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Suggested problems from text (6 th edition) Chapter 3.1 p85 Problems 1, 4, 9, 10 Computer problems 1, 2, 4, 7 Chapter 3.2 p101 Problems 4, 15, 17, 19 Computer problems 1, 2, 4, 8, 9, 14 Chapter 3.3 p119 Problems 2, 3, 5 Computer problems 1, 3, 5, 6, 7

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p3.1-1: find intersection of y = 3x and y = exp(x) How is this problem related to finding roots of a function?

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p3.1-4 p85: solve the equation ln(x+1) + tan(2x) = 0

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X f(x) = ln(x+1) + tan(2x) Where are the roots?

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P3.1-9 p86 f(x) = cos(x) - cos(3x) Which roots will be easy to find? Which roots will be hard to find? Write a MabLab program for graphical solution.

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X f(x) = cos(x) – cos(3x) What are the multiplicities of these roots?

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y = inline(‘log(x+1)/(1-x.*x)’); x = linspace(-0.5, 2, 100); plot(x,y(x)) What is wrong with this MatLab code? With fplot we don’t need to create a vector of x values. P3.1-10 p86: y=log(x+1)/(1-x 2 ); Where are the roots?

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What is wrong with this plot? >> p1031=inline('log(1+x)/(1-x^2)'); >> fplot(p1031,[0,2]);

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X f1031(x) = ln(x+1)/(1-x 2 ) x(1)=-0.5; y(1)=f1031(x(1)); for i=2:149 x(i)=x(i-1)+0.01; y(i)=f1031(x(i)); end x(150)=1.01; y(150)=f1031(x(150)); for i=151:199 x(i)=x(i-1)+0.01; y(i)=f1031(x(i)); end plot(x,y)

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Computer problems 1, 2, 4, 7 will give you practice with your bisection code. Ignore details like those in #7 “one run”, “full machine precision”, “correct number of steps” and “false positive method” How can you be sure that your code is giving correct answers?

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Suggested problems from text Chapter 3.1 p85 Problems 1, 4, 9, 10 Computer problems 1, 2, 4, 7 Chapter 3.2 p101 Problems 4, 15, 17, 19 Computer problems 1, 2, 4, 8, 9, 14 Chapter 3.3 Problems 2, 3, 5 Computer problems 1, 3, 5, 6, 7

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>> f324=@(x) x.^4+2*x.^3-7*x.^2+3'; >> df324=@(x) 4*x.^3+6*x.^2-14*x; >> fplot(f324,[-4,2]) >> [r,fr,~]=newtfun(f324,df324,-1); >> disp([r,fr]) -0.6180 -0.0000 >> [r,fr,~]=newtfun(f324,df324,0.75); >> disp([r,fr]) 0.7913 -0.0000 >> [r,fr,~]=newtfun(f324,df324,1.7); >> disp([r,fr]) 1.6180 -0.0000 >> [r,fr,~]=newtfun(f324,df324,-4); >> >> disp([r,fr]) -3.7913 -0.0000 Problem 3.2-4 page 101: find all the zeros of x 4 +2x 3 -7x+3

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P3.2-15: f(x) = x 3 - x +1 if x 0 in newton’s method is 1, what is x 1 ? P3.2-17 f(x) = x 5 – x 3 +3 if x n in newton’s method is 1, what is x n+1 ? See slide 10 Lecture on finding zeros

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P3.2-19 f(x) = x 2 /(1 + x 2 ) from what starting values will newton’s method converge?

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f(x) = x 2 /(1+x 2 ) X Symmetry can cause cycling of Newton’s method Find analytical solution for when Newton’s method will converge.

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From solution manual for text

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Computer problems 1, 2, 4, 8, 9, 14 will give you practice with your newton code.

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Suggested problems from text Chapter 3.1 p85 Problems 1, 4, 9, 10 Computer problems 1, 2, 4, 7 Chapter 3.2 p101 Problems 4, 15, 17, 19 Computer problems 1, 2, 4, 8, 9, 14 Chapter 3.3 p119 Problems 2, 3, 5 Computer problems 1, 3, 5, 6, 7

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P3.3-2: f(x) = x 3 - 2x +2 if x 0 = 0 and x 1 = 1 in secant method, what is x 2 ? P3.3-3 f(x) = x 5 + x 3 +3 if x n-2 = 0 and x n-1 = 1 in secant method, what is x n ? See slide 33 in Lectures on finding zeros.

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P3.3-5: find the largest positive root of x 3 - 5x +3 by bisection, newton’s and secant methods

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Computer problems 1, 3, 5, 6, 7 will give you practice with your secant code. I think cp3.3-3 is referring to functions on p77 of text. I think cp3.3-5 is referring to example 1 on p113 of text f(x) = x 5 + x 3 +3

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