enjoy:
consider the function y = 1/x
let A be a point on the function
the tangent through A intersects the coordinate axes at C(c,0) and D(0,d)
prove that the area of this triangle is the same no matter where A is located and find that area
enjoy:
consider the function y = 1/x
let A be a point on the function
the tangent through A intersects the coordinate axes at C(c,0) and D(0,d)
prove that the area of this triangle is the same no matter where A is located and find that area
Hello,
Did you draw a sketch ?
The area of the triangle OCD is defined by $\displaystyle \frac{OD \cdot OC}{2}$. Do you agree ?
Distance OD is d and distance OC is c, so $\displaystyle \mathbb{A}=\frac{cd}{2}$
Now, you may know the equation of the tangent to a function. Let m be the absciss of A.
The equation will be :
$\displaystyle y=f'(m)(x-m)+f(m)$
Knowing that $\displaystyle f'(x)=\frac{-1}{x^2}$, determine the equation.
You know that C and D are on this tangent. Hence, replace their coordinates in it and isolate c and d.
So you will have $\displaystyle c=\dots$ and $\displaystyle d=\dots$ with respect to m.
Substitute in the expression of $\displaystyle \mathbb{A}$ and conclude
It's long because I've given all the elements, so try to do and when you can't move on, post