You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
$\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}$
$$y=x^{3}-3 x^{2}-1, \quad 2 \leq x \leq 5, \quad x_{0}=\frac{27}{10}$$