I was just watching session 2 Clip 2, where the professor solves for the area of the triangle formed by any tangent line to y = 1/x and the y- and x-axes. While he finds the x-intercept of the tangent in a straightforward manner, using the equation y - y0 = -(1/(x0)^2) * (x - x0), from the fact that the derivative of y = 1/x is y = -1/x0^2, making the equation of a tangent line to y = 1/x at some point (x0,y0) what I listed above in point-slope form. But when finding the y-intercept, he takes a shortcut using the fact that y = 1/x is its own inverse function. Can anyone explain this shortcut
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If you rename x to y and y to x, you end up with the same form of equation, and that equation will have the same value for its intercept (but for x).
Let me use an example in physics to explain inverse function:
We know a formula that converts F temperature to C temperature
C = 5/9 (F-32)
To find C, what we do is subtract 32 from F and then multiply by 5/9.
To find F, we have to reverse the process.
Multiply C by 9/5 and then add 32.
Therefore, F = (9/5 * C) +32
Similarly, the inverse function of y = 1/x is x =1/y (replacing y with x and x with y)
Now, you know the equation of the tangent line as y-1/xo = -1/xo^2 (x-xo)
Interchange x and y, xo and yo
Plugging the value y=0, you get y = 2yo
In simple words, if y= 1/x you get x=2xo. If x = 1/y you get y = 2yo
Hope you understand this.