If anyone could take some time to answer any or all of the problems below, it would be a massive help. I've gone through my homework and highlighted a few problems I've had as I've gone through. Much of it involves lack of mechanical skill on my part...if someone would like to explain exactly how the textbook is arriving at the sol'ns below, again, it would be a huge help.

Thanks,

Graeme

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1.What's going on when the textbook/sol'n manual busts out the absolute value symbols, such as in:

Since 12x^2 + 4 > for all x, we have

f''(x) > 0 <-> x^2 - 1 > <-> |x| > 1

2.Mechanics;I feel like I'm limited by my ability to manipulate functions. Could someone explain how the sol'n manual as gone from point A to point B in the following examples:

(a) $\displaystyle f'(x) = \frac{2x\sqrt{x+1} - x^2 * 1/(2\sqrt{x+1)}}{x+1} = \frac{x(3x+4)}{2(x+1)^{3/2}} $

(b) $\displaystyle f'(x) = \frac{\sqrt{x^2 +1} - x*\frac{2x}{2\sqrt{x^2+1}}}{(x^2 + 1)^{1/2})^2} = \frac{x^2 + 1 - x^2}{(x^2 +1)^{3/2}} $

(c) $\displaystyle f'(x) = x * \frac{1}{2}(5-x)^{-1/2}(-1)+(5-x)^{1/2} *1 = \frac{1}{2}(5-x)^{-1/2}[-x+2(5-x)] = \frac{10-3x}{2\sqrt{5-x}}$

3.And I'm most likely missing something but how the hell does $\displaystyle x^2$ end up as 1:

lim $\displaystyle \frac{x/x}{\sqrt{x^2 + 1}/x}$ =

x->inf

lim $\displaystyle \frac{x/x}{\sqrt{x^2 + 1}/\sqrt{x^2}}$ =

x->inf

lim $\displaystyle \frac{1}{\sqrt{1 + 1/x^2}}$

x->inf

lim $\displaystyle \frac{1}{\sqrt{1 + 0}}$ = 1

x->inf

4.How do you know when there are noasymptotes? Are there just no HAs if there denominational power is smaller than the numerator highest power? How about VAs?

e.g. $\displaystyle y = x-3x^{1/3}$

5.How do you when to make acommon denominator? For example, why can't I just evaluate the intervals of increase and decrease right off the bat with the following function, as opposed to doing what the book does:

e.g. $\displaystyle y' = 1 - x^{-2/3} = 1 - \frac{1}{x^{2/3}} = \frac{x^{2/3} - 1}{x^{2/3}}$

6.Could someone take me through how to findinterceptsfor the following function (I don't get the whole x=(2n+1)pi/2 thing):

$\displaystyle f(x) = \frac{cosx}{2+sinx}$