ࡱ> Y O*bjbjWW ޭ==%s]tttt8,tp"$b0 :@4@@7@ֿttNamed Ranges 3 Animation Drawing a moving Tangent Open Excel, select File ( Open, and open Name3 from your floppy disk. Highlight Cells from B4 to C31 (Notice that Cell B4 remains clear) and then move mouse pointer so that it lies on the top line of Cell B4. You will notice that the pointer icon changes from being a large cross into an arrow as its position changes from being inside the Cell B4 to being on the top (or left) border of the Cell. See pictures below.  Once the mouse pointer changes into an arrow click and hold down the left mouse button. Still holding the left button down, drag the mouse pointer so that it lies within the Cell B6. This has the effect of dragging the whole range of x and y values down by two Cells. All links to Cells, Names, Graph Ranges, etc, will automatically have their addresses updated. In Cell B3 type Step, Cell C3 type k and Cell D3 type t, pressing Return after each. Highlight Cells B3 to D4 and select Insert ( Name ( Create, and press OK. This next step sets up the spreadsheet to deal with circular references, which we utilise in order to animate graphs. If it not done first then an error message of the type below will be shown during the proceeding work and then Cancel should be clicked. However, to avoid this message, do the following. Select Tools ( Options( Calculation Tab, and set up the Dialogue Box as shown below: Notice however, 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. Now, in Cell B4 type 0.05 and press return, Cell C4 type = k + step (This would have been the point to cause the above error message without our change in the Options Dialogue Box) and press Return. Before we type into Cell D4 it will be convenient to double-label the first and last values of x. We do this using the Name Box. Click into Cell B8 (The first x value), then click into the Name Box and type first.x and remember to press Return to enter this 2nd Name. Similarly for the last x value, click in Cell B33, and then click in the Name Box and type last.x. Now may be a good time to click on the Down Arrow in the Name Box to see just what Names have been defined, notice that clicking on a particular name will highlight its range on the Worksheet. Back to Cell D4, type in = k + first.x and press Return In Cell E7 type a and in Cell F7 type b, pressing Return after each. Highlight Cells E7 to F8 and select Insert ( Name ( Create, and press OK. In Cell E8 type = t and press Return and in Cell F8 type = 0.3*a^2-3*a-6 and press Return. This is duplicating the equation of the curve defined in column C. Cells E8 and F8 therefore represent a single point on the curve. To copy this point onto the same axes we use the same technique as met above in the Name2 exercises. Highlight Cells E7 to F8 and press Ctrl+C to copy them, then click anywhere on the graph and then select Edit ( Paste Special, etc. Fill out the Paste Special Dialogue Box as shown below. NOTICE that even though we shall only be entering a single point on to the graph, we DO NOT choose the New point(s) option from the Add cells as frame but still choose the New series option as before.  Remember, to remove the flickering border around the copied Cells simply press the Escape Key. Next, click once anywhere on the graph. Something will be selected but it will most likely not be the point that weve just copied to the graph. Therefore, to find our single point we can cycle through all of the graphical objects contained within the graph by pressing either the UP or DOWN arrow keys on the Keyboard. Eventually the single point will be highlighted (probably situated at the leftmost point of the curve), if you now move the mouse on top of this point it will display its details after a moment. Once you are sure that it is the actual point then carefully double click it with the left mouse button, this will show the Format Data Series Dialogue Box. In the Patterns Tab go into the Marker frame and check the Custom option, then make your own changes to the Style, Foreground, Background and Size Text Boxes. Press OK to see the outcome on the graph. Animation We can now make the point move simply by pressing the Special Function Key F9 at the top of the Keyboard. Try it what happens? Well what you should have observed is that the point moves off the end of the curve and appears to have its abscissa forever increasing, thereby causing the axis to automatically rescale. This is confirmed by a glance at the value of either t or a, which have increased past the maximum value of x in column B. To overcome this problem we alter the formula in Cell C4 (which is what I call the driver Cell for the worksheet animation). Click in Cell C4 and then click anywhere in the Formula Bar, which should be displaying the formula = k + step . Write instead = Mod (k + step, last.x first.x) , and press Return. Again hold down the Special Function Key, F9, and observe the motion. The point (a, b) now cycles over the shown curve. To alter the speed of the point simply alter the value of step in Cell B4. Tangent to the Curve at (a, b) - Theory This is dependent upon us determining, numerically, the gradient of the tangent at the point (a, b). We use the Central Difference Formula for approximating the first derivative of the function  EMBED Equation.3 , namely for small h in preference to the better known form  EMBED Equation.3 , ---- (2). which is less robust numerically. Therefore, at the point (a, b), the gradient is given by  EMBED Equation.3 . ---- (3). Moreover, to draw the tangent line through (a, b) we require only two points. We choose points P and Q with abscissas first.x and last.x respectively, for reasons that will become apparent. At first.x The straight-line form  EMBED Equation.3 , through (a, b), gives the coordinates of point P as  EMBED Equation.3  ---- (4). At last.x Similarly, the coordinates of point Q are given by  EMBED Equation.3  ---- (5). Calculation of Numerical Gradient In this section we use our first bit of Visual Basic for Applications (VBA) programming. Press the keys Alt+F11 (F11 being one of the Special Function Keys at the top of the keyboard). This opens the VBA worksheet associated with the Excel Worksheet. Select Insert ( Module, and type the following text as shown below in the newly opened Module window. It will look something like this: You will notice as you type that the editor tries to complete words, if a word is highlighted from an automatic fill list and you dont want to have to type it then simply press the Tab Button from the keyboard and this will enter the full text. To return to Excel, click on the Excel Icon in the Standard Toolbar. Click into Cell G3 and type h, then click into Cell G4 and type 0.05. Highlight Cells G3 to G4 and select Insert ( Name ( Create, and click OK to name this value h. Click into Cell G7 and type m, then click into Cell G8 and type the Numerical Central Difference formula as defined in (3) above, viz: = (f(a + h) - f(a - h))/(2*h) and press Return. Highlight Cells G7 to G8 and select Insert ( Name ( Create, and click OK to name this value m. In Cell E11 type Point P and in Cell E12 type Point Q In Cell F11 type = first.x and press Return, and in Cell F12 type = last.x and press Return. Entering formulae (4) and (5) above give us the end-points of the gradient line through (a, b) that correspond to the minimum and maximum within the range of x. In Cell G11 type = b + m*(first.x - a) , and press Return. In Cell G12 type = b + m*(last.x - a) , and press Return. Next we highlight the four Cells containing the two end-points of the tangent line, P and Q, by dragging the mouse over them and then press Ctrl+C to copy them. Click once on the graph and select Edit ( Paste Special. Select from the Dialogue Box the following items shown below and press OK. What should have been produced is a straight line which passes through points P and Q and also though the point (a, b) the Point of Tangency. By repeated pressing/holding down of the Special Function Key F9 we see the tangent move along the curve. However, what is confusing is that the vertical axis scale is dynamically changing to best accommodate the varying maximum and minimum heights of the tangent line. This can be prevented by first placing the mouse directly over the vertical axis, double clicking to show the Format Axis Dialogue Box. Next, click on the Scale Tab and, for example, alter the Minimum value to 50 and the Maximum value to 60. At this point it is important to uncheck the boxes opposite these values otherwise they will continue to automatically adjust. However, altering a particular value should uncheck it for you. Furthermore, to avoid the graph flickering whilst pressing F9, it is helpful to do a single right mouse button click anywhere within the chart area and from the drop down menu choose Format Plot Area and then choose None in the Area frame. Gridlines can also be removed by a right mouse button click on any one of them and then choosing Clear from the drop down menu. Finally, click back into Cell F8, type in = f(a) and press Return. What happens? For a more detailed discussion on some of the advantages and possible pitfalls of such VBA, User Defined Functions, see the next Exercise, Programming 1. Your worksheet should resemble the one shown below.  PAGE  PAGE 12 Cross Arrow  EMBED Equation.DSMT4  ---- (1). Dont type xs in here 5NO  D E N O  Ua os+,jCJEHUjD< CJUVmH  jCJUjCJUmH6CJCJH*>*CJ5CJjCJUmH jCJCJ 5>*CJ 5>*CJ(D56~WX    M 56~WX  L M ST½}xsnid_A? "#M N  L M ST[\*+ D%[\*+%&_BCN6XYL k l ~  !½ꐋ|wr /0R:EF)bc56@++%&_BCN6XYL k l ~  $x D%`atuvwCMefyz{|()*+6WfgL ໲ঝysss jCJjCJUmH 5>*CJj CJEHUj{< CJUVmH jl CJEHUjk< CJUVmH jLCJEHUjd< CJUVmH >*CJjCJEHUjW< CJUVmH CJ jCJUjCJEHUjL< CJUVmH *L M ""^#_#v&}&C)Y)))))))))))))))))****5*:*N*O*>* jUEHUj!< CJUVmH  jU0JmH0J j0JU6CJjCJUmH jCJCJjCJUmH% !y!z!"W"""##''(()))))))))))))&`#$!y!z!"W"""##''(())))))))))))))N*O*o))))))))))**4*5*L*M*N*O*$&`#$301hP. 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