Consider the function $$F(x) = \int_0^x \sin(-1.0657 t^2) dt$$

We now compare the graphs of $y=F'(x)$ and $y=\sin(-1.0657 x^2)$. Recall that $$F'(x) = \lim_{h\to 0}\frac{F(x+h)-F(x)}{h}$$ Since we have only computed values of $F(x)$ in steps of size 0.1, the smallest value of $h$ we can use is $h=0.1$ giving us an approximation: $$F'(x)\approx\frac{F(x+0.1)-F(x)}{0.1}$$

Start with the spreadsheet you saved from the previous part, whose first column contains $x$ values from -10 to 10 going up in increments of 0.1. The second column contains the corresponding values $F(x)$. Now start a third column computing approximations to $F'(x)$ using the above approximation.

The third column should end one row early, corresponding to $x=9.9$, since computing cell C201 $=F'(10)$ by this method, would require knowing $F(10+0.1)=F(10.1)$, which we haven't computed.

Finally fill in the fourth column by computing the values of $\sin(-1.0657 x^2)$ corresponding to the $x$ values in the first column. This completes the Table 3 spreadsheet, which you are supposed to hand in (in two forms, one displaying numbers, the second displaying formulas).

You will notice that the values in the third column $F'(x)$ roughly match those in the fourth column $\sin(-1.0657 x^2)$ for $x$ values in the range $-4\le x\le 4$, but tend to diverge from each other outside this interval. Select this region $-4\le x\le 4$ in your table and paste it into a new spreadsheet using the "Paste Special$\longrightarrow$ Values" command. Then delete the second column (containing $F(x)$ values), leaving a spreadsheet containing $x$ values (from -4 to 4) in the first colum, corresponding $F'(x)$ values in the second column, and corresponding $\sin(-1.0657 x^2)$ values in the third column. Select all the cells in this spreadsheet and, using the appropriate menu or task bar icon, create a chart plotting the selected cells fas in the first part of problem.

The following questions are designed to tell you whether your methods are correct, and to debug your spreadsheets. Answer them by referring to Tables 1, 2 and 3 prepared according to the above instructions. You will need to submit a printout of the response from Webwork indicating that your answers are correct.

When $x=1.1$ then $F'(x) \approx$
When $x=2.2$ then $F'(x) \approx$
When $x=-0.8$ then $F'(x) \approx$
When $x=-1.7$ then $F'(x) \approx$