For the function y=(e^2x)+(3x)-(10), use Newton's method and the calculator method to find the x value for which y=15. Please show your work so that I can understand the question! Thank you so much!! it means alot!!

Answer 1

so you are solving
e^(2x) + 3x - 25 = 0

let y = e^(2x) + 3x - 25
y' = 2e^(2x) + 3

newtons's formula says
X = x - f(x)/f'(x)

where x is your starting x value and X is the new value.
Hopefully your X will approach x, when that happens you have the solution.

X = x - (e^(2x) + 3x - 25)/(2e^(2x) + 3)
= (2xe^(2x) - e^(2x) + 25)/(2e^2(2x) + 3)

I made a rough sketch of y = e^(2x) + 3x - 25
and noticed that there was an x-intercept, which would be your solution, at appr x = 1.5
So I will make that my starting x
X ---- x
-------- 1.5
1.5096 ---1.50915695
1.50951695 -- 1.509515688
1.509515688 -- 1.50951688

Wow, got the answer correct to 9 decimal places after only 3 iterations.

The key thing is that you start with an initial guess close to the real number.