Find the area of a triangle enclosed by axes and tangent to y = 1/x

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- Apr 2nd 2012, 02:09 PMjoshuaaFind the area of a triangle.
Find the area of a triangle enclosed by axes and tangent to y = 1/x

(Crying) - Apr 2nd 2012, 02:13 PMMathVideosRe: Find the area of a triangle.
Where do you run into your problem? Finding the tangent line? Also, there's an infinite amount of triangles that could be made. Do they specify anything else?

- Apr 2nd 2012, 02:35 PMjoshuaaRe: Find the area of a triangle.
Could this picture help you?

Attachment 23501 - Apr 2nd 2012, 06:10 PMProve ItRe: Find the area of a triangle.
Well, let's see. If the function is $\displaystyle \displaystyle \begin{align*} y = \frac{1}{x} \end{align*}$, the derivative is $\displaystyle \displaystyle \begin{align*} \frac{dy}{dx} = -\frac{1}{x^2} \end{align*}$. This tells us the gradient of the tangent line at any point x.

So the tangent line is of the form $\displaystyle \displaystyle \begin{align*} y = \left(-\frac{1}{x^2}\right)x + c = -\frac{1}{x} + c \end{align*}$.

c is the y intercept. To find the x intercept, we'll let y = 0, and we find that

$\displaystyle \displaystyle \begin{align*} 0 &= -\frac{1}{x} + c \\ \frac{1}{x} &= c \\ x &= \frac{1}{c} \end{align*}$.

So the height of the triangle is c, the length is $\displaystyle \displaystyle \begin{align*} \frac{1}{c} \end{align*}$, which means the area is

$\displaystyle \displaystyle \begin{align*} A &= \frac{1}{2}(c)\left(\frac{1}{c}\right) \\ &= \frac{1}{2} \end{align*}$

So the area is $\displaystyle \displaystyle \begin{align*} \frac{1}{2}\,\textrm{unit}^2 \end{align*}$ - Apr 2nd 2012, 11:01 PMbiffboyRe: Find the area of a triangle.
Don't think that is correct. Example consider tangent at (1,1) Gradient is -1 Equation of tangent is y=-x+2 Intercepts are both 2 Area of triangle is 2

In general consider tangent at (k,m) Gradient of tangent is -1/k^2 Get the equation of the tangent and the intercepts. Knowing also that km=1 wil get area of triangle=2 - Apr 2nd 2012, 11:16 PMalexmahoneRe: Find the area of a triangle.
- Apr 3rd 2012, 02:11 AMjoshuaaRe: Find the area of a triangle.
Thank you very much. That was helpful :)