I have the solution as shown in the photo but I don`t know how did they get the numbers to compare, such as $8 \times 10^{-3}$ and $1 \times 10^{-3}$. Obviously, the upper numbers were obtained by adding the lower numbers. So how to get the lower numbers? I tried to find the relative error between every 2 numbers in the table such as $|\frac{0.705 - 0.686}{0.705}|$ but didn`t work.
Please see the photo:
Thank YOU
I think this is a terrible question. How can you say what value of $f'(1.399)$ is worst if you are not provided with any? I plotted the data with a linear trendline below. It looks to me like the value for $f(1.399)$ is incorrect because everything else fits a line so neatly. I think the values you are to consider for $f'(1.399)$ are the various values available of secants $\frac {f(x)-f(x')}{x-x'}$. Maybe you are supposed to fix $x'$ at $1.399,$ maybe not. Using $x=1.398, x'=1.399$ you get a slope of $1$, while if you use $x=1.400,x'=1.399$ you get a slope of $5$. If we think the true slope is about $2.5$ from the trendline, $5$ is further away and I would cite that. No other pair of points makes a slope that far from $2.5$. Probably you are expected to assess $f''(1.399)$ by using the three point formula centered at $1.399$ and will find a positive value. My look at the data claims $f'(1.399)=0$ and the best estimate of $f'(1.400)$ is also $2.5$ but unless you are given more specific instructions I would say many answers are possible. Good luck guessing what the writer was thinking. The provided solution does not answer the question at all unless there is some text to go with it.