1、2 一元微積分實驗2.1 曲線繪圖練習題2.1 會出下列常見曲線的圖形（其中a=1,b=2,c=3）.立方拋物線syms x y; ezplot(y=x(1/3),-5,5) title(y=x(1/3)高斯曲線syms x y; ezplot(y=exp(-x2),-5,5) title(y=exp(-x2)笛卡爾曲線 syms x y; ezplot(x3+y3=3*x*y,-2,2) title(x3+y3-3*x*y=0)蔓葉線 syms x y; ezplot(y2*(1-x)=x3,-10,10) title(y2=x3/(1-x)擺線 syms t; x=t-sin(t); y=

2、2-2*cos(t); ezplot(x,y)星形線 syms t; x=cos(t)3; y=sin(t)3; ezplot(x,y)螺旋線 t=0:0.1:30; x=cos(t); y=2*sin(t); z=3*t; plot3(x,y,z);阿基米得螺旋線 theta=0:0.1:6; r=theta; plot(r,theta)2.2極限與導數1.求出下列極限：（1） syms n; limit(n3+3n)(1/n),n,inf)ans =3（2）syms n; limit(n+2)(1/2)-2*(n+1)(1/2)+n(1/2),n,3) ans = 5(1/2)-4+3(1

3、/2) vpa(ans,20) ans = -.318812149313330101e-1（3）syms x; limit(x*cot(2*x),x,0) ans = 1/2（4）clear syms x m; limit(cos(m/x)x,x,inf) ans = 1（5）syms x; limit(1/x-1/(expx-1),x,1) ans = (exp-2)/(exp-1) (6) clear syms x; limit(x2+x)(1/2)-x,x,inf) ans = 1/22.有個客戶看重某套面積為180，每平方米7500的房子.他計劃首付30%，其余70%用20年按揭貸款（

4、貸款年利率5.04%）.按揭貸款中還有10萬元為公積金貸款（貸款年利率4.05%），請問他的房屋總價、收付款額和月付款額分別為多少？function c=fukuan(m),c=(7500*180*(1+0.0504)20+100000*(1+0.0405)20)*m總付款：m=1;y=fukuan(m)c = 3.8306e+006y = 3.8306e+006首付：m=0.3;c=fukuan(m)c = 1.1492e+006c = 1.1492e+006月付款額：clear m=0.7/(20*12);c=fukuan(m)c = 1.1172e+004c = 1.1172e+0043

5、.作出下列函數及其導函數的圖形，觀察極值點、最值點的位置并求出所有駐點以及對應的二階導數值，求出函數的單調區(qū)間.(1)函數的圖形：fplot(x2*sin(x2-x-2),-2,2) grid on title(f(x)=x2*sin(x2-x-2),-2,2)導函數的圖形：clear syms x y y=x2*sin(x2-x-2); yx=diff(y,x)yx =2*x*sin(x2-x-2)+x2*cos(x2-x-2)*(2*x-1) fplot(yx,-2,2)極值1：f=inline(-x2*sin(x2-x-2),x); x,f=fminsearch(f,-1.5)x = -

6、1.5326f = -2.2364極值2： f=inline(x2*sin(x2-x-2),x); x,f=fminsearch(f,-0.6)x = -0.7315f = -0.3582極值3：f=inline(x2*sin(x2-x-2),x); x,f=fminsearch(f,0)x = 1.5951f = -2.2080最值1：x=-2:0.1:-1.8; y=x.2.*sin(x.2-x-2);m k=min(y)m = -3.0272k = 1 x(k)ans =-2最值2：x=-1.8:0.1:2; y=x.2.*sin(x.2-x-2);m k=max(y)m = 2.214

7、0k =4 x(k)ans = -1.5000駐點1及相應的二階導數值：f=inline(-x2*sin(x2-x-2),x); x,f=fminsearch(f,-1.5)x = -1.5326f = -2.2364 syms x y y=x2*sin(x2-x-2); yxx=diff(y,x,2); x=-1.5326;eval(yxx)ans = -44.1089駐點2及相應的二階導數值：clear f=inline(x2*sin(x2-x-2),x); x,f=fminsearch(f,-0.6)x = -0.7315f = -0.3582 syms x y y=x2*sin(x2-

8、x-2); yxx=diff(y,x,2); x=-0.7315;eval(yxx)ans =6.9830駐點3及相應的二階導數值：clear f=inline(x2*sin(x2-x-2),x); x,f=fminsearch(f,1.5)x = 1.5951 f = -2.2080 syms x y y=x2*sin(x2-x-2); yxx=diff(y,x,2); x=1.5951;eval(yxx)ans = 18.3287函數的單調區(qū)間為：單調遞增區(qū)間:-2x-1.5326 -0.7315x0以及1.5951x2;單調遞減區(qū)間：-1.5326x-0.7315以及0 x fplot(

9、3*x5-20*x3+10,-3,3) grid on title(f(x)=3*x5-20*x3+10,x,-3,3)導函數的圖形：clear fplot(3*x5-20*x3+10,-3,3) grid on title(f(x)=3*x5-20*x3+10,x,-3,3) syms x y y=3*x5-20*x3+10; yx=diff(y,x) yx = 15*x4-60*x2 fplot(yx,-3,3)極值點1：ff=inline(-(3*x5-20*x3+10),x); x,f=fminsearch(ff,-2)x = -2f = -74極值點2：ff=inline(3*x5-

10、20*x3+10,x); x,f=fminsearch(ff,2)x = 2f = -54最值1：clear x=-3:0.1:-1; y=3.*x.5-20.*x.3+10; m k=min(y)m = -179k = 1 x(k)ans =-3最值2：x=1:0.1:3; y=3.*x.5-20.*x.3+10; m k=max(y)m = 199k = 21 x(k)ans = 3駐點1及相應的二階導數值：clear syms x y y=3*x5-20*x3+10; yxx=diff(y,x,2); x=-2;eval(yxx)ans = -240駐點2及相應的二階導數值：clear

11、syms x y y=3*x5-20*x3+10; yxx=diff(y,x,2); x=2;eval(yxx)ans = 240函數的單調區(qū)間為：單調遞減區(qū)間為：-2x0以及0 x2;單調遞增區(qū)間為：-3x-2以及2x grid on title(f(x)=abs(x3-x2-x-2),x,-3,3)導函數的圖形為：syms x y y=abs(x3-x2-x-2); yx=diff(y,x) yx = abs(1,x3-x2-x-2)*(3*x2-2*x-1) fplot(yx,-3,3) grid on極值點：ff=inline(abs(x3-x2-x-2),x); x,f=fminse

12、arch(ff,1)x = 2.0000f = 1.5543e-014最大值的相反數為：x=-3:0.1:-2; y=x.3-x.2-x-2; m k=min(y)m = -35k = 1 x(k)ans =-3（注：最大值為35）最小值是：x,f=fminsearch(abs(x3-x2-x-2),2)x = 2f = 0駐點及相應的二階導數值：clear syms x y y=x3-x2-x-2; yxx=diff(y,x,2); x=2;eval(yxx)ans = 10 syms x y y=-(x3-x2-x-2); yxx=diff(y,x,2); x=2;eval(yxx)ans

13、 = -10函數的單調區(qū)間為：單調遞增區(qū)間為：-3x2單調遞減區(qū)間為2x fplot(x4-2x,-2,2) grid onf=inline(x4-2x,x); y1=fzero(f,-1)y1 = -0.8613 y2=fzero(f,1,1.5)y2 =1.2396（2）fplot(x*log(sqrt(x2-1)+x)-sqrt(x2-1)-0.5*x,1,10) grid onf=inline(x*log(sqrt(x2-1)+x)-sqrt(x2-1)-0.5*x,x); y=fzero(f,2)y = 2.11552.農夫老李有一個半徑為10m的牛欄，里面長滿了草，老李要將家里的一

14、頭牛拴在牛欄邊界的一根欄桿上，要求只讓牛吃到圓形牛欄中一一半的草，請問拴牛鼻子的繩子應該為多長？fplot(100*acos(1-1/200*(r2)+r2*acos(1/20*r)-10*sqrt(r2-1/400*r4)-50*pi,10,20) grid onf=inline(100*acos(1-1/200*(r2)+r2*acos(1/20*r)-10*sqrt(r2-1/400*r4)-50*pi,r); y=fzero(f,12)y = 11.58733.求解下列非線性方程組在遠點附近的根：clear syms x y z x y z=solve(9*x2+36*y2+4*z2-

15、36,x2-2*y2-20*z,16*x-x3-2*y2-16*z2,x,y,z) x = .13416411205540333669211943944519 .13416411205540333669211943944519 3.6882488593362642482287401666476 3.6882488593362642482287401666476 111.77622956892053765218612881478+32.680303502566776107753058941785*i 111.77622956892053765218612881478+32.68030350256

16、6776107753058941785*i -4.8047917540141686806150019329857 -4.8047917540141686806150019329857 -8.5700803552185742086781153026662 -8.5700803552185742086781153026662 111.77622956892053765218612881478-32.680303502566776107753058941785*i 111.77622956892053765218612881478-32.680303502566776107753058941785*

17、i y = .99720662260461349181073758666936 -.99720662260461349181073758666936 1.5801464927409783709908110788805*i -1.5801464927409783709908110788805*i 78.346264134912844032563308403253+41.639045100647464014558398318904*i -78.346264134912844032563308403253-41.639045100647464014558398318904*i 2.253547634

18、3185085513549721744346*i -2.2535476343185085513549721744346*i 4.5905484614647522902202849261298*i -4.5905484614647522902202849261298*i 78.346264134912844032563308403253-41.639045100647464014558398318904*i -78.346264134912844032563308403253+41.639045100647464014558398318904*i z = -.985421043684692628

19、46330807189218e-1 -.98542104368469262846330807189218e-1 .92984527627192419820957388402960 .92984527627192419820957388402960 130.86346028082967813004883465010-287.16461448768476745583792307227*i 130.86346028082967813004883465010-287.16461448768476745583792307227*i 1.6621488839863822158515099085012 1.

20、6621488839863822158515099085012 5.7796273824508065886875777144674 5.7796273824508065886875777144674 130.86346028082967813004883465010+287.16461448768476745583792307227*i 130.86346028082967813004883465010+287.16461448768476745583792307227*i 4.畫出下面兩個橢圓的圖形，并求出他們所有的交點坐標, ezplot(x-2)2+(y+2*x-3)2=5) grid on hold on ezplot(18*(x-3)2+y2=36)title(x-2)2+(y+2x-3)2=5) title(x-2)2+(y+2x-3)2=5以及18(x-3)2+y2=36)function f=fun(t)x=t(1);y=t