# 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.

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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

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

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

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

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.

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

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?

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

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)

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?

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

>> 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

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

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

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.

From solution manual for text

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

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

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.

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

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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