Tangent To A Function - Limits

[Plutonium88]

Homework Statement

The tangent to the function y=3x(x-3) at point P(2,-6) is the hypotenuse of a right triangle that forms with the coordinate axes. Find Area

The Attempt at a Solution

First of all, i know that i A=BxH/2 so i need the opposite and adjacent sides of this triangle.

Okay so personally i graphed the function at first, and then drew the tangent at the point 2,-6. I then connected the line to the x and y axes. I did this because it says it forms a triangle with the coordinate axes.I read that the coordinate axis are respectively the x and y axis, so i thought that this made sense. Can some one reassure me about this?

Next i thought that i would need the slope of the tangent at that point because it would relate to the slope of hypotenuse of the triangle.

So i took the limit.

Lim (f(x+h)-f(x))/H
H->0

where P(2,-6), so x=2

solving this limit i got

Lim 6x+h-9 = 6x-9
H->0
and when x=2
the limit is = 3

therefore the slope of the tangent is 3. mt=3

so I'm curious, does this mean that the hypotenuse also = 3 because it says the the tangent at the point is the hypotenuse. Or would the hypotenuse side be the tangent line (6x-9)?

So yea this is my dilemma..

 

[LCKurtz]

The line through (2,-6) with slope 3 is the hypotenuse. Figure out its equation (it isn't y = 6x-9) and find where its x and y intercepts are to get the legs of the triangle.

 

[Plutonium88]

The line through (2,-6) with slope 3 is the hypotenuse. Figure out its equation (it isn't y = 6x-9) and find where its x and y intercepts are to get the legs of the triangle.

okay say i have the point (2,-6) and mt=3

here is my diagram.
http://s18.postimage.org/3r9iyue7d/New_Bitmap_Image.png

y= mx+b
-6=3(2)+b
b=-12

So therefore the height or the y value of the triangle is 12. cause it is the y intercept

y=3x-12

to find x intercept, set y=0

3x = 12
x= 4

So there for the base of the triangle is 4, and the height of the triangle is twelve.

so area=bxh/2=12x4/2=24units^2

 

[LCKurtz]

okay say i have the point (2,-6) and mt=3

here is my diagram.
http://s18.postimage.org/3r9iyue7d/New_Bitmap_Image.png

y= mx+b
-6=3(2)+b
b=-12

So therefore the height or the y value of the triangle is 12. cause it is the y intercept

y=3x-12

to find x intercept, set y=0

3x = 12
x= 4

So there for the base of the triangle is 4, and the height of the triangle is twelve.

so area=bxh/2=12x4/2=24units^2

Yes. You have it correct.