Various Questoins

1. Jul 6, 2004

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)]

2. Jul 6, 2004

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)).

Last edited: Jul 6, 2004
3. Jul 6, 2004

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)

Last edited: Jul 7, 2004
4. Jul 6, 2004

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.

Last edited: Jul 7, 2004
5. Jul 6, 2004

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?)?

6. Jul 7, 2004

JonF

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

7. Jul 7, 2004

Vance

$$sin(x+y) = ?$$
$$sin(sin^-^1(x))= ?$$

8. Jul 7, 2004

Vance

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

9. Jul 7, 2004

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)...

10. Jul 7, 2004

Vance

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

Uhmm, the same!

11. Jul 7, 2004

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?

Last edited: Jul 7, 2004