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Calculus

Math Review with Matlab:. Calculus. Taylor’s Series. S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn. Series Operations. Symbolic Summation Taylor Series Taylor Command Taylor Series Example Approximation and Comparison Example.

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Calculus

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  1. Math Review with Matlab: Calculus Taylor’s Series S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn

  2. Series Operations • Symbolic Summation • Taylor Series • Taylor Command • Taylor Series Example • Approximation and Comparison Example

  3. Symbolic Summation • Find the sum of the following series s1 and s2 if they converges Diverges! • Example 1: » s1=symsum(1/x^2,1,inf) s1 = 1/6*pi^2 Converges • Example 2 » num = 4*x*x-x-3 » den = x^3+2*x » s2=symsum(num/den,1,inf) s2 = inf

  4. Summation Examples » s3=symsum(1/(x-1.5)^2,1,inf) s3 = 4+1/2*pi^2 » eval(s3) ans = 8.9348 • Example 3: » s4=symsum((1/x)*(-1)^(x+1),1,inf) s4 = log(2) » eval(s4) ans = 0.6931 • Example 4:

  5. Finite Summation Example • Example 5: » syms x N; » s5=symsum((x+3)*(x+1),1,N) s5 = 7/6*N-11/6+3/2*(N+1)^2+1/3*(N+1)^3 » s5=simple(s5) s5 = 1/6*N*(31+15*N+2*N^2)

  6. Taylor Series • Taylor Series approximation is defined as: • MacLaurin Series is the Taylor series approximation with a=0:

  7. Taylor Command • taylor(f) is the fifth order MacLaurin polynomial approximation to f • taylor(f,n) is the (n -1)-st order MacLaurin polynomial • taylor(f,n,a) is the Taylor polynomial approximation about point a with order (n -1).

  8. Taylor Series Example • Given the function: 1) Find the first 6 Taylor Series Terms (a=0) 2) Find the first 4 terms about the point a=2 » sym x; » f=log(1+x) % Matlab's Natural Log f = log(1+x)

  9. Taylor Series Terms • Find the first 6 Taylor Series Terms (a=0) » taylor(f) %Default is 5th order ans = x-1/2*x^2+1/3*x^3-1/4*x^4+1/5*x^5 • Find the first 4 terms about the point a=2 • Note that this is 3rd order » taylor(f,4,2) ans = log(3)+1/3*x-2/3-1/18*(x-2)^2+1/81*(x-2)^3

  10. Taylor Series Approximation and Comparison Example • Given the function: 1) Plot f(x) from -2p to 2p 2) Find the first 8Taylor Series Terms (a=0) 3) Plot the approximation and compare against the original function f(x)

  11. Plot f(x) • The easiest way to generate a graph is to use ezplot • ezplot leaves the axes unlabeled » syms x » f=1/(5+4*cos(x)); » ezplot(f,-2*pi,2*pi); » grid on » xlabel('x');ylabel('f(x)')

  12. Plot of f(x)

  13. Taylor Approximation • To find the first 8 terms of the Taylor series approximation: » ft_8=taylor(f,8) ft_8 = 1/9+2/81*x^2+5/1458*x^4+49/131220*x^6

  14. Taylor approximation Original f(x) Comparison • Plot approximation: » hold on » ezplot(ft_8) » axis([-2*pi 2*pi 0 5]) • Approximation is only good for small x

  15. Summary • The symbolic toolbox can be used to analyze definite and indefinite series summations • Taylor series can be used to approximate functions • MacLaurin series is a special case of the Taylor series approximated around x=0 • Increase the number of terms to increase approximation accuracy

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