Newton's Method (1 Viewer)

[fashionista]

hi everybodee
i have a couple of newtonian method questions im stuck on, mainly because i don't think you can apply newton's method to them but thats just me.
can anyone please help me with these?
1. (i) show that y=x^3+x-5 is monotonic increasing for all values of x.
for this question i know what monotonic increasing means and i know that this function is monotonic increasing for all values of x i just dont know how to answer this question.
(ii) explain how you know this y=x^3+x-5 has only one real root.
I can solve this using 4 unit techniques but seeing as we're in 3 unit i need help solving it this way.

and this question
6 (ii) (there's a diagram showing the graphs y=lnx and y=e^(-x) and where they intersect...its close to x=1 (but not on the x- axis..duh)) Find an approximation to the root of y=e^(-x)-lnx by starting with X1=1 and using Newton's method once.

please help meeeeeeee
luv me

 

[George W. Bush]

Originally posted by fashionista
1. (i) show that y=x^3+x-5 is monotonic increasing for all values of x.

First derivative.

(ii) explain how you know this y=x^3+x-5 has only one real root.

f(x) < 0 as x-> - infinity, f(x) > 0 as x-> infinity - crosses the axis and is cts therefore has a root
As monotonic increasing only one root

6 (ii) (there's a diagram showing the graphs y=lnx and y=e^(-x) and where they intersect...its close to x=1 (but not on the x- axis..duh)) Find an approximation to the root of y=e^(-x)-lnx by starting with X1=1 and using Newton's method once.

Use Newton's method......

 

[untamedanimal]

for the last one
(subbing in x = 1)
let f(x) = e^(-x) - In x -----> f(1) = e^(-1)

f'(x) = -e^(-x) - 1/x -----> f'(1) = -e^(-1) + 1

X2 = 1 - [e^(-1)] / [-e^(-1) + 1]
= 0.42 (2 decimal places)

I have a feeling i might have made a mistake but atleast the methods right.

 

[Xayma]

untamedanimal you should have picked a point where you expect the value to be negative, and then the approx to the root is whatever value is closer to 0.

 

[fashionista]

thank u!!

 

[fashionista]

ok i must just be crap at this method but im not getting through much of my sheet (maybe because my 3 u book is not with me anymore whoever took it give it back)
this question is still with the one ....gah ill quit explaining

6 (i)(diagram of graphs y=lnx and y=e^(-x) and where they intersect) show that the point of intersection of these two curves has an x value between 1 and 1.5
please help me with this one

 

[Xayma]

If there is an intercept of f(x) and g(x) between x₁ and x₂

Then

f(x₁) > g(x₁)
f(x₂) < g(x₂)

 

[fashionista]

thank u so much!

 

[untamedanimal]

untamedanimal you should have picked a point where you expect the value to be negative, and then the approx to the root is whatever value is closer to 0.

Does it make a difference? It asked to use x = 1 as a first estimate

 

[Xayma]

yes but if there isnt a negative value you cant deduce that it is a root (it could be an asymptote etc)