Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

Accepted Answer

Identify the given function: $$f(x) = 2 - x - x$$^{3}$$, with the domain -2$$ \leq x \leq 2$$, and the point of interest x_0$ = \frac{3}{2}. $$Calculate the value of the function at $$x_0$$: compute $$f\left(\frac{3}{2}\right) to $$find the point $$(x_0$, f(x_0$)) on $$the graph of $$f. $$Find the inverse function $$g = f$$^{-1}$$ implicitly or using a CAS, and evaluate g$ at f$$($$x_0$$)$$ to get the point (f$$($$x_0$$), $$x_0$$)$$ on the graph of g$. Determine the derivatives f$$'(x)$$ and g$$'(x)$$ at the points x_0$ and f$$($$x_0$$)$$ respectively, to find the slopes of the tangent lines to f$ and g$ at these points. Plot the following on the same coordinate system: the function f$, its inverse g$, the identity line y = x$, the tangent line to f$ at $$($$x_0$$, f($$x_0$$))$$, the tangent line to g$ at (f$$($$x_0$$), $$x_0$$)$$, and the line segment joining the points $$($$x_0$$, f($$x_0$$))$$ and (f$$($$x_0$$), $$x_0$$)$$. Observe and discuss the symmetry of f$ and g$ about the line y = x$.

Risposta video verificata per un problema simile:

Concetti chiave

Inverse Functions and Their Graphical Relationship

Derivatives and Tangent Lines

Symmetry About the Line y = x