Find all lines that are tangent to the curve y = x^3 and are also parallel to the line 3x - y = 5

Any help will be appreciated!

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- Mar 12th 2008, 08:20 AMhasanbalkanTangent and parallel lines
Find all lines that are tangent to the curve y = x^3 and are also parallel to the line 3x - y = 5

Any help will be appreciated! - Mar 12th 2008, 08:42 AMwingless
Firstly,

$\displaystyle y=x^3$

$\displaystyle y'=3x^2$

The lines must be parallel to 3x-y = 5, $\displaystyle y=3x-5$. So their slope must be $\displaystyle m=3$.

Let our line be $\displaystyle y=m\cdot x + C$, m is the slope and C is a constant. As $\displaystyle m=3$ it'll become $\displaystyle y=3x + C$

At the tangent point, the line must satisfy $\displaystyle m=3=3x^2$.

$\displaystyle 3x^2 = 3$

$\displaystyle x^2=1$

$\displaystyle x= \pm 1$

So the line passes the points (-1,f(-1)) and (1,f(1)).

(-1,-1), (1,1)

The line is $\displaystyle y=3x + C$

And it makes $\displaystyle C=2$ for (-1,1)

and $\displaystyle C=-2$ for (-1,1)

The lines are then

$\displaystyle y=3x + 2 $ and $\displaystyle y=3x - 2$

*bu arada, türk müsün? =)*