restart;

f:=(x)->if x < 0 then -a else a end;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlQCUyOSQiIiEsJEkiYUdGJSEiIkYxRiVGJUYl

f(-3);

NiMsJEkiYUc2IiEiIg==

f(3);

NiNJImFHNiI=

f1:=(x)->signum(x);

NiM+SSNmMUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJ3NpZ251bUdGJTYjOSRGJUYlRiU=

plot(f1(x),x=-10..10,y=-10..10);

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

limit(f1(x),x=0,left);

NiMhIiI=

limit(f1(x),x=0,right);

NiMiIiI=

Ist f(x) stetig an der Stelle x0 = 0?-> Nein, die Funktion springt bei 0 und ist deshalb nicht stetig.

Ist f(x) stetig auf R\{0}?-> Ja, die Spr\374nge wurden ausgeschlossen.

g:=(x,a,b)->signum(x)*a*(1-exp(-signum(x)*x/b));

NiM+SSJnRzYiZio2JUkieEdGJUkiYUdGJUkiYkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigtSSdzaWdudW1HNiRJKnByb3RlY3RlZEdGMkkoX3N5c2xpYkdGJTYjOSQiIiI5JUY2LCZGNkY2LUkkZXhwR0YxNiMsJCooRi9GNkY1RjY5JiEiIkY/Rj9GNkYlRiVGJQ==

plot(g(x,5,.1),x=-10..10,y=-10..10);

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

Die Funktion ist auf ganz R stetig.

g(x,a,b) kann mit der Variabel b der Funktion f(x) \344ndlich gemacht werden. Der einzige Unterschied ist, dass g(x,a,b) auch bei x0=0 stetig ist.

restart;

f:=(x)->0.1*(x^4)-0.5*(x^3)-1.5*(x^2)+0.5;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoqJiQiIiIhIiJGLyokOSQiIiVGL0YvKiYkIiImRjBGLyokRjIiIiRGL0YwKiYkIiM6RjBGLyokRjIiIiNGL0YwRjVGL0YlRiVGJQ==

plot(f(x),x=-10..10,y=-50..50);

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

f1:=D(f);

NiM+SSNmMUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoKiQ5JCIiJCQiIiUhIiIqJEYuIiIjJCEjOkYyRi4kISNJRjJGJUYlRiU=

f2:=D(f1);

NiM+SSNmMkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoKiQ5JCIiIyQiIzchIiJGLiQhI0lGMkYzIiIiRiVGJUYl

X:=sort([solve(f1(x)=0,x)]);

NiM+SSJYRzYiNyUkISs3JnpSVyIhIiokIiIhRiskIis3JnpSPiZGKQ==

Es gibt drei Stellen die potentielle Maximas sein k\366nnen.f2(X[1]);

NiMkIitMMi5NUSEiKg==

Die Stelle -1.444 ist ein m\366gliches Minimumf2(X[2]);

NiMkISNJISIi

Die Stelle 0 ist ein m\366gliches Maximumf2(X[3]);

NiMkIitFcDR6OCEiKQ==

Die Stelle 5.19r ist ein m\366gliches Minimumf(X[1]);

NiMkISomZmN1byEiKg==

f(X[2]);

NiMkIiImISIi

f(X[3]);

NiMkIStFITRbcyQhIik=

Die Stelle (5,194 / -37.248) ist ein globales Minimum.Um die Stelle (0 / 0.5) m\374sste ein Interval gesetzt werden.

g1:=(x)->(2-exp(x)+2*x)/3;

NiM+SSNnMUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoIyIiIyIiJCIiIi1JJGV4cEdGJTYjOSQjISIiRi9GNEYtRiVGJUYl

plot(g1(x),x=0..1);

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

g11:=D(g1);

NiM+SSRnMTFHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiM5JCMhIiIiIiQjIiIjRjYiIiJGJUYlRiU=

g12:=D(g11);

NiM+SSRnMTJHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJC1JJGV4cEc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiM5JCMhIiIiIiRGJUYlRiU=

X:=sort([solve(g11(x)=0,x)]);

NiM+SSJYRzYiNyMtSSNsbkc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMiIiM=

Es gibt nur ein Maxima und das wird auch das max(|g(x)|) sein.g12(X[1]);

NiMjISIjIiIk

g1(X[1]);

NiMsJC1JI2xuRzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2IyIiIyNGKyIiJA==

evalf(g1(X[1]));

NiMkIisvNyk0aSUhIzU=

Das absolute Maximum ist 2/3 * ln(2)

g2:=(x)->1+exp(-cos(x-1));

NiM+SSNnMkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmIiIiRi0tSSRleHBHNiRJKnByb3RlY3RlZEdGMUkoX3N5c2xpYkdGJTYjLCQtSSRjb3NHRjA2IywmOSRGLSEiIkYtRjpGLUYlRiVGJQ==

plot(g2(x),x=1..2);

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

g21:=D(g2);

NiM+SSRnMjFHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiMsJjkkIiIiISIiRjVGNS1JJGV4cEdGLzYjLCQtSSRjb3NHRi9GMkY2RjVGJUYlRiU=

g22:=D(g21);

NiM+SSRnMjJHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJiomLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjFJKF9zeXNsaWJHRiU2IywmOSQiIiIhIiJGNkY2LUkkZXhwR0YwNiMsJEYuRjdGNkY2KiYtSSRzaW5HRjBGMyIiI0Y4RjZGNkYlRiVGJQ==

X:=sort([solve(g21(x)=0,x)]);

NiM+SSJYRzYiNyMiIiI=

Es gibt nur ein Maxima und das wird auch das max(|g(x)|) sein.g22(X[1]);

NiMtSSRleHBHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjISIi

g2(X[1]);

NiMsJiIiIkYkLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRihJKF9zeXNsaWJHNiI2IyEiIkYk

evalf(g2(X[1]));

NiMkIitUJXp5TyIhIio=

Das absolute Maximum ist 1+e ^ (-1)

g3:=(x)->(ln(x)^x);

NiM+SSNnM0c2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSktSSNsbkc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiM5JEYzRiVGJUYl

plot(g3(x),x=4..5);

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

g31:=D(g3);

NiM+SSRnMzFHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUqJiktSSNsbkc2JEkqcHJvdGVjdGVkR0YxSShfc3lzbGliR0YlNiM5JEY0IiIiLCYtRi82I0YuRjUqJEYuISIiRjVGNUYlRiVGJQ==

g32:=D(g31);

NiM+SSRnMzJHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJiomKS1JI2xuRzYkSSpwcm90ZWN0ZWRHRjJJKF9zeXNsaWJHRiU2IzkkRjUiIiIsJi1GMDYjRi9GNiokRi8hIiJGNiIiI0Y2KiZGLkY2LCYqJkY1RjtGL0Y7RjYqJkYvISIjRjVGO0Y7RjZGNkYlRiVGJQ==

X:=sort([solve(g31(x)=0,x)]);

NiM+SSJYRzYiNyMqJC1JJGV4cEc2JEkqcHJvdGVjdGVkR0YrSShfc3lzbGliR0YlNiMqJC1JKUxhbWJlcnRXR0YqNiMhIiJGMkYy

Es gibt kein reelees Maxima und daher k\366nnen wir die Intervalgrenzen verwenden.g3(4);

NiMsJCokLUkjbG5HNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYjIiIjIiIlIiM7

g3(5);

NiMqJC1JI2xuRzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2IyIiJkYr

Der Funktionswert ist bei 5 gr\366sser und es ist daher das Maximum.Das absolute Maximum ist 1+e ^(-1)

restart;

f:=x->(x^3)-(x^2)+x;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIkIiIiKiRGLiIiIyEiIkYuRjBGJUYlRiU=

genau:=evalf(int(f(x),x=0..5));

NiM+SSZnZW5hdUc2IiQiK0xMJDNGIiEiKA==

Das genaue Resultat lautet: 127.08333333

Wir machen den Versuch mit 30, 100, 1000, 10000Approx := proc (f, a, b, n) local xi, i, tot, delx; tot:=0; delx:=(b-a)/n; for i from 1 to n do xi:=a+i*delx; tot:=tot+f(xi); end do; evalf(tot*delx); end proc:f:=x->(x^3)-(x^2)+x;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIkIiIiKiRGLiIiIyEiIkYuRjBGJUYlRiU=

n30:=Approx(f, 0, 5, 30);

NiM+SSRuMzBHNiIkIitqeiQpZjghIig=

n100:=Approx(f, 0, 5, 100);

NiM+SSVuMTAwRzYiJCIrXSg9c0giISIo

n1000:=Approx(f, 0, 5, 1000);

NiM+SSZuMTAwMEc2IiQiKylvZk1GIiEiKA==

n10000:=Approx(f, 0, 5, 10000);

NiM+SSduMTAwMDBHNiIkIitaZTRyNyEiKA==

Abweichung von n30 zu genaun30-genau;

NiMkIilJWSsqKSEiKA==

Abweichung von n100 zu genaun100-genau;

NiMkIik8YVFFISIo

Abweichung von n1000 zu genaun1000-genau;

NiMkIihiamkjISIo

Abweichung von n10000 zu genaun10000-genau;

NiMkIic5REUhIig=

Die Abweichung nimmt mit h\366herer Approximation ab.

restart;

Wir setzten den Start des Intervalls und das Ende des Intervalls ein, und wenn eines > 0 und das andere < 0 ist, dann hat es dazwischen eine L\366sung (Mittelwertsatz).

f:=(x)->((x-2)^2)-ln(x);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJCwmOSQiIiIhIiNGMCIiI0YwLUkjbG5HNiRJKnByb3RlY3RlZEdGNkkoX3N5c2xpYkdGJTYjRi8hIiJGJUYlRiU=

f(1);

NiMiIiI=

f(2);

NiMsJC1JI2xuRzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2IyIiIyEiIg==

Es gibt eine L\366sungevalf(f(exp(1)));

NiMkIStjQHJTWyEjNQ==

evalf(f(4));

NiMkIitSY3E4RSEiKg==

Es gibt eine L\366sung

g:=(x)->2*x*cos(2*x)-((x-2)^2);

NiM+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJjkkIiIiLUkkY29zR0YlNiMsJEYuIiIjRi9GNCokLCZGLkYvISIjRi9GNCEiIkYlRiVGJQ==

evalf(g(2));

NiMkISslW3VYaCMhIio=

evalf(g(3));

NiMkIis/PC1oWiEiKg==

Es gibt eine L\366sungevalf(g(3));

NiMkIis/PC1oWiEiKg==

evalf(g(4));

NiMkIStxLStrXiEiKg==

Es gibt eine L\366sung

Wir plotten die Funktion und sch\344tzen das Intervall ab.

h1:=(x)->x-(3^(-x));

NiM+SSNoMUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmOSQiIiIpIiIkLCRGLSEiIkYyRiVGJUYl

plot(h1(x),x=0..1);

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

Das Intervall ist [0,1]

h2:=(x)->4*(x^2)-exp(x);

NiM+SSNoMkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmKiQ5JCIiIyIiJS1JJGV4cEc2JEkqcHJvdGVjdGVkR0Y0SShfc3lzbGliR0YlNiNGLiEiIkYlRiVGJQ==

plot(h2(x),x=-2..6,y=-10..5);

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

Das Intervall ist [-1,5]

h3:=(x)->(x^3)-ln(x)-5;

NiM+SSNoM0c2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoKiQ5JCIiJCIiIi1JI2xuRzYkSSpwcm90ZWN0ZWRHRjRJKF9zeXNsaWJHRiU2I0YuISIiISImRjBGJUYlRiU=

plot(h3(x),x=-1..3,y=-5..5);

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

Das Intervall ist [0,2]

restart;

f:=(x)->2*x*cos(x)-((x-2)^2);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJjkkIiIiLUkkY29zR0YlNiNGLkYvIiIjKiQsJkYuRi8hIiNGL0YzISIiRiVGJUYl

Das genaue Resultatgenau:=f(0.4);

NiM+SSZnZW5hdUc2IiQhKzA3OkI9ISIq

Taylor-Funktion von Mapletaylor(f(x),x=0,4);

NiMrLUkieEc2IiEiJSIiISIiJyIiIiEiIiIiI0YqIiIkLUkiT0dJKnByb3RlY3RlZEdGLzYjRikiIiY=

Taylor-Funktion von HandNiM+LSZJIlBHNiI2I0kibkdGJzYjSSJ4R0YnLUkkc3VtR0YnNiQqJi0pSSJmR0YnSSJrR0YnNiNJI3gwR0YnIiIiLUkqZmFjdG9yaWFsR0kqcHJvdGVjdGVkR0Y5NiNGMyEiIi9GMzsiIiFGKQ==NiM+LSZJIlBHNiI2IyIiJDYjSSJ4R0YnLCotSSJmR0YnNiNJI3gwR0YnIiIiKiYtSSNmMUdGJ0YvRjEsJkYrRjFGMCEiIkYxRjEqKC1JI2YyR0YnRi9GMS1JKmZhY3RvcmlhbEdJKnByb3RlY3RlZEdGPDYjIiIjRjZGNUY+RjEqKC1JI2YzR0YnRi9GMS1GO0YoRjZGNUYpRjE=P:=x->f(x0)+f1(x0)*(x-x0)+(f2(x0)/(2)!)*((x-x0)^2)+(f3(x0)/(3)!)*((x-x0)^3);

NiM+SSJQRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCotSSJmR0YlNiNJI3gwR0YlIiIiKiYtSSNmMUdGJUYvRjEsJjkkRjFGMCEiIkYxRjEqKC1JI2YyR0YlRi9GMS1JKmZhY3RvcmlhbEdJKnByb3RlY3RlZEdGPTYjIiIjRjdGNUY/RjEqKC1JI2YzR0YlRi9GMS1GPDYjIiIkRjdGNUZFRjFGJUYlRiU=

x0:=0;

NiM+SSN4MEc2IiIiIQ==

f1:=D(f);

NiM+SSNmMUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwqLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiU2IzkkIiIjKiZGMyIiIi1JJHNpbkdGL0YyRjYhIiMiIiVGNkYzRjlGJUYlRiU=

f2:=D(f1);

NiM+SSNmMkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoLUkkc2luRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiU2IzkkISIlKiZGMyIiIi1JJGNvc0dGL0YyRjYhIiNGOUY2RiVGJUYl

f3:=D(f2);

NiM+SSNmM0c2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiU2IzkkISInKiZGMyIiIi1JJHNpbkdGL0YyRjYiIiNGJUYlRiU=

P(x);

NiMsKiEiJSIiIkkieEc2IiIiJyokRiYiIiMhIiIqJEYmIiIkRis=

Das approximierte Resultatapprox:=P(0.4);

NiM+SSdhcHByb3hHNiIkISVDPSEiJA==

Restgliet R3(x) f\374r Sch\344tzung:NiM+LSZJIlJHNiI2I0kibkdGJzYjSSJ4R0YnKigtKUkiZkdGJywmRikiIiJGMUYxNiMtSSN4aUdGJ0YqRjEtSSpmYWN0b3JpYWxHSSpwcm90ZWN0ZWRHRjc2I0YwISIiKSwmRitGMUkjeDBHRidGOUYwRjE=NiM+LSZJIlJHNiI2IyIiJDYjSSJ4R0YnKigtKiRJImZHRiciIiVGKiIiIi1JKmZhY3RvcmlhbEdJKnByb3RlY3RlZEdGNDYjLCZGKUYxRjFGMSEiIkYrRjA=R:=(x)->f4(x)/(3+1)!*(x^4);

NiM+SSJSRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigtSSNmNEdGJTYjOSQiIiItSSpmYWN0b3JpYWxHSSpwcm90ZWN0ZWRHRjQ2IyIiJSEiIkYwRjZGJUYlRiU=

f4:=D(f3);

NiM+SSNmNEc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmLUkkc2luRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiU2IzkkIiIpKiZGMyIiIi1JJGNvc0dGL0YyRjYiIiNGJUYlRiU=

R(x);

NiMsJComLCYtSSRzaW5HNiRJKnByb3RlY3RlZEdGKUkoX3N5c2xpYkc2IjYjSSJ4R0YrIiIpKiZGLSIiIi1JJGNvc0dGKEYsRjAiIiNGMEYtIiIlI0YwIiND

sin(x) und cos(x) werden auf den Maximalwert 1 gesetzt. Bei anderen Funktionen kann sonst ein negativer Wert entstehen. Durch die Ersetzung hat man sicher das Maximum des Fehlers.R:=x->1/24*(8+2*x)*x^4;

NiM+SSJSRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJiwmIiIpIiIiOSQiIiNGMEYxIiIlI0YwIiNDRiVGJUYl

Gesch\344tzter Fehler:geschaetzt:=R(0.4);

NiM+SStnZXNjaGFldHp0RzYiJCIrbm1tJ1EqISM3

tatsaechlich:=abs(genau-approx);

NiM+SS10YXRzYWVjaGxpY2hHNiIkIicmelspISIq

verhaeltnis:=geschaetzt/tatsaechlich;

NiM+SSx2ZXJoYWVsdG5pc0c2IiQiK3Y5KWU1IiEiKQ==