Hi

Could someone please show me how to do question 12?

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- Apr 3rd 2010, 12:57 AMxwrathbringerxImplicit Differentiation
Hi

Could someone please show me how to do question 12? - Apr 3rd 2010, 01:19 AMSudharaka
Dear xwrathbringerx,

By differentiation with respect to x,

$\displaystyle 3x^2+3y^{2}\frac{dy}{dx}=3\left[y+x\frac{dy}{dx}\right]$

$\displaystyle [3y^{2}-3x]\frac{dy}{dx}=3y-3x^2$

$\displaystyle [y^2-x]\frac{dy}{dx}=y-x^2$

$\displaystyle \frac{dy}{dx}=\frac{y-x^2}{y^2-x}$

Can you do it from here?? If you need more help don't hesitate to ask.

Hope this helps you. - Apr 3rd 2010, 03:29 AMxwrathbringerx
Hmm how exactly do I know that the tangents are horizontal and vertical?

Also could you please show me how to do (b) and (c)...I have no idea how to do them (Crying) - Apr 3rd 2010, 04:47 AMHallsofIvy
A horizontal line has slope 0, a vertical line does not have a slope.

For (b) you are to show that $\displaystyle \frac{1+ (y/x)^3}{y/x}= 3/x$.

Starting from the original equation, $\displaystyle x^3+ y^3= 3xy$, divide both sides by $\displaystyle x^3$ to get $\displaystyle 1+ (y/x)^3= 3y/x^2= 3(y/x)/x$. Now multiply both sides by y/x.

For (c) you need to know that $\displaystyle x^3+ y^3= (x+ y)(x^2- xy+ y^2)= (x+y)(xy)(x/y- 1+ y/x)$. - Apr 3rd 2010, 05:03 AMxwrathbringerx
hmmm for (b), when x = infinity, shouldnt y/x = infinity too? (since y/x = x/3 + y^2/3x^2?)

- Apr 3rd 2010, 05:43 AMSudharaka
Dear xwrathbringerx,

How did you get, $\displaystyle \frac{y}{x}=\frac{x}{3}+\frac{y^2}{3x^2}~??$

This is obviously a wrong equation since, when multiplied by $\displaystyle 3x^2$ you get, $\displaystyle 3xy=x^3+y^2$ which contradicts the original equation. - Apr 3rd 2010, 05:46 AMxwrathbringerx
the 1st part of (b) says "show ...". I cross multiplied this to obtain it with y/x =

- Apr 3rd 2010, 05:53 AMSudharaka
- Apr 3rd 2010, 05:57 AMxwrathbringerx
Ooops

Bur Sudharaka, doesnt that still give infinity? - Apr 3rd 2010, 06:09 AMSudharaka
Dear xwrathbringer,

Suppose when $\displaystyle x\rightarrow{\infty}\Rightarrow{y\rightarrow{-\infty}}$. Therefore generally we cannot say that $\displaystyle \frac{y}{x}\rightarrow{\infty}$ since we don't know what happens to y when x goes to infinity. Does this solve your problem?? - Apr 3rd 2010, 06:16 AMxwrathbringerx
OH

But I dont get it.Why did the question say y/x -->**-1**? - Apr 3rd 2010, 06:24 AMSudharaka
Now consider, $\displaystyle \frac{1+\left(\frac{y}{x}\right)^{3}}{\frac{y}{x}} =\frac{3}{x}$

When, $\displaystyle \frac{y}{x}\rightarrow{-1}\Rightarrow{}\frac{1+\left(\frac{y}{x}\right)^{3 }}{\frac{y}{x}}\rightarrow{0}\Rightarrow{\frac{3}{ x}\rightarrow{0}\Rightarrow{x\rightarrow{\infty}}}$

Get the idea??? If you have any questions please don't hesitate to ask. I'll try my best to answer them.