Find (f^ -1) '(296) ; In this context I'm not sure what the inverse of the derivative means.

Question

I was given that f(2005) = 296

and I found f^−1(296)=2005

then I was given f '(2005) = 2.65

and I was asked to find (f^ −1) '(296).

I don't know how to find this exactly, but I tried many approaches. I first tried 2005, but I realized that was wrong. Then I tried 1/2.65, but that was marked wrong. then I tried just 2.65, but I was marked wrong on that too. I'm not sure what it is asking for so I am in a bit of a pickle as to what to do now.

Answer 1

f(2005) = 296   contains the point  (2005, 296)

 

So the inverse of the function will just switch place with the x and y coordinate of the point.

 

f^-1(x) = y    ---->   f(y) = x

 

f^-1(296) = 2005

 

 

We know that the slope of the tangent line at x=2005 is 2.65.  So the equation of the tangent line is

 

f(x) = 2.65(x - 2005) + 296

 

f(x) = 2.65x - 5313.25 + 296

 

f(x) = 2.65x - 5017.25

 

 

Find the inverse of this bolded function, by switching places with f(x) and x.  From there, solve for f(x) to get your inverse function.  Then take the derivative of the resulting function.  After that, evaluate that derivative at x=296.