Question

These two q's for my homework I am very confused on how to do:

First Question:
Using Newton’s method, approximate the value of √5 up to 2 decimal points
starting with x1 = 3.


2nd Question:
Thomas Malthus was an economist that predicted that the population grows exponentially while the food supply grows linearly. If p(t) gives the population and f(t) gives the number of people that can be supported by the amount of food produced at t years, we would be concerned about when p(t) = f(t). After this point, the pop. would be beyond capacity that the environment can support. Suppose p(t)= 1000e^(0.02t) and f(t)= 30t + 2000. Using Newton’s method with x1 = 60, find x2 to approximate when the functions intersect.

Answer 1

let x = √5

then x^2 - 5

let y = x^2 - 5
y' = 2x

Newton said:
newx = x - y/y'
= x - (x^2 - 5)/(2x)
= (2x^2 - x^2 + 5)/2x
= (x^2 + 5)/(2x)

x newx
3 2.3333....
2.333... 2.23809...
2.23809... 2.23606... could have stopped here
2.23606.. 2.236067878
2.236067978.. 2.236067977


2. let y = 1000 e^(.02t) - 30t - 2000
find y' and follow the same steps as I did in #1

Since you only have to find x2, (only one step)
I would not bother simplifying the expression
and just do one step of
newx = x = y/y'