Consider the function $$F(x) = \int_0^x \sin(-1.0657 t^2) dt$$ Find a numerical approximation to this function using midpoint Riemann sums and plot it using a spreadsheet. Use equally spaced partitions of size $\Delta x = \pm 0.1$. Then numerically compute an approximation to the derivative $F'(x)$ and compare the resulting graph to that of $\sin(-1.0657 x^2)$.

To get credit for this problem you need to submit separate printouts (during lecture or office hours). of each of the following: \begin{enumerate} \item The spreadsheet Tables 1, 2 and 3 with numerical values as indicated below. To conserve paper, just print the first two pages of each table. \item The spreadsheet Tables 1, 2 and 3 with spreadsheet formulas displayed instead of numbers. To conserve paper, just print the first two pages of each table. \item The graph of $y=F(x)$ using the spreadsheet chart facility. \item The graphs of $y=F'(x)$ and $y=\sin(-1.0657 x^2)$ plotted together, also using the spreadsheet chart facility. \item A printout of this page, after you have clicked the Submit button, indicating that you have answered the questions below correctly. (Don't bother submitting anything else without a WeBWorK printout showing you have at least computed the F(x) values correctly.) \end{enumerate}

The spreadsheet Tables 1 and 2 should have the following format: \begin{itemize} \item The A column should contain $x$ values starting with $x=0$ going up to $x=\pm 10$ in increments of the step size $\pm 0.1$. \item The B column should contain the step size (interval width) $\Delta x$. \item The C column should contain the midpoint of the interval whose left endpoint is in column A and whose width is in column B. \item The D column should contain the value of $f(x)=\sin(-1.0657 x^2)$ at the midpoint in column C \item The E column should contain the (signed) area of the approximating rectangle over the interval whose left endpoint is in column A. \item The F column should contain an approximation to the value of $y=F(x)$ corresponding to the $x$ value in column A. This approximation is obtained by summing up the areas of the approximating rectangles in column E in all the rows above the current one (EXCLUDING the current row). \end{itemize} You spreadsheet should consist of two separate tables, Table 1 using a step size of $\Delta x = 0.1$ (stepping forward), Table 2 using a step size of $\Delta x = -0.1$ (stepping backward).

For the plot chart of $y=F(x)$ follow these basic instructions, modifying as needed to suit your spreadsheet software. \begin{enumerate} \item Select all the cells in your spreadsheet and "Copy" \item Open a new blank worksheet \item Click on the first cell and "Paste Special$\longrightarrow$ Values" into the new worksheet. \item Delete columns B, C, D, and E. The resulting spreadsheet will have $x$ values in column A and corresponding $y=F(x)$ values in column B. \item Select all the cells in your new spreadsheet and "Sort" them in ascending order according to column A. \item Again select all the cells in your spreadsheet and, using the appropriate menu or task bar icon, create a chart plotting the selected cells \item When asked for the type of chart you want, select XY scatter chart with X values in column A. Then select the choice of chart which connects points by smooth curves, without dotting the points. \item The $x$ values on your chart should range from -10 to 10. \item Save the spreadsheet from Step 5 above to start Table 3 for the second part of the problem. \end{enumerate} 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 $F(x) \approx$