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1、数学软件第四章微积分本章考虑用Maple系统求解极限、导数、微分、积分、级数展开、级数求和等问题。4.5导数的应用1、Rolle定理f:=x->x^5-2*x^2+x;diff(f(x),x);iscont(f(x),x=0..1,closed);evalb(f(0)=f(1));fsolve(f(x)=0,x=0..1);plot(f(x),x=0..1);例1.证明方程在(0,1)内至少有一个根.2、函数的单调性例2.确定在(-∞,+∞)内的单调性.f:=x->exp(x)-x-1;f1:=diff(f(x),x);fsolve(f1=0,x);assume(x>0);is(f1
2、>0);assume(x<0);is(f1<0);plot(f(x),x=-10..5);3、函数的极值例3.求函数的极值.restart:f:=x->2*x^3-6*x^2-18*x+7;f1:=diff(f(x),x);sols:=fsolve(f1=0,x);assume(x<-1);is(f1>0);assume(x>-1,x<3);is(f1<0);assume(x>3);is(f1<0);plot(f(x),x=-5..10);例4.利用高阶导数求函数的极值.restart:f:=x->2*x^3-6*x^2-18*x+7;f1:=diff(f(x),x);f2:=dif
3、f(f1,x);sols:=[fsolve(f1=0,x)];subs(x=sols[1],f2);subs(x=sols[2],f2);例5.求函数在区间[-5,1]上的最大值和最小值.restart:f:=x+sqrt(1-x);eq:=diff(f,x);sols:=[solve(eq,x)];evalf(subs(x=sols[1],f));evalf(subs(x=-5,f));evalf(subs(x=1,f));例6.判断下列曲线的凹凸性restart:f:=x+sqrt(1-x);diff(f,x$2);g:=x^3;g1:=diff(g,x$2);assume(x>
4、0);is(g1>0);assume(x<0);is(g1<0);restart:f:=3*x^4-4*x^3+1;f1:=diff(f,x$2);eq:=f1=0;solve(eq,x);例7.求函数的拐点.assume(x<0);is(f1>0);assume(x>0,x<2/3);is(f1<0);assume(x>2/3);is(f1>0);subs(x=0,f);subs(x=2/3,f);plot(f,x=-1/2..1);4.6不定积分调用形式:int(表达式,积分变量);1、不定积分的计算例:int(x^2-2*x+3,x);int(exp(x+1),x);不定积分表
5、Int(k,x)=int(k,x);Int(x^mu,x)=int(x^mu,x);Int(1/x,x)=int(1/x,x);Int(1/sqrt(1-x^2),x)=int(1/sqrt(1-x^2),x);Int(cos(x),x)=int(cos(x),x);Int(sin(x),x)=int(sin(x),x);Int(1/cos(x)^2,x)=int(1/cos(x)^2,x);Int(1/sin(x)^2,x)=int(1/sin(x)^2,x);Int(sec(x)*tan(x),x)=int(sec(x)*tan(x),x);Int(csc(x)*cot(x),x
6、)=int(csc(x)*cot(x),x);Int(a^x,x)=int(a^x,x);例1.计算下列不定积分Int(1/(x*x^(1/3)),x)=int(1/(x*x^(1/3)),x);Int((1+x+x^2)/(x*(1+x^2)),x)=int((1+x+x^2)/(x*(1+x^2)),x);Int(1/(sin(x/2)^2*cos(x/2)^2),x)=int(1/(sin(x/2)^2*cos(x/2)^2),x);Int(1/(3+sin(x)^2),x)=int(1/(3+sin(x)^2),x);可利用combine对表达式进行化简Int(1/(sin(
7、x/2)^2*cos(x/2)^2),x)=int(combine(1/(sin(x/2)^2*cos(x/2)^2)),x);(1)换元积分法2、积分方法with(student):ut:=Int((cos(x)+1)^3*sin(x),x);changevar((cos(x)+1)=u,ut);ut1:=value(%);subs(u=(cos(x)+1),ut1);restart:with(student):assume(a>0);assume(t>
