# Two Analysis II Problems

• Apr 13th 2009, 12:43 PM
JoanF
Two Analysis II Problems
Hi!
I need some help in these exercices:

Exercice 1:
1.1 - Prove that tan(x+y)= [tan(x)+tan(y)] / [1−tan(x).tan(y)] and then use this to prove that arctg(x) + arctg(y) = arctg ( [x+y] / [1-xy] ) indicating any possible restriction of arguments.

1.2 - Conclude that: pi / 4 = arctg (1 / 2) + arctg (1 / 3) and pi / 4 = 4.arctg (1 / 5) - arctg (1 / 239)

1.3 - Use this last equation and Taylor's
polynomials of arctg(x) to prove that: pi = 3.14159...

Exercice 2:
2.1 - Considering a function such that f''(x) + f(x) = 0 for all x belonging to lR, and that f(0) = 0, f'(0) = 0. Prove that all derivatives exist.

2.2 - Calculate Taylor's polynomial of this fucntion at the point 0 and the respective rest.

2.3 - Conclude that any function satisfying these conditions is necessarily null.

I hope someone can help me!

Thank you very much (Nod)
• Apr 13th 2009, 01:46 PM
Jhevon
This is analysis II? seems like it is more calc 3 or something
Quote:

Originally Posted by JoanF
Hi!
I need some help in these exercices:

Exercice 1:
1.1 - Prove that tan(x+y)= [tan(x)+tan(y)] / [1−tan(x).tan(y)]

try using the fact that tan(x + y) = sin(x + y)/cos(x + y), which you know the addition formulas for. expand them, and simplify and change everything to tangents.

Quote:

and then use this to prove that arctg(x) + arctg(y) = arctg ( [x+y] / [1-xy] ) indicating any possible restriction of arguments.
plug in arctan(x) for x and arctan(y) for y in the previous formula. you should be able to take it from there.

Quote:

1.2 - Conclude that: pi / 4 = arctg (1 / 2) + arctg (1 / 3) and pi / 4 = 4.arctg (1 / 5) - arctg (1 / 239)
just plug in the relevant values for x and y in the equation from the previous part

Quote:

1.3 - Use this last equation and Taylor's
polynomials of arctg(x) to prove that: pi = 3.14159...

we know that $\frac {\pi}4 = \arctan (1/2) + \arctan (1/3)$. and we can write $\arctan x + \arctan y$ as a single arctangent function. use the Taylor series of this function. plug in the relevant x and y values to get $\frac {\pi}4$. you can estimate this using the partial sums up to 5 or 6 decimal places accuracy. then, multiply that answer by 4

Quote:

Exercice 2:
2.1 - Considering a function such that f''(x) + f(x) = 0 for all x belonging to lR, and that f(0) = 0, f'(0) = 0. Prove that all derivatives exist.

2.2 - Calculate Taylor's polynomial of this fucntion at the point 0 and the respective rest.

2.3 - Conclude that any function satisfying these conditions is necessarily null.

did you do differential equations? begin by solving the differential equaiton. you will perhaps know what to do after that