Q. 6 - Calculus

Question

Use the graphs of A(x) and f (x) that you drew earlier to illustrate the relationship between A'(x) and f(x) that you just discovered. Use the words increasing, decreasing, positive, and negative in your

discussion.

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{The} \mathrm{function} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{concave} \mathrm{up}. \\ \mathrm{The} \mathrm{first} \mathrm{derivative} \mathrm{of} \mathrm{A}\left(\mathrm{x}\right) \mathrm{shows} \mathrm{a} \mathrm{concave} \mathrm{down} \mathrm{on} \mathrm{the} \mathrm{x}<0 \mathrm{and} \mathrm{a} \mathrm{concave} \mathrm{up} \mathrm{on} \mathrm{the} \mathrm{x}>0. \end{gathered}$$

Step 1. Given information is;

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2 \\ \mathrm{A}\text{'}\left(\mathrm{x}\right)\approx \sum _{}^{\mathrm{n}}\mathrm{f}\text{'}\left(\frac{1}{\mathrm{x}}\right)\frac{1}{\mathrm{x}}-\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)\frac{1}{{\mathrm{x}}^2} \end{gathered}$$

Step 2. Graph and interpretation

$$\begin{gathered} \mathrm{A}\text{'}\left(\mathrm{x}\right)\approx \sum _{}^{\mathrm{n}}\mathrm{f}\text{'}\left(\frac{1}{\mathrm{x}}\right)\frac{1}{\mathrm{x}}-\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)\frac{1}{{\mathrm{x}}^2} \mathrm{is} \mathrm{the} \mathrm{slope} \mathrm{of} \mathrm{graph} \mathrm{A}\left(\mathrm{x}\right). \\ \mathrm{Hence}, \mathrm{using} \mathrm{above} \mathrm{graph}, \mathrm{as} \mathrm{explained} \mathrm{the} \mathrm{function} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{concave} \mathrm{up}. \\ \mathrm{The} \mathrm{first} \mathrm{derivative} \mathrm{of} \mathrm{A}\left(\mathrm{x}\right) \mathrm{shows} \mathrm{a} \mathrm{concave} \mathrm{down} \mathrm{on} \mathrm{the} \mathrm{x}<0 \mathrm{and} \mathrm{a} \mathrm{concave} \mathrm{up} \mathrm{on} \mathrm{the} \mathrm{x}>0. \end{gathered}$$