the method will converge to the correct solution. I would also recommend using a computer program or calculator to help with the calculations and iterations.