KeytoMatlabexErcise7SchoolofMathematicalSciencesXiamenUniversityhttp:/gdjpkc.xmu.edu.cn 1+2*x+2*x^2+4/3*x^3+2/3*x^4+4/15*x^5+445*x^6+8/315*x^7+2/315*x^8+4/2835*x^9+4/14 175*x^10+8/155925*x^11+4/467775*x^12+8/6081075*X^13+8/42567525*x^14 >> clear; syms x f, f-exp(2*x); exp(-2)2*exp(-2)(x+1)+2exp(-2)°(x+1)2+4/3exp(-2)(x+1)^3+2/3°exp(-2)*(x+1)^4+4/ 15*exp(-2)*(x+1)^+4/45*exp(-2)*(x+1)^6+8/315*eXp(-2)(x+1)^7+2/315*exp(-2)°(x+1)^8 3) > clear; syms x y f; fexp(2*y > taylor(f, 5, x) ans 1+2*x*y+2*y^2*x^2+4/3*y3*x3+2/3*y4*x4 > clear; format short e, syms a x; fcos(x)+2*x, f int=int(f) >>b=a+10*pi, a+5 pi, a+pi, a+ 1/2*pi, a+1/16*pi, a+ 1/1024 pi]: >>fork=1:6 y(kint(f,a, b(k)); y app(k =(subs(f int, b(k)-subs(f int, a)*(b(k)-a) >>y, y app -2*sin(a)+10*a*pi+25pi^2, 2*sin(a)+2°a*p+pi^2 cos(a)+a*pi+1/4*pi"2-sin(a) sin(a+l/6°p1)+l/8°a*pi+1/256pi^2-sina) sin(a+1/l024°p)+1/512*api+1/048576*p^2-sin(a) 10°(a+10*p)2-a^2)* a+5*p)^2-a^2)pi, (-2*sin(a)+(a+pi)^2-a2)*pl, l/2*(cos(a)+(a+1/2°pi)^2-sin(a)a^2)°pi 1/16*(sin(a+1/16*pi)+(a+ 1/16 *pir2-sin(a)-ar2)*pi, l/1024(sin(a+1/1024*p)+(a+1/l024p)y2-sn(a)a^2)°pl >>y 0=subs(y, 0); y app 0=subs(y app, 0); error y=y o-y app 0 -3.0019e+004-3.6290e+0032.1137e+001-1.9792e+0001.8777e-0013.0679e-003 (a+ Conclusion While b closes to a 2(b-a)approaches >>clear; format long e; syms x xi; f1/(x+1)2: >> f int=int(f, 0, 1); Xi subs( inverse(f), f int)Key to MATLAB Exercise 7  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex7­6  1+2*x+2*x^2+4/3*x^3+2/3*x^4+4/15*x^5+4/45*x^6+8/315*x^7+2/315*x^8+4/2835*x^9+4/14  175*x^10+8/155925*x^11+4/467775*x^12+8/6081075*x^13+8/42567525*x^14  2) >> clear; syms x f; f=exp(2*x);  >> taylor(f,9,­1) ans =  exp(­2)+2*exp(­2)*(x+1)+2*exp(­2)*(x+1)^2+4/3*exp(­2)*(x+1)^3+2/3*exp(­2)*(x+1)^4+4/  15*exp(­2)*(x+1)^5+4/45*exp(­2)*(x+1)^6+8/315*exp(­2)*(x+1)^7+2/315*exp(­2)*(x+1)^8  3) >> clear; syms x y f; f=exp(2*x*y);  >> taylor(f,5,x) ans =  1+2*x*y+2*y^2*x^2+4/3*y^3*x^3+2/3*y^4*x^4  12.  >> clear; format short e; syms a x; f=cos(x)+2*x; f_int=int(f);  >> b=[a+10*pi, a+5*pi, a+pi, a+1/2*pi, a+1/16*pi, a+1/1024*pi];  >> for k=1:6  y(k)=int(f,a,b(k));  y_app(k)=(subs(f_int, b(k))­subs(f_int, a))*(b(k)­a);  end  >> y, y_app  y =  [ 20*a*pi+100*pi^2,  ­2*sin(a)+10*a*pi+25*pi^2,  ­2*sin(a)+2*a*pi+pi^2,  cos(a)+a*pi+1/4*pi^2­sin(a),  sin(a+1/16*pi)+1/8*a*pi+1/256*pi^2­sin(a),  sin(a+1/1024*pi)+1/512*a*pi+1/1048576*pi^2­sin(a)] y_app =  [ 10*((a+10*pi)^2­a^2)*pi,  5*(­2*sin(a)+(a+5*pi)^2­a^2)*pi,  (­2*sin(a)+(a+pi)^2­a^2)*pi,  1/2*(cos(a)+(a+1/2*pi)^2­sin(a)­a^2)*pi,  1/16*(sin(a+1/16*pi)+(a+1/16*pi)^2­sin(a)­a^2)*pi,  1/1024*(sin(a+1/1024*pi)+(a+1/1024*pi)^2­sin(a)­a^2)*pi]  >> y_0=subs(y, 0); y_app_0=subs(y_app, 0); error_y=y_0­y_app_0  error_y =  ­3.0019e+004 ­3.6290e+003 ­2.1137e+001 ­1.9792e+000 1.8777e­001  3.0679e­003  Conclusion: While b closes  to a,  ( ) sin 2 ( ) 2 2 a b a b  b a Ê Ê + ˆ + ˆ Á ˜ + - Á ˜ Ë Ë ¯ ¯ approaches  to  (cos 2 ) b  a  x + x dx Ú .  13.  >> clear; format long e; syms x xi; f=1/(x+1)^2;  >> f_int=int(f, 0, 1); xi= subs(finverse(f),f_int)