Should be done in google spreadsheet. Please help
In our first example, we will consider the function /(r). We will list the values of the function at numbers between 0 and 1, spaced apart by h = 0.01. 1. Enter: . x in cell A1 . f (x) in cell B1 . . h in cell E1 2. Enter 0.01 in cell E2 . This is the spacing between the numbers. 3. Fill the cells A2-A102 with the numbers 0, 0.01, 0.02. .... Enter O in cell A2 and enter -A2+E$2 in cell A3 . Select cell A3 and drag down the blue dot to cell 4102. 1. Fill the cells B2-B102 with the values of f(x). Enter =42-2 in cell B2 and drag down the blue dot to cell B102 . The cells C2-C101 should contain the increments of the values in column B. 5. Enter =B3-B2 in cell C2 . Select cell C2 and drag down the blue dot to cell C101 . i Note that there is one fewer increment than values of the function, because you need the next value to compute the increment. Enter a spacing of 1 in cell E2. You will see the sequence of all odd numbers in ON column C. Explain why this is the case. What does this observation have to do with Exercise 2 in Lab 2 (An interesting fact about polynomials)? 6. Graph the function f and the increments by selecting cells A2-C101 , and clicking on Insert and then Chart. In the panel on the right, you may need to change Chart type from Column chart to Line chart. You also need to click on Add X-axis and select x. Lastly, click on the three little dots next to x under Series and select Remove. x. f(x) and Delta f The blue graph is the graph of the function f(x) while the red graph shows the increment between two adjacent values. Note that the red graph is very close to the r-axis, because each increment is computed for two very close values, for example at a = 0.3 we compute / (0.31) - (0.30). In order to obtain the derivative, we need to divide by h, so for example, we need to compute f(0.31) - f(0.30) 0.01 7. Enter =C2/E$2 in cell D2 . Select cell D2 and drag down the little blue dot to cell D101 . 8. Add column C to the graph. Select the graph, click on the three little dots and Edit chart to open the Chart editor. Click on Add Series and then on the grid symbol. Enter D2: D101 and select OK. x, 1(x) and Delta The new, yellow graph is the graph of the derivative of f. Note that the numbers in column D are just 100 times as large as those in column C. The main advantage of considering the numbers in column D, as opposed to the mere increments in column C, is that these numbers are not very sensitive to the choice of h. For example, if choose a very small number for , like h = 0.00001, then the values for a = 0.3 in columns C and D change from f(0.31) - f(0.3) = 0.0061, f(0.31) - f(0.3) ! = 0.61 0.01 to f (0.3001) - f(0.3) = 0.000060001, f(0.3001) - f(0.3) = 0.60001 0.0001 As a result, the red graph corresponding to a smaller h would be even closer to the z-axis, but the yellow graph would remain almost the same. Exercise 2 Repeat the same construction for the function /(x) = 2". Select File, Download, PDF and Export to save your work as a PDF. Upload this PDF to Gradescope
