o Mathematics by T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel (chapter 10: Differential calcusus)restart; DerivativesThe operator D is defined for differentiation. Thus we may compute the derivative of a function, for examplef:=x->sin(exp(x/a));D(f);The command diff differentiates expressions with respect to certain variablesdiff(f(x),a);The second argument of the command has to contain the according variable.Now we are able to analize a curve for example with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj4vJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPkYrRlhGZW5GPg== .f1:=unapply(subs(a=1,f(x)),x);The symbol @ indicates the composition of operators (functions).First we search for roots.lsg:=solve(f1(x)=0,x);No root is found. But be careful: roots do exist! In this case Maple may find all roots, if we allow a set of multiple solutions. Therefore we allocate_EnvAllSolutions := true:Now we obtainlsg:=solve(f1(x)=0,x);Mind, Maple always computes in the complex field (if no specifications are set). Maple introduced three help variables identified by the additional ~ symbol. The about command returns the conditions and restrictions on these variables. Hence the unknows are isolated from the expression by the indets command.about(indets(lsg));The asymptotic behaviour of the function for 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 or 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 is another important characteristic.limit(f1(x),x=-infinity); limit(f1(x),x=infinity);The second limit does not exist, though Maple returns an interval of all limit values according to any chosen sequence x_n. Another question remains: Are there extremal points, thus roots of the first derivative?lsg:=solve(D(f1)(x)=0,x);
about(indets(lsg));As a matter of fact we also consider the secound derivative.ddf1:=(D@@2)(f1);Note, by @@ powers of operators may be written in a short form. D@@2 is just another notation for the command D(D(f1));By evalb( evalf(ddf1(ln(1/2*Pi))) < 0);we make sure that the first positive extremal is a maximum of the function.The command evalb logically evaluates an expression (datatype boolean).To determine derivatives of higher order, the following modification of the diff command is also optionaldiff(f(x),a$5);as a shortcut for diff(diff(diff(diff(diff(asdr,a),a),a),a),a);Finally we regard the plot of f1 .plot(f1,-3..4); The Newton methodVarious options for solving none linear equations have already been mentioned. In most cases Maple's solve command is the first choice. Nevertheless we like to examine the Newton method more detailed. For example, solving the 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 the interval [0,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/2] , or in other words - finding roots of the function f:=x->x^2-cos(x);The plot of f shows that there exists a root in the intervalwith(plots): with(plottools):
p1:=plot(f(x),x=0..Pi/2,y=-1..1,color=blue,thickness=3, scaling=constrained):
p2:=circle([0.8241,0],0.04,color=red,thickness=3):
display(p1,p2);We are going to code a short program that executes some Newton-steps starting from an initial value to approximate a root. To write a procedur, Maple offers the command proc . By local and global variables are defined. Local variables only exist inside the programm, while global variables remain allocated after running the programm. Of course common program concepts, like loops (while , for ) and conditional commands ( if ) are available. newtappr := proc (f, x0, N)
local i, xn, fp;
global newsteps;
xn := x0;
fp := D(f);
newsteps := xn;
for i to N do
xn := evalf(simplify(xn-f(xn)/fp(xn)));
newsteps := newsteps, xn
od;
newsteps
end:Remark: To get a new line without activating the command use [Shift]+[Enter].With the starting value 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 we compute the first 10 iteration steps:l2:=newtappr(f,1,10);p1:=plot(f(x),x=0.6..1.1,y=-0.5..0.5,color=blue,scaling=constrained):
p2:=circle([l2[1],f(l2[1])],0.01,color=red,thickness=3):
p3:=plot(f(l2[1])+D(f)(l2[1])*(x-l2[1]),x=0.6..1.1,color=red,linestyle=DASH):
p4:=circle([l2[2],f(l2[2])],0.01,color=red,thickness=3):
p5:=plot(f(l2[2])+D(f)(l2[2])*(x-l2[2]),x=0.6..1,color=black,linestyle=DASH):
display(p1,p2,p3,p4,p5);Setting 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 , we obtainl2:=newtappr(f,-0.1,10);p1:=plot(f(x),x=-4..1,y=-3..3,color=blue,scaling=constrained):
p2:=circle([l2[1],f(l2[1])],0.01,color=red,thickness=3):
p3:=plot(f(l2[1])+D(f)(l2[1])*(x-l2[1]),x=-4..0.4,color=red,linestyle=DASH):
display(p1,p2,p3);Choosing this starting value , the method converges to the second root of the function. Taylor polynominalsTo create a Taylor polynominal the command taylor is reserved. f:=x->1/(1+x);
p2:=taylor(f(x),x=0,3);The first argument holds the function as an expression. The second argument contains the variable and the center of expansion. By the third argument the number of evaluated terms is set. All further terms of a higher degree, are merged in the Landau symbol, here O(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQUYrRkZGSUZB). Now we can plot the Taylor approximation of the function. First the tangent (as an expression)p1:=taylor(f(x),x=0,2);
a1:=convert(convert(p1,polynom),exp);and even the second and tenth Taylorpolynominala2:=convert(convert(p2,polynom),exp);
p10:=taylor(f(x),x=0,11):
a10:=convert(convert(p10,polynom),exp);plot([f(x),a1,a2,a10],x=-0.5..0.5,color=[black,red,blue,green]);Note, in complicated cases it is impossible to determine a Taylorpolynominal (by Maple). In the book we presented the significant example of a piecewise defined function (piecewise) , where the Taylorseries did not converge to the function. f:=x->piecewise(x>0, exp(-1/x));
plot(f,-0.5..0.5);taylor(f(x),x=0,3);Even the more general command series does not help in such cases. Spline-InterpolationThe Spline-Interpolation is included in Maple by a command. Let us revise the example of the book, the 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
.We allocate two lists with the base points and their according values of the functionxdata:=[seq(j,j=-3..3)];
ydata:=[seq(1/(1+j^2),j=-3..3)];thus the spline command yields a piece-wise defined functionspline(xdata,ydata,t,linear);The first argument lists the base points and the second argument lists the values at these points. The variable of the function is set in the third argument, while the fourth argument of the spline command is optional and determines the degree of the polynominals. If nothing is specified, a natural cubical spline is determined. Valid numbers are integers 1,2,3, ... or names like linear, quadratic or cubic for the last parameter.First we regard the plot of the linear spline:f:=x->1/(1+x^2);
g:=t->spline(xdaten,ydaten,t,linear);
plot([f(x),g(x)],x=-3..3,color=[black,red]);For the according natural cubical spline we obtaing:=t->spline(xdaten,ydaten,t,3);
plot([f(x),g(x)],x=-3..3,color=[black,blue]);The spline command is part of the package CurveFitting , however in most actual Maple versions it is available without explicitly loading the package. Besides we find the PolynomialInterpolation command in this package, that computes the polynominal interpolation of chapter 4.with(CurveFitting):
PolynomialInterpolation(xdaten,ydaten, t);with the plotsg:=t->PolynomialInterpolation(xdaten,ydaten, t);
plot([f(t),g(t)],t=-3..3,color=[black,red]);The option Lagrange provides the Lagrange representation of the polynominal interpolation: PolynomialInterpolation(xdaten,ydaten, t, form=Lagrange); Exercises(To work on the exercises, please open a new worksheet to trial an error your commands. You will get a suggestion for the solution, if you open up the solution. Before you look at them try on your own.)1. Consider the 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Determine the domain, its roots, its extremal points and the asymptotic behaviour for increasing absolute values of x.SolutionFirst we define the operator f:f:=x->(2*x+1)/(x^2+x+1);We plot the graph of the function to get an overview:plot(f(x),x=-3..3);The domain is determined by the poles, the roots of the denominator.solve(denom(f(x))=0,x);The imaginary number I appears. Since both roots are complex numbers, the equation has no real solution and the funciton is well defined for any real x value. (To examine the denominator of the expression f(x) individually, the command denom is useful.) Now the roots of f .solve(f(x)=0,x);To determine extremal points we require the first and second derivatives.d1f:=D(f);
d2f:=D(d1f);Forming expressions in a more clear view by (normal) :d1:=normal(d1f(x));
d2:=normal(d2f(x));Roots of the first derivative:minmax:=[solve(d1f(x)=0,x)];Working with single elements of the list, we definee1:=minmax[1]; e2:=minmax[2];The second derivative tells us about the type of extremaevalb(evalf(d2f(e1))<0);
evalb(evalf(d2f(e2))<0);Hence there is a minima at e1 and a maxima at e2 with the following values of the functionevalf(f(e1),8);
evalf(f(e2),8);Finally we obtain the asymptotic behaviour of f withlimit(f(x),x=infinity);
limit(f(x),x=-infinity);2. Evaluate the Taylor polynominals of tenth order for the following functions in the surrounding of 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 ;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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictRiM2OUYrLUYjNjctRiw2NVEiZkYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUqdW5kZXJsaW5lR0Y+LyUqc3Vic2NyaXB0R0Y+LyUsc3VwZXJzY3JpcHRHRj4vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGPi8lK2V4ZWN1dGFibGVHRj4vJSlyZWFkb25seUdGPi8lKWNvbXBvc2VkR0Y+LyUqY29udmVydGVkR0Y+LyUraW1zZWxlY3RlZEdGPi8lLHBsYWNlaG9sZGVyR0Y+LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y+LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0ZpblEnbm9ybWFsRicvJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZhcC1JKG1mZW5jZWRHRiQ2NS1GIzY1LUYsNjVRInhGJ0Y2RjlGPEY/RkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRjZGOUY8L0ZARj5GQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX28tRlxvNj5RIj1GJ0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0Zhb0Zjb0Zlb0Znb0Zpb0ZbcEZdcC9GYHBRLDAuMjc3Nzc3OGVtRicvRmNwRmFxLUYjNjctRiw2NVElY29zaEYnRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vRltvLUZlcDY1LUYjNjUtSSZtZnJhY0dGJDYqRmdwLUYjNjUtSSNtbkdGJDY1USIyRidGNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX28vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmpyLyUpYmV2ZWxsZWRHRj4vJStmb3JlZ3JvdW5kR0ZKLyUrYmFja2dyb3VuZEdGTUY2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0YrRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vRitGNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX28=, 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 SolutionThe Taylorpolynominals are simply available by:taylor(sin(3*x),x=0,11);taylor(cosh(x/2),x=0,11);taylor(sqrt(1+x),x=0,11);3. Code a programm that provides the Halley method of exercise 10.5 and compare your programm with the Newton method to find one root for each of the two 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 with starting value 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 ; 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 with starting value 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SolutionFirst we enter the procedur similar to the Newton method.halleyappr := proc(f, x0, N)
local i, xn, fp, fpp, fx, fpx;
global halleysteps;
xn := x0;
fp := D(f);
fpp := D(fp);
halleysteps := xn;
for i to N do
fx := f(xn);
fpx := fp(xn);
xn := evalf( xn - fx*fpx/(fpx^2-.5*fx*fpp(xn)) );
halleysteps := halleysteps, xn
od;
halleysteps
end:We examine the first functionf:=x->x^(3)-x+3;
plot(f(x),x=-3..2);With the initial value 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 we evaluate the first ten interation steps of the Newton method (see above) and of the Halley method. This yieldsnewtappr(f,-1,10);and regarding the Halley methodhalleyappr(f,-1,10);Faster convergency is achieved by the Halley method, though we have to pay for this advantage with addtional effort since the Halley method requires the second derivative.A remarkable difference of both methods appears if 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 is chosen as the starting value. Let us review the plot. Convergency of the Newton method cannot be expected. But the local quadratic approximation of the Halley method will return the root in this situation.newtappr(f,0,20);
halleyappr(f,0,8);In the second example the root at 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 is as well an extremal point:g:=x->exp(10*x)-1;
plot(g(x),x=-1..0.1);
newtappr(g,-0.5,150);
halleyappr(g,-0.5,8);