ࡱ> a7 -8bjbjUU 7|7|2*l((($L&&&Pv<4L#<"8l.l.l.;;;;;;;$= ??n;(l.+l.l.l.;58~;555l."p(8;5l.;55V;r(V;8 !L &2DV;V;L;0#<V;?D4fQ@V;5LL MACROBUTTON MTEditEquationSection Equation Section 1 SEQ MTEqn \r \h \* MERGEFORMAT  SEQ MTSec \r 1 \h \* MERGEFORMAT Named Ranges 5 Animating the Construction of a Tangent Graph Open Excel and save the new file as Tangent on your floppy disk. Centre Justify all Cells. In Cell B35 type x, in Cell C35 type y and in Cell D35 type dy. Then in Cell B36 type -4 and press Return, then select Edit ( Fill ( Series, and, for the series in columns, choose a step value of 0.1 to finish at a value of 9, which should be in Cell B166 Name the range B36 to B166 x, the range C36 to C166 y and the range D36 to D166 dy. In Cell B28 type h and in Cell B29 type 0.05, then name Cell B29 as h. Also, in Cell C28 type n and in Cell C29 type 0.1, then name Cell C29 as n. We will see later that h will be used as part of the argument in the calculation of a numerical approximation to the gradient value, and n will be used to control the speed at which a point of tangency moves along the curve in question. Next select Tools ( Options( Calculation Tab, and if not already done, set up the Dialogue Box as shown below which allows the spreadsheet to deal with circular references: Notice again, that the value in the Maximum change text box is unimportant to the present set up, as are the Workbook options, so they can be left with whatever default values they have. In Cell B31 type in k and in Cell C31 type in t. Highlight Cells B31:C32 and then select Insert ( Name ( Create, and press OK. Now click once in B36 and then in the Name Box and call this first value of x first.x not forgetting to press Return to enter it! Similarly, name the Cell B166 last.x, also using the Name Box. Next, in B32 type = MOD(k+n,last.x-first.x) and in C32 type = k+first.x. Thus the driver Cell is B32, which is calling itself, and Cell C32 is adjusting the driver Cells output for the required x range that happens to start at a negative value rather than at zero. Now press Alt+F11 to open up the VBA Editor and then select Insert ( Module. We will actually be inserting five functions within the same module, some of which you will notice call one another. Type in the function code shown below: (Do not attempt to write in the horizontal lines they are put in automatically by the VBA Editor!)  After typing in the above code return to the Excel worksheet and in Cell C36 type in = f(x+0) ,and press Return (Reasons for this rather strange argument to the function will be given on Page 2 of the next exercise: Programming 1, under the Title: Second Fixit.). Cell C36 should now contain the value 5.56 to 2 d.p. Double click on the fill handle of Cell C36 to automatically fill this function down alongside the corresponding x values. However, this will only produce an entire column of 5.56s!, the reason being that the worksheet has been put in Manual Calculation mode Therefore, simply press the Special Function Key F9 once in order to recalculate the worksheet. Similarly, in Cell D36 type in either of the following forms: = (f(x+h) f(x-h))/(2*h) or = df(x+0,h). Either of which will calculate the value of the tangent at the respective x value by using the Central Difference approximation to the Derivative with the value of the small increment h from Cell B29. The value in the Cell D36 should now be -7.04 to 2 d.p. Double click on the fill handle of Cell D36 to automatically fill down this derivative function alongside the corresponding x values, and again press the Special Function Key F9 once in order to recalculate the worksheet. Highlight Cells F28:F30 and right justify them. In Cell F28 type y value =, in Cell F29 type dy/dx value = and in Cell F30 type d2y/dx2 = Now in Cell G28 type in =f(t), in Cell G29 type =df(t,h) and in Cell G30 type =d2f(t,h) Notice here that we do not need to write the argument as =f(t+0), or =df(t+0,h), etc. refer again to Page 2 of the next exercise: Programming 1, under the Title: Second Fixit for details. The theory contained within this next section is given in an Appendix at the end of this booklet. Next, click in Cell I28 and left justify, then type into it Tangent Coords. In Cell I29 type in = t and in Cell J29 type in = f(t). Then in Cell I30 type in = xgradpoint(t,h)+t and in Cell J30 type in = ygradpoint(t,h)+f(t) Similarly, click in Cell L28 and left justify, then type into it Tangent Graph. In Cell L29 type in = t and in Cell M29 type in 0. Then in Cell L30 type in = t and in Cell M30 type in = df(t,h) Now without any of the x, y or dy ranges selected click on the Chart Wizard icon in the Standard Toolbar. Next, in the 1st Chart Wizard window Standard Types Tab, double click on the XY (Scatter) picture and then on the smooth curve without data points present. Click Next and in the Data Range Tab of the following Dialogue Box in the Data Range Input Box just type x,y and press Next. This will automatically select the required x-y range for the graph. Now continue to set the remaining graph options so that no titles, keys or axes names are showing, just leaving the graph alone. Next right click anywhere on the background shading of the graph and select Format Plot Area, from the drop down menu. Then in the Area frame click on None to remove the background shading, also right click on any single horizontal, graph Gridline and from the drop down menu select Clear. Next put the mouse over the vertical, y-axis and right click once, select Format Axis, Scale Tab and alter the Minimum y-axis value to 5, the Maximum to 8, the Major unit to 1 and the Minor unit to 0.5. Similarly with the x-axis but choose values Minimum value to 4, and the Maximum to 9. Make sure that all check boxes alongside the range boxes are clear or else the graph will continue to automatically adjust its axis. Next, resize the graph so that it covers the Cells A3:G26 Notice that if you hold down the Alt button whilst moving or resizing then the graph will Snap-to the spreadsheet gridlines. Now highlight Cells I29:J29 and move the mouse pointer back into Cell I29 and towards the top of this Cell but still inside the Cell until the pointer changes from a large cross back to the more familiar pointer shape. Then left click the mouse and whilst holding down this button drag these Cells anywhere into the graph area and let go of the mouse button this is just a quick way of Copying and Special Pasting this coordinate point onto the graph. You will then be presented with a Paste Special Dialogue Box that should be filled out as shown below.  Click on OK and notice that the graph border should be highlighted. We need to find the point we have just copied to the graph and recall that the easiest way to do this is to use the cursor buttons on the keyboard that said, just press the Down Arrow cursor button once and that should highlight the single point. Notice that this should be confirmed by the border around Cells I28:J29 becoming coloured. Now we can double click on this single point on the graph, and then fill out the Format Data Point Dialogue Box as shown below: After pressing OK and clicking on any empty Cell to de-select this point, press the Special Function Key F9 to see the point moving along the graph. Notice how adjusting the value of n in Cell C29 adjust the speed of the point along the section of the graph being displayed. Now we plot the variable sized tangent line onto this graph in the following, similar manner to the last point above. Highlight Cells I29:J30 and as above move the mouse pointer to the inside top border of Cell I29 whilst the block remains highlighted. When the pointer actually looks like a pointer hold the left mouse button down and drag the block to anywhere within the graph. Let go of the mouse button and then fill out the Paste Special Dialogue Box, which will have automatically appeared, in as shown to the right You may then see a fine line coming out from the previously formed point of tangency, if you can clearly double click on it as it is then do so, if not use the cursor key arrows to highlight it before you double click it. Then fill out the Format Data Series Dialogue Box as shown below:  Once again after pressing OK and first clicking on an empty Cell, press the Special Function Key F9 repeatedly to see the whole tangent moving along the graph and changing its size accordingly. Next we plot the Tangent Graph of the function f(x). Once again without any of the x, y or dy ranges selected click on the Chart Wizard icon in the Standard Toolbar. Next, in the 1st Chart Wizard window Standard Types Tab, double click on the XY (Scatter) picture and then on the smooth curve without data points present. Click Next and in the Data Range Tab of the following Dialogue Box in the Data Range Input Box just type x,dy and press Next. This will automatically select the required x-dy range for the graph. Again, continue to set the remaining graph options so that no titles, keys or axes names are showing, just leaving the graph alone. Next right click anywhere on the background shading of the graph and select Format Plot Area, from the drop down menu. Then in the Area frame click on None to remove the background shading, then click OK. Also right click on any single horizontal, graph Gridline and from the drop down menu select Clear. It is important that we use the same axes settings as in the graph of the function, to remind you they were: Minimum y-axis value equal to 5, Maximum y-axis value to 8. Minimum x-axis value to 4 and the Maximum x-axis value to 9. Change them now. Next, resize the graph so that it covers the Cells H3:N26 Again notice that if you hold down the Alt button whilst moving or resizing then the graph will Snap-to the spreadsheet gridlines. This puts the two graphs right against one another. To show exactly how the Tangent Graph is constructed we need to place a vertical line, having the same length as that of the tangent, onto the x-axis in the Tangent Graph. The tip of this line represents the height of a point that must be plotted on the tangent curve for the particular value of x at the point of tangency in the previous graph of the function under consideration. To do this we highlight Cells L29:M29 and move the mouse pointer back into Cell L29 and, remaining inside the Cell, move the pointer towards the top of this Cell but until it changes from a large cross back to the more familiar pointer shape. Then click and hold down the left mouse button and whilst holding it down drag these Cells anywhere into the Tangent Graph area and then let go of the mouse button again this is just a quick way of Copying and Special Pasting this coordinate point onto the Tangent Graph. You will then be presented with a Paste Special Dialogue Box, which should be filled out in the same manner as shown for the single moving point (of tangency) in the previous graph. Next, highlight Cells L29:M30 and as above move the mouse pointer to the inside top border of Cell L29 whilst the block remains highlighted. Drag these Cells onto the Tangent Graph and, as before after using the appropriate Dialogue Boxes, colour the line red and increase its weight or thickness. Click OK and then select any empty Cell to de-select the graphical object. Now press and hold down the Special Function Key F9 to see both graphs animated. Your worksheet should look something like this one below:  Appendix Here it is shown how to locate the coordinates of the end points of the tangent line to any function given that the length of the tangent from  EMBED Equation.DSMT4 is given by the gradient value at the point  EMBED Equation.DSMT4  Consider the function graph of  EMBED Equation.DSMT4  as shown below     Let  EMBED Equation.DSMT4  be the value of the gradient evaluated at  EMBED Equation.DSMT4 , that is  EMBED Equation.DSMT4 . Then by Pythagoras we have:  EMBED Equation.DSMT4 . (1) However,  EMBED Equation.DSMT4 , that is:  EMBED Equation.DSMT4  (2) Subst. (2)  EMBED Equation.DSMT4  (1):  EMBED Equation.DSMT4 , giving:  EMBED Equation.DSMT4 . On inspection we see that the negative root can be dropped since the sign of  EMBED Equation.DSMT4  will give the correct position of  EMBED Equation.DSMT4 . Hence  EMBED Equation.DSMT4  (3) Subst. (3)  EMBED Equation.DSMT4  (2):  EMBED Equation.DSMT4  (4) Therefore, the end-points of the variable length tangent line at the point  EMBED Equation.DSMT4  are  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . A quick check will show that the distance between these two points is  EMBED Equation.DSMT4 , the gradient value at  EMBED Equation.DSMT4 . 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