GraphingCalculator 4; Window 74 51 809 1340; PaneDivider 0; FontSizes 18; Slider 0 1; SliderControlValue 40; 2D.BottomLeft -2.328125 -2.921875; 2D.Axes 0; 2D.GraphPaper 0; Text "================================= Matemagi® by Ambjörn Naeve ©Dialectica ================================= Given a function F and its derivative f:"; Color 17; Expr function(F,x)=sin(x),function(f,x)=function(optotal(x),function(F,x)); Text "and another function G and its derivative g:"; Color 17; Expr function(G,x)=x^3,function(g,x)=function(optotal(x),function(G,x)); Text "The composite function H = G o F and its derivative h are given by:"; Color 17; Expr function(H,x)=function(G,function(F,x)),function(h,x)=function(optotal(x),function(H,x)); Text "We will illustrate the chain rule, which states that h(a) = g(F(a)) f(a) We draw F in blue in the left window:"; Color 17; Expr y=function(F,x); Color 7; Expr vector(x,y)=vector(2*pi*t,function(F,2*pi*t)); Text "and G in grey (top left "; Color 7; Expr vector(x,y)=vector(2*t-1,function(G,2*t-1))+vector(0,5); Text " In order to increase clarity, we position the axis cross (origin) in the point (0,5) and draw the corresponding x-axis as the black line:"; Color 17; Expr x=0; Color 17; Expr y=0; Color 17; Expr y=5; Text " We also draw a vertical axis through the point (9, 5):"; Color 17; Expr x=9; Text "Then we can draw the composite function H as "; Color 7; Expr vector(x,y)=vector(2*pi*t,function(H,2*pi*t))+vector(9,5); Text "Now, let the variable point x = a be described by ""the animation parameter"" n :"; Color 17; Expr a=2*pi*n; Text "The variation of the independent variable is described by the green point:"; Color 4; Expr vector(a,0); Text "The corresponding point on the curve y = F(x) is given by the light-blue point:"; Color 5; Expr vector(a,function(F,a)); Text "and the y-value F(a) plotted on the y-axis is given by the light-blue point:"; Color 5; Expr vector(0,function(F,a)); Text " Plotting the value of the combination H(a) as a function of F(a) we have "; Color 5; Expr vector(function(F,a),0)+vector(0,5); Expr vector(function(F,a),function(H,a))+vector(0,5); Text "and projected on the G-axis:"; Expr vector(0,function(H,a))+vector(0,5); Text " Observe that ""the x-axis"" has been displaced five units upwards and is given by the line y = 5. The composite function value y = G(F(a)) plotted on the y-axis is given by the purple point:"; Expr vector(a+9,function(H,a)+5); Text " and projected on the H-axis"; Expr vector(9,function(H,a)+5); Text "This point is now expressed as the functional value y(a) = G(F(a)) = H(a), i.e., as a function of the green point"; Color 4; Expr vector(a+9,5); Text "The tangent in the corresponding points of the three curves are given by the red lines:"; Color 3; Expr [y-function(F,a)]=function(f,a)*[x-a],a-kk, where"; Color 17; Expr k=1/2; Text "The slope of the tangent is the value of the derivative at the point of tangency These derivative values, f(a), g(F(a)) and h(a) are represented by the three red points:"; Color 17; Expr vector(a,function(f,a)); Color 17; Expr vector(function(F,a),function(g,function(F,a))+5); Color 17; Expr vector(a+9,function(h,a)+5); Text "The curves for the variation of the derivatives when we vary the independent variable a is described by the red curves:"; Color 17; Expr vector(x,y)=vector(a*t,function(f,a*t)); Color 17; Expr vector(x,y)=vector(function(F,a*t),function(g,function(F,a*t)))+vector(0,5); Color 17; Expr vector(x,y)=vector(a*t+9,function(h,a*t)+5); Text " The identity map u maps to u :"; Color 17; Expr vector(x,y)=vector(2*pi*t,2*pi*t)+vector(9,0); Color 17; Expr vector(a,function(F,a))+vector(9,0); Color 4; Expr vector(a,0)+vector(9,0); Color 17; Expr [y-function(F,a)]=function(f,a)*[[x-9]-a],a-k<[x-9]G(F(x) = (G˚F)(x) we have the same end value. Hence the purple point and the green point have the same y-coordinate and therefore both lie on this line:"; Color 17; Expr y=function(H,a)+5; Color 17; Expr vector(x,y)=t*vector(function(F,a),function(G,function(F,a))+5)+[1-t]*vector(a+9,function(H,a)+5); Text " The representation of ∆u :"; Color 2; Expr '∆'_p=s; Color 17; MathPaneSlider 99; Expr s=slider([0,1]); Color 2; Expr vector(a+'∆'_p,0); Color 2; Expr vector(a+'∆'_p+9,0); Color 2; Expr vector(a+'∆'_p+9,5); Color 2; Expr vector(a-'∆'_p,0); Color 2; Expr vector(a-'∆'_p+9,0); Color 2; Expr vector(a-'∆'_p+9,5); Text " Mapping the red points with the function F: "; Color 2; Expr vector(a+'∆'_p,function(F,a+'∆'_p)); Color 2; Expr vector(a-'∆'_p,function(F,a-'∆'_p)); Text " Mapping the F-curve segment between these points"; Color 2; Expr vector(a+[2*t-1]*'∆'_p,function(F,a+[2*t-1]*'∆'_p)); Text " Projecting these two curve points onto the vertical F(u) axis"; Color 2; Expr vector(0,function(F,a+'∆'_p)); Color 2; Expr vector(0,function(F,a-'∆'_p)); Text " Laying out these points on the horizontal G-axis (top left):"; Color 2; Expr vector(function(F,a+'∆'_p),5); Color 2; Expr vector(function(F,a-'∆'_p),5); Text " Finding the corresponding points on the G-curve (top left) "; Color 2; Expr vector(function(F,a+'∆'_p),function(G,function(F,a+'∆'_p))+5); Color 2; Expr vector(function(F,a-'∆'_p),function(G,function(F,a-'∆'_p))+5); Text " Mapping the G-curve segment between these points:"; Color 2; Expr vector(function(F,a+[2*t-1]*'∆'_p),function(G,function(F,a+[2*t-1]*'∆'_p))+5); Text " The combination H = G˚F :"; Color 2; Expr vector(a+'∆'_p+9,function(G,function(F,a+'∆'_p))+5); Color 2; Expr vector(a-'∆'_p+9,function(G,function(F,a-'∆'_p))+5); Text " Mapping the G˚F-segment between these points"; Color 2; Expr vector(a+[2*t-1]*'∆'_p+9,function(G,function(F,a+[2*t-1]*'∆'_p))+5); Text " Axes:"; Color 4; Expr vector(-0.5,0),vector(7,0); Color 4; Expr vector(8.5,0),vector(16,0); Color 5; Expr vector(0,-0.5),vector(0,2); Color 5; Expr vector(-0.5,5),vector(7,5); Expr vector(0,4.5),vector(0,7); Text " ///"; Color 4; Expr vector(8.5,5),vector(16,5); Expr vector(9,4.5),vector(9,7); PageMargins 72 72 72 72;