Course:MATH102/Question Challenge/2008 December Q2a

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Question

The following problems require little or no computation. Put the answer in the box provided. You may use space here for supporting reasoning or computations, if any, for which part-marks would be awarded.

Suppose $y=f(x)$ is a function having the following properties: $f(1)=2,f^{\prime }(1)=3$. Suppose $g(x)$ is the inverse function for $f(x)$. Then the slope of the tangent line to $g(x)$ at the point $x=2$ is:

Hints

Hint
At corresponding points, the slopes of inverse functions are related.. how?

Solutions

Solution

Given: $f(1)=2$, we know that the point $(1,2)$ is on the graph of the function $f(x)$, and hence the point $(2,1)$ is on the graph of the function $g(x)$, i.e. $g(2)=1$. We also use the fact that at these corresponding points, the slopes of the tangent lines to the function and its inverse are reciprocals. Hence, since $f^{\prime }(1)=3$, we conclude that $g^{\prime }(2)=1/3$. For a summary of such properties of inverse functions, see slides 55-59 in [1]