Find the area of a triangle.

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April 2nd 2012, 02:09 PM
joshuaa

Find the area of a triangle.

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

(Crying)
April 2nd 2012, 02:13 PM
MathVideos

Re: 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?
April 2nd 2012, 02:35 PM
joshuaa

1 Attachment(s)

Re: Find the area of a triangle.

Could this picture help you?

Attachment 23501
April 2nd 2012, 06:10 PM
Prove It

Re: Find the area of a triangle.

Quote:

Originally Posted by joshuaa

Could this picture help you?

Attachment 23501

Well, let's see. If the function is $\displaystyle \begin{align*} y = \frac{1}{x} \end{align*}$ , the derivative is $\displaystyle \begin{align*} \frac{dy}{dx} = -\frac{1}{x^2} \end{align*}$ . This tells us the gradient of the tangent line at any point x.

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

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

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

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

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

So the area is $\displaystyle \begin{align*} \frac{1}{2}\,\textrm{unit}^2 \end{align*}$
April 2nd 2012, 11:01 PM
biffboy

Re: 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
April 2nd 2012, 11:16 PM
alexmahone

Re: Find the area of a triangle.

Quote:

Originally Posted by Prove It

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

This isn't the equation of a line. (See post #5.)
April 3rd 2012, 02:11 AM
joshuaa

Re: Find the area of a triangle.

Thank you very much. That was helpful :)