Question

Use the equation for A(x) that you just found to find A'(x). What do you notice about A'(x) and f (x)?

Short answer

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

Step-by-step solution

Step 1. Given information is:

Suppose A(x) is a function that for each x>0 is equal to area under the graph of f(t)=t² from 0 to x.

Step 2. Calculating A'(x)

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