Show that every triangle formed by the coordinate axes and a tangent line to y = 1/x ( for x > 0) always has an area of 2 square units.

Hint: Find the equation of the tangent line at x = a. (It will contain a’s as well as x and y.) Then find the

x-and y-intercepts for that line to find the lengths of sides of the right triangle.

Step-by-step explanation:

given a point the equation of a line with slope m that passes through the  given point is

or equivalently

.

Recall that a line of the form , the y intercept is b and the x intercept is .

So, in our case, the y intercept is and the x  intercept is .

In our case, we know that the line is tangent to the graph of 1/x. So consider a point over the graph . Which means that

The slope of the tangent line is given by the derivative of the function evaluated at . Using the properties of derivatives, we get

. So evaluated at we get

Replacing the values in our previous findings we get that the y intercept is

The x intercept is

The triangle in consideration has height and base . So the area is

So regardless of the point we take on the graph, the area of the triangle is always 2.

d i think

step-by-step explanation:

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this