GraphingCalculator 4; Window 60 62 896 863; PaneDivider 75; FontSizes 18; BackgroundType 0; Slider -1 1; SliderControlValue 0; 3D.X -4 4; 3D.Y -4 4; 3D.Z -4 4; 3D.Axes 0; 3D.View -0.569106990654812 0.750125777347833 0.336791554743204 -0.642516028637555 -0.661283504243274 0.387139871312384 0.513118296430605 0.00393003489279179 0.858308900509679; 3D.Speed 0; Text "================================= Matemagi® av Ambjörn Naeve ©Dialectica ================================= Consider a function f : R2 -----------> R (x, y) ------> f(x, y) where"; Color 17; Expr function(f,x,y)=[sin(x)]*[cos(y)]; Text " The FIRST order partial derivatives of the function f(x, y) are given by"; Color 17; Expr function(p,x,y)=function(oppartial(x),function(f,x,y)); Color 17; Expr function(q,x,y)=function(oppartial(y),function(f,x,y)); Text " The SECOND order partial derivatives of the function f(x, y) are given by"; Color 17; Expr function(u,x,y)=function(oppartial(x),function(oppartial(x),function(f,x,y))); Color 17; Expr function(v,x,y)=function(oppartial(y),function(oppartial(x),function(f,x,y))); Color 17; Expr function(w,x,y)=function(oppartial(y),function(oppartial(y),function(f,x,y))); Text " The THIRD order partial derivatives of the function f(x, y) are given by"; Color 17; Expr function(g,x,y)=function(oppartial(x),function(oppartial(x),function(oppartial(x),function(f,x,y)))); Color 17; Expr function(h,x,y)=function(oppartial(y),function(oppartial(x),function(oppartial(x),function(f,x,y)))); Color 17; Expr function(j,x,y)=function(oppartial(y),function(oppartial(y),function(oppartial(x),function(f,x,y)))); Color 17; Expr function(k,x,y)=function(oppartial(y),function(oppartial(y),function(oppartial(y),function(f,x,y)))); Text " The FOURTH order partial derivatives of the function f(x, y) are given by"; Color 17; Expr function(c,x,y)=function(oppartial(x),function(oppartial(x),function(oppartial(x),function(oppartial(x),function(f,x,y))))); Color 17; Expr function(d,x,y)=function(oppartial(y),function(oppartial(x),function(oppartial(x),function(oppartial(x),function(f,x,y))))); Color 17; Expr function(l,x,y)=function(oppartial(y),function(oppartial(y),function(oppartial(x),function(oppartial(x),function(f,x,y))))); Color 17; Expr function(m,x,y)=function(oppartial(y),function(oppartial(y),function(oppartial(y),function(oppartial(x),function(f,x,y))))); Color 17; Expr function(s,x,y)=function(oppartial(y),function(oppartial(y),function(oppartial(y),function(oppartial(y),function(f,x,y))))); Text " We will compare the function f(x, y) with its Taylor polynomials of the FIRST, SECOND, THIRD och FOURTH degree around the point p = (a, b), where"; Color 17; Expr a=cos(pi*n); Color 17; Expr b=sin(pi*n); Text " The graph of the function f(x, y) is described by the checkered surface"; Color 10; Opacity 0.7; Expr z=function(f,x,y); Text " The point (a, b), around which we are about to approximate f(x, y) with a sum of its Taylor polynomials of successively higher degree, corresponds to the 3-dimensional point (a, b, f(a, b)):"; Color 2; Grain 0.1; Expr vector(a,b,function(f,a,b)); Text " The graph of the Taylor polynomial Taylor(1, f(x, y), a, b) of the first degree of the function f(x, y) at the point (a, b) is described by the dark-grey surface "; Color 8; Opacity 0.7; Expr z=function(f,a,b)+function(p,a,b)*[x-a]+function(q,a,b)*[y-b]; Text " NOTE: 1. Since the relationship between x, y och z is LINEAR, the dark-grey surface is a PLANE. 2. This plane is the TANGENT PLANE of the checkered surface z = f(x, y) at the point (a, b, f(a, b)). 3. The affine function z = Taylor(1, f(x, y), a, b) is the AFFINE (= First degree) APPROXIMATION of the function z = f(x, y) in the neighborhood of the point p = (a, b), which means that it is the polynomial mapping of the first degee R2 -----> R which best describes the action of the function f : R2 -----> R close to the point p = (a, b). ===================== The graph of the Taylor polynomial Taylor(2, f(x, y), a, b) of the second degree for the function z = f(x, y) at the point p = (a, b) is described by the rainbow-colored surface"; Color 13; Opacity 0.7; Expr z=function(f,a,b)+function(p,a,b)*[x-a]+function(q,a,b)*[y-b]+1/2*function(u,a,b)*[x-a]^2+function(v,a,b)*[x-a]*[y-b]+1/2*function(w,a,b)*[y-b]^2; Text " NOTE: 1. Since this relationship between x, y and z contains first and second degree terms in x och y but only first degree terms in z, the rainbow-colored surface is a PARABOLOID. 2. This paraboloid is the OSCULATING PARABOLOID for the surface z = f(x, y) at the point (a, b, f(a, b)). 3. The osculating paraboloid z = Taylor(2, f(x, y), a, b) has the same tangent plane and the same curvature as the original surface z = f(x, y) at the point (a, b, f(a, b)). 4. The second degree polynomial function Taylor(2, f(x, y), a, b) is the SECOND-DEGREE POLYNOMIAL FUNCTION R2 -----> R which best describes the behaviour of the function f : R2 -----> R in the neighborhood of the point (a, b). 5. The osculating paraboloid is ELLIPTIC if and only if the original surface has POSITIVE GAUSSIAN CURVATURE (i.e., it is shaped like a bowl) around the point (a, b, f(a, b)). 6. The osculating paraboloid is HYPERBOLIC at the point p = (a, b) (A HYPERBOLOC PARABOLOID is also called a SADDLE SURFACE ) if and only if the original surface has NEGATIVE GAUSSIAN CURVATURE (i.e., it is shaped like a SADDLE) around the point (a, b, f(a, b)). 7. The osculating paraboloid is A CYLINDER if and only if the original surface has GAUSSIAN CURVATURE = ZERO at the point (a, b, f(a, b)). ==================== The graph of the Taylor polynomial Taylor(3, f(x, y), a, b) of the third degree for the function f(x, y) around the point (x, y) = (a, b) is described by the yellow surface "; Color 6; Opacity 0.7; Expr z=function(f,a,b)+function(p,a,b)*[x-a]+function(q,a,b)*[y-b]+1/2*function(u,a,b)*[x-a]^2+function(v,a,b)*[x-a]*[y-b]+1/2*function(w,a,b)*[y-b]^2+1/6*function(g,a,b)*[x-a]^3+1/2*function(h,a,b)*[x-a]^2*[y-b]+1/2*function(j,a,b)*[x-a]*[y-b]^2+1/6*function(k,a,b)*[y-b]^3; Text " NOTE: 1. The Taylor polynomial Taylor(3, f(x, y), a, b) is a third-degree polynomial in x and y and the surface z = Taylor(3, f(x, y), a, b) could be called the OSCULATING THIRD DEGREE POLYNOMIAL for the surface z = f(x, y). 2. The osculating third-degree polynomial surface z = Taylor(3, f(x, y), a, b) has OPTIMAL CONTACT OF THE THIRD DEGREE with the original surface z = f(x, y) at the point (a, b, f(a, b)). 3. The third-degree Taylor polynomial Taylor(3, f(x, y), a, b) is THE BEST THIRD-DEGREE POLYNOMIAL APPROXIMATION of the function f(x, y) close to the point (a, b), which means that it is the third-degree polynomial function R2 -----> R that closest describes the action of the function f : R2 -----> R in the viscinity of the point (a, b). ===================== The graph of the Taylor polynomial Taylor(4, f(x, y), a, b) of the fourth degree for the function f(x, y) around the point (x, y) = (a, b) is described by the light-blue surface"; Color 5; Opacity 0.7; Expr z=function(f,a,b)+function(p,a,b)*[x-a]+function(q,a,b)*[y-b]+1/2*function(u,a,b)*[x-a]^2+function(v,a,b)*[x-a]*[y-b]+1/2*function(w,a,b)*[y-b]^2+1/6*function(g,a,b)*[x-a]^3+1/2*function(h,a,b)*[x-a]^2*[y-b]+1/2*function(j,a,b)*[x-a]*[y-b]^2+1/6*function(k,a,b)*[y-b]^3+1/24*function(c,a,b)*[x-a]^4+1/6*function(d,a,b)*[x-a]^3*[y-b]+1/4*function(l,a,b)*[x-a]^2*[y-b]^2+1/6*function(m,a,b)*[x-a]*[y-b]^3+1/24*function(s,a,b)*[y-b]^4; Text " NOTE: 1. The Taylor polynomial Taylor(4, f(x, y), a, b) is a fourth-degree polynomial in x och y and it could be called THE OSCULATING FOURTH DEGREE POLYNOMIAL for the surface z = f(x, y). 2. The osculating fourth-degree polynomial surface z = Taylor(3, f(x, y), a, b) has OPTIMAL CONTACT OF THE FOURTH DEGREE with the original surface z = f(x, y) at the point (a, b, f(a, b)). 3. The fourth-degree Taylor polynomial Taylor(4, f(x, y), a, b) is THE BEST FOURTH-DEGREE POLYNOMIAL APPROXIMATION of the function f(x, y) close to the point (a, b), which means that it is the fourth-degree polynomial function R2 -----> R that closest describes the action of the function f : R2 -----> R in the viscinity of the point (a, b). ";