HP 40gs HP 39gs_40gs_Mastering The Graphing Calculator_English_E_F2224-90010.p - Page 320

Area Under Curves This topic is most easily handled using an aplet from The HP HOME View web site (at http://www.hphomeview.com). This aplet, called "Curve Areas" will draw rectangles either over or under a curve or use trapezoids. A number of curves are supplied pre-set but the user can also enter their own. The user can nominate the interval width and the number of rectangles. Most importantly, a worksheet is bundled with the aplet which will lead the student through the process of deducing an area function and hence to the anti-differentiation of xn . Fields of Slopes and Curve Families One of the concepts which students find quite difficult to come to grips with is that of sketching a field of slopes from a derivative function and, from this, sketching a family of curves. An aplet from The HP HOME View web site (at http://www.hphomeview.com), called "Slope Fields", will assist with this process. In this aplet the user enters the derivative function into F1(X) and then uses the VIEWS menu to produce a field of slopes. A cross-hair is projected onto the field which the user can move around. When the user presses ENTER, a curve is drawn, starting at that point and projecting to the right and then the left, and following the field of slopes. Repetition of this will illustrate the fact that there are a family of curves, separated by a constant, which all fit the 'description' of the function stored in F1(X). The screen shots to the right are the result of F1(X)=X2+1 ⎛ dy ⎜⎝ dx = x 2 +1⎞⎟⎠ , F1(X)=X2*Y ⎛ dy ⎜⎝ dx = x2 y ⎞ ⎟ ⎠ and F1(X)=(X+1)*Y ⎛ dy ⎜⎝ dx = ( x + 1) y ⎞ ⎟⎠ respectively. 320