Question

Use Newton's Method to find the root of 2x^2+5=e^x accurate to six decimal places in the interval [3,4]

Answer 1

Let f(x)=2x²+5-eˣ

f'(x)=4x-eˣ

x₁=x₀-f(x₀)/f'(x₀).

Let x₀=3.5 (midrange), 

x₁=3.310862334,

x₂=x₁ (stable result).

So the solution to 6 decimal places is 3.310862.