Various Questoins

[JonF]

I just took a take home calc test this weekend, turned it in this morning. There were a few questions I couldn’t answer. Will you guys tell me (or even better show me) how to solve these?

#1 \int \frac{dx}{3sinx - 4cosx}

#2 Find g’(x) where g is an inverse function of f(x)
f(x) = 3 + x^2 + sin([pi]x) -0.4 < x < 0.4

#3 find the exact value of sin[arcsine(1/3) + arcsine(2/3)]

[Muzza]

#1. You can rewrite 3sin(x) - 4cos(x) as k * sin(x + v), where k and v are constants. Then you just have integrate 1/k * csc(x + v).

#2. Hmm, I wonder if you can leave g(x) in the answer?

#3. Simpify using the addition formula for sine, and find a formula for cos(arcsin(x/y)).

[Zurtex]

Here is always a useful substitution for integrals of that and similar form:

t = \tan \frac{x}{2}


\cos x = \frac{1-t^2}{1+t^2}


\sin x = \frac{2t}{1+t^2}


\frac{dx}{dt} = \frac{2}{1+t^2}

(Not 100% sure I have that last one right)

[Parth Dave]

#2 - f(x) = 3 + x^2 + sin([pi]x)
g(x) is the inverse of f(x). Remember, just because it is written as g(x) doesnt mean that you have to make g a function of x. You are not asked to find the inverse of the function. You are rather being asked to find dy/dx of the inverse.

The inverse of f(x) is:
x = 3 + y^2 + sin([pi]y)

now can you find dy/dx? (think of it as implicit differentiation)

EDIT: nevermind you can't do this. I just realized they don't want dy/dx they instead want g'(x) which is not the same thing.

[Parth Dave]

How does this take-home test system work? What is stopping you from asking these same questions during the test period (other than your conscience - but who ever listens to it anyways?)?

[JonF]

(other than your conscience)

thats about it. It was 25 questions and these are the only 3 that i didn't get right i'm pretty sure.

[Vance]

#3 find the exact value of sin[arcsine(1/3) + arcsine(2/3)]
sin(x+y) = ?
sin(sin^-^1(x))= ?

[Vance]

forget to say it is just another way to solve the problem because the answer given by Muzza can also be applied..

[Muzza]

Seems to me like we were thinking of the same solution. Since you know, sin(x + y) will not only include sin(x) and sin(y), but also cos(x) and cos(y)...

[Vance]

Yeah, i should have thought about what you wrote more deeper....Nothing big right ? --lol

Uhmm, the same!

[JonF]

Thanks for the help guys, those problems were a lot easier then I was making them out to be…

i have a new question…

Is the equation y = \lim_{n \rightarrow \infty} \pm(1 - x^{2n})^{1/2n} a square?

Is \lim_{n \rightarrow \infty} \int \pm(1 - x^{2n})^{1/2n} dx = to 4? i.e. a 2x2 square?