matlab 第七章课后题答案

第一题分解因式

syms x y z

>> A=x^9-1;

>> factor(A)

ans =

(x-1)*(x^2+x+1)*(x^6+x^3+1) 解（1）

>> B=x^4+x^3+2*x^2+x+1;

>> factor(B)

ans =

(x^2+1)*(x^2+x+1) 解（2）

> C=125*x^6+75*x^4+15*x^2+1;

>> factor(C)

ans =

(5*x^2+1)^3 解（3）

> D=x^2+y^2+z^2+z*(x*y+y*z+z*x);

>> factor(D)

ans =

x^2+y^2+z^2+z*x*y+y*z^2+z^2*x 解（4）

第二题化简表达式

syms x y a b

>> s=y/x+x/y;

>> simplify(s)

ans =

(x^2+y^2)/x/y 解（1）s=sqrt(a+sqrt(a^2-b))/2+sqrt(a-sqrt(a^2-b))/2;

ans =

1/2*(a+(a^2-b)^(1/2))^(1/2)+1/2*(a-(a^2-b)^(1/2))^(1/2) 解（2）

s=2*cos(x)^2*x-sin(x)^2*x;

>> simplify(s)

ans =

x*(3*cos(x)^2-1) 解（3）s=sqrt(3+2*(sqrt2))

第三题求函数的极限

> syms x

>> f=(x^2-6*x+8)/(x^2-5*x+4);

> limit(f,x,4)

ans =

2/3 解（1）

>> f=abs(x)/x;

>> limit(f,x,0)

ans =

NaN 解（2）

f=(sqrt(1+x^2)-1)/x;

>> limit(f,x,0)

ans =

0 解（3）

f=(x+1/x)^x;

>> limit(f,x,inf,'left')

ans =

Inf 解（4）

第四题求函数的符号导数

f=3*(x^2)-5*x+1;

>> diff(f)

ans =

6*x-5 解(1)y’

>> diff(f,x,2)

ans =

6 解（1）

> y=sqrt(x+sqrt(x+sqrt(x)));

>> diff(y)

ans =

1/2/(x+(x+x^(1/2))^(1/2))^(1/2)*(1+1/2/(x+x^(1/2))^(1/2)*(1+1/2/x^(1/2))) 解（2）

diff(y,x,2)

ans =

-1/4/(x+(x+x^(1/2))^(1/2))^(3/2)*(1+1/2/(x+x^(1/2))^(1/2)*(1+1/2/x^(1/2)))^2+1/2/(x+(x+x^(1/2) )^(1/2))^(1/2)*(-1/4/(x+x^(1/2))^(3/2)*(1+1/2/x^(1/2))^2-1/8/(x+x^(1/2))^(1/2)/x^(3/2)) 解(2)

y=sin(x)-x^2/2;

> diff(y)

ans =

cos(x)-x 解（3）

>> diff(y,x,2)

ans =

-sin(x)-1 解（3）

syms x y z

>> z=x+y-sqrt(x^2+y^2);

>> diff(z,x,y)

ans =

1-1/(x^2+y^2)^(1/2)*y 解（4）

>> diff(y,x)

ans =

0 解（4）

第五题求不定积分

x=sym('x');

>> f=1/(x+a);

>> int(f)

ans =

log(x+a) 解（1）

>> f=(1-3*x)^3;

>> int(f)

ans =

-1/12*(1-3*x)^4 解（2）

>> f=(1/(sin(x)^2*cos(x)^2));

>> int(f)

ans =

1/sin(x)/cos(x)-2/sin(x)*cos(x) 解（3）

>> f=x^2/(sqrt(a^2+x^2));

>> int(f)

ans =

1/2*x*(a^2+x^2)^(1/2)-1/2*a^2*log(x+(a^2+x^2)^(1/2)) 解（4）

第六题求定积分

> x=sym('x');

> int((x*(2-sin(x)^2))^12,0,1)

ans =

-13072167041243000966100527033032931/1439431206610157332070400000000000*sin(1)^19 *cos(1)-63988617583073709724938474490679159346183608452999323027852007/820274272 498737105178830959457441284892917760000000000*cos(1)*sin(1)+417844027386435896683 78350956709518241555640967463723429593/1230411408748105657768246439186161927339 376640000000000*sin(1)^3*cos(1)-6287598784304532394386769772554886862775718358540 017487607/1794349971090987417578692723813152810703257600000000000*sin(1)^7*cos(1)-117903417317/3522410053632*sin(1)^23*cos(1)+9312911877102093870852677177232352401 4639/1529945519744217245425650892800000000000*sin(1)^17*cos(1)-2677966496932891906 2789407028617562008014032541099/6567698857381017597043504106365255680000000000* sin(1)^10-90936661567370530014104030508869215332048771345315401/53375584364747317 6140678428961747763200000000*cos(1)^2+2541573211/146767085568*sin(1)^24+940325057 70279736611460220749/16522396770089041920000000000*sin(1)^20+2704734082846637530 0822998906838403821858600396601/4670363631915390291230936253415292928000000000* sin(1)^8+5542192477209543230894137867604219553503/255755814666761099367961067520 00000000*sin(1)^16+6290548805350754451916704658025155325352197570086424587493/20 18643717477360844776029314289796912041164800000000000*sin(1)^9*cos(1)-74888453896 484988301898479573506809/1756135812378578105401344000000000*sin(1)^18-66275838868 9551809679364987359346538465101954279/84285468669723059162058302698354114560000 0000*sin(1)^14+4741716006420769418428944378170996543275038000761225969993/153801 4260935132072210308048982702409174220800000000000*sin(1)^5*cos(1)-273698005143037 11474211657412466731397670944577/100266509581957021396215456910540800000000000* sin(1)^15*cos(1)+18044178399358284974551495/22109663333532016620601344*sin(1)^21*co s(1)+159018588498544047612814017616772807534595903672788230889/18504234076875807 7437802687143231383603773440000000000*sin(1)^13*cos(1)-42990929319556261053136947 04999192498272936575705473777243/2220508089225096929253632245718776603245281280 000000000*sin(1)^11*cos(1)+72644795857216400572361363249893595045872001672201/160 1267530942419528422035286885243289600000000*sin(1)^4-179168559345113148705406926 00563709435304263998599/3002376620517036615791316162909831168000000000*sin(1)^6-3 228431702614231399553/6979060395685611307008*sin(1)^22+1809464903223467961506769 2014871302547716716561173/8669362491742943228097425420402137497600000000*sin(1)^1 2+908212034006674534482628671295497731536658010905190650989336891/1066356554248 3582367324802472946736703607930880000000000

解（1）

int(x/(x^2+x+1),-1,1)