1. This question deals with the derivatives of inverse functions.
(a) Suppose $m(x)$ is an invertible function, that the point $(3, -2)$ lies on the graph of $y = m(x)$, and
that the tangent at the point $(3, -2)$ has slope $1/5$. From this information, what point must be
on the graph of $y = m^{-1}(x)$? And what is the slope of the tangent to $y = m^{-1}(x)$ at that point?
Now that the warmup in part (a) is done, for the rest of this problem suppose $f$ is continuous
and invertible on $\mathbb{R}$, and that $\lim_{x \to 5} \frac{f(x) - 7}{x - 5} = -12$.
(b) Find $f(5)$ and $f'(5)$. Explain how you know their values, and use those values to write an equation
of the line tangent to $y = f(x)$ at $x = 5$.
(c) Find an equation of the line tangent to $y = f^{-1}(x)$ at $x = 7$. Explain how you found the slope of
the tangent line and the $y$-value of the point.
(d) Let $g(x) = f(x^2 - 4)$. Use the chain rule, and facts given above, to find the slope of the line
tangent to $y = g(x)$ at $x = 3$.
(e) Use your answer to part (1d) to find an equation of the line tangent to $y = g^{-1}(x)$ at $x = 7$.
