Ans. to the q. no-1(b)
Matlab command: clear all close all x=-10:0.01:10; fx=((cos(x)).^2-x+6); plot(x,fx); ylim([-5,5]); grid on
Figure:

Ans. to the q. no-1(c)
Matlab command: clear all close all tol=10^-8; a=6; b=7; f=@(x) ((cos(x)).^2-x+6); r=f(a).*f(b); while (b-a)/2>tol if r<0 c=(a+b)./2; s=f(c); else ('Product is positive') end if f(c)==0 break end if f(c).*f(a)<0 b=c; else a=c; end end c Result: c = 6.776092305779457
Ans. to the q. no-1(d)
Solution: I have fixed tol=10^-8 so that I can find the solution is correct to seven decimal places and the results of MATLAB command using tol=10^-8 : c =

6.776092320680618

c =

6.776092305779457
Ans.to.the.q.no-2(b)
clear all close all x=-10:0.01:10; f=log(x)+(x.^2)-3; plot(x,f) grid on ylim([-10,10]) xlabel('x'); ylabel('f(x)'); Graph:

Ans.to.the.q.no-2(c): Bisection using for loop:
MATLAB COMMAND: clear all close all format long a=1; b=2; f=@(x) log(x)+(x.^2)-3; r=f(a).*f(b); c=zeros(28,1); for n=1:28; if r<0 c(n)=(a+b)/2; s=f(c(n)); else ('Product is positive') end if f(c(n))==0 break end if f(c(n))*f(a)<0 b=c(n); else a=c(n); end end c Result: c =

1.500000000000000 1.750000000000000 1.625000000000000 1.562500000000000 1.593750000000000 1.578125000000000 1.585937500000000 1.589843750000000 1.591796875000000 1.592773437500000 1.592285156250000 1.592041015625000 1.592163085937500 1.592102050781250 1.592132568359375 1.592147827148438 1.592140197753906 1.592144012451172 1.592142105102539 1.592143058776856 1.592142581939697 1.592142820358276 1.592142939567566 1.592142879962921 1.592142909765244 1.592142924666405 1.592142932116985 1.592142935842276
Ans.to.the.q.no-2(d): Setting tolerance in above for loop: clear all close all format long tol=10^-8; a=1; b=2; f=@(x) log(x)+(x.^2)-3; r=f(a).*f(b); c=zeros(28,1); while (b-a)/2>tol for n=1:28; if r<0 c(n)=(a+b)/2; s=f(c(n)); else ('Product is positive') end if f(c(n))==0 break end if f(c(n))*f(a)<0 b=c(n); else a=c(n); end end end c

c =

1.500000000000000 1.750000000000000 1.625000000000000 1.562500000000000 1.593750000000000 1.578125000000000 1.585937500000000 1.589843750000000 1.591796875000000 1.592773437500000 1.592285156250000 1.592041015625000 1.592163085937500 1.592102050781250 1.592132568359375 1.592147827148438 1.592140197753906 1.592144012451172 1.592142105102539 1.592143058776856 1.592142581939697 1.592142820358276 1.592142939567566 1.592142879962921 1.592142909765244 1.592142924666405 1.592142932116985
If I set the tolerance via while loop just above of for loop then I can relate the tolerance and step number. Here I have observed that when I change tolerance less than 10^-8 result changes and also when I change the step number than result got change.
Ans. To. The. q. no-3(b) clear all close all x=-10:0.01:10; g=((2.8).*x)-(x.^2); plot(x,g) grid on; hold on; y=x; plot(x,y) ylim([-2,3]) Graph:

Rearranging the equation for which fixed point is a solution: clear all close all x=-10:0.01:10; g=((2.8).*x)-(x.^2)-x; plot(x,g) grid on; hold on; y=x; plot(x,y) ylim([-2,3]) graph:

Ans. to the q.no- 3(c):
MATLAB Command: clear all close all format long x=0.5; x(1)=x; g=@(x) (((2.8).*x)-(x.^2)); for i=1:151; if x(i)==g(x(i)) break else x(i+1)=g(x(i)); end end i x(i)
Result:
i =

151

ans =

1.800000000000003
Ans.To.the q. no-4(a): clear all close all x=-10:0.01:10; g=(cos(x)-sin(x)+x); plot(x,g) grid on; hold on; y=x; plot(x,y) ylim([-20,20]) Graph:

In above curve I can observe there are lots of fixed point.
Ans.to.the.q.no- 4(c)
Matlab command: clear all close all format long x=0.5; x(1)=x; g=@(x) (cos(x)-sin(x)+x); for i=1:39; if x(i)==g(x(i)) break else x(i+1)=g(x(i)); end end i x(i) result: i =

39

ans =

0.785398163397448 as in 4(a) we can see we have find lots of fixed point then considering initial guess=0.5 we can find a fixed point for corresponding guess. And 39 step are need to reach to fixed point.