University:
Kentucky State University
Course:
COS 232| Programming I
Academic year:

2020
Views:

388

Pages:

11
Author:

lankstoetp15

4.18 f(x) = x-1-1/2*sin(x) f '(x) = 1-1/2*cos(x) f ''(x) = 1/2*sin(x) f '''(x) = 1/2*cos(x) f IV(x) = -1/2*sin(x) Using the Taylor Series Expansion (Equation 4.5 in the book), we obtain the following 1st, 2nd, 3rd, and 4th Order Taylor Series functions shown below in the Matlab program-f1, f2, f4. Note the 2nd and 3rd Order Taylor Series functions are the same. From the plots below, we see that the answer is the 4th Order Taylor Series expansion. x=0:0.001:3.2; f=x-1-0.5*sin(x); subplot(2,2,1); plot(x,f);grid;title('f(x)=x-1-0.5*sin(x)');hold on f1=x-1.5; e1=abs(f-f1); %Calculates the absolute value of the difference/error subplot(2,2,2); plot(x,e1);grid;title('1st Order Taylor Series Error'); f2=x-1.5+0.25.*((x-0.5*pi).^2); e2=abs(f-f2); subplot(2,2,3); plot(x,e2);grid;title('2nd/3rd Order Taylor Series Error'); f4=x-1.5+0.25.*((x-0.5*pi).^2)-(1/48)*((x-0.5*pi).^4); e4=abs(f4-f); subplot(2,2,4); plot(x,e4);grid;title('4th Order Taylor Series Error');hold off f(x )= x -1-0.5*s in(x ) 1s t O rder Tay lor S eries E rror 3 0.8 2 0.6 1 0.4 0 0.2 -1 0 0 1 2 3 4 2nd/3rd O rder Tay lor S eries E rror 0.2 0 1 2 3 4 4th O rder Tay lor S eries E rror 0.015 0.15 0.01 0.1 0.005 0.05 0 0 0 1 2 3 4 0 1 2 3 4 4.19 EXCEL WORKSHEET AND PLOTS x -2.000 -1.750 -1.500 -1.250 -1.000 -0.750 -0.500 -0.250 0.000 0.250 0.500 0.750 1.000 1.250 1.500 1.750 2.000 f(x) 0.000 2.141 3.625 4.547 5.000 5.078 4.875 4.484 4.000 3.516 3.125 2.922 3.000 3.453 4.375 5.859 8.000 f(x-1) -2.891 0.000 2.141 3.625 4.547 5.000 5.078 4.875 4.484 4.000 3.516 3.125 2.922 3.000 3.453 4.375 5.859 f(x+1) f'(x)-Theory f'(x)-Back f'(x)-Cent f'(x)-Forw 2.141 10.000 11.563 10.063 8.563 3.625 7.188 8.563 7.250 5.938 4.547 4.750 5.938 4.813 3.688 5.000 2.688 3.688 2.750 1.813 5.078 1.000 1.813 1.063 0.313 4.875 -0.313 0.313 -0.250 -0.813 4.484 -1.250 -0.813 -1.188 -1.563 4.000 -1.813 -1.563 -1.750 -1.938 3.516 -2.000 -1.938 -1.938 -1.938 3.125 -1.813 -1.938 -1.750 -1.563 2.922 -1.250 -1.563 -1.188 -0.813 3.000 -0.313 -0.813 -0.250 0.313 3.453 1.000 0.313 1.063 1.813 4.375 2.688 1.813 2.750 3.688 5.859 4.750 3.688 4.813 5.938 8.000 7.188 5.938 7.250 8.563 10.891 10.000 8.563 10.063 11.563 First Derivative Approximations Compared to Theoretical 14.0 12.0 10.0 8.0 Theoretical Backward Centered Forward f'(x) 6.0 4.0 2.0 0.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 -2.0 -4.0 x-values x f(x) -2.000 -1.750 -1.500 -1.250 -1.000 -0.750 -0.500 -0.250 0.000 0.250 0.500 0.750 1.000 1.250 1.500 1.750 2.000 f(x-1) 0.000 2.141 3.625 4.547 5.000 5.078 4.875 4.484 4.000 3.516 3.125 2.922 3.000 3.453 4.375 5.859 8.000 f(x+1) -2.891 0.000 2.141 3.625 4.547 5.000 5.078 4.875 4.484 4.000 3.516 3.125 2.922 3.000 3.453 4.375 5.859 f(x-2) 2.141 3.625 4.547 5.000 5.078 4.875 4.484 4.000 3.516 3.125 2.922 3.000 3.453 4.375 5.859 8.000 10.891 f(x+2) 3.625 -2.891 0.000 2.141 3.625 4.547 5.000 5.078 4.875 4.484 4.000 3.516 3.125 2.922 3.000 3.453 4.375 3.625 4.547 5.000 5.078 4.875 4.484 4.000 3.516 3.125 2.922 3.000 3.453 4.375 5.859 8.000 10.891 14.625 f''(x)Theory -12.000 -10.500 -9.000 -7.500 -6.000 -4.500 -3.000 -1.500 0.000 1.500 3.000 4.500 6.000 7.500 9.000 10.500 12.000 f''(x)- f''(x)-Cent Back 150.500 -12.000 -12.000 -10.500 -10.500 -9.000 -9.000 -7.500 -7.500 -6.000 -6.000 -4.500 -4.500 -3.000 -3.000 -1.500 -1.500 0.000 0.000 1.500 1.500 3.000 3.000 4.500 4.500 6.000 6.000 7.500 7.500 9.000 9.000 10.500 10.500 12.000 Approximations of the 2nd Derivative 15.0 10.0 f''(x) 5.0 0.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 -5.0 -10.0 -15.0 x-values 0.5 1.0 1.5 2.0 2.5 f''(x)-Theory f''(x)-Backward f''(x)-Centered f''(x)-Forward f''(x)Forw -10.500 -9.000 -7.500 -6.000 -4.500 -3.000 -1.500 0.000 1.500 3.000 4.500 6.000 7.500 9.000 10.500 12.000 13.500