study guide

Number of Qs = 82 =82= 21 (level 1) + 21 (level 2) + 20 (level 3) + 20 (level 4)

(Question 1)What is the output of the following MATLAB expression? (a) A=[1 3; 4 5]; (b) C=eye(2)-A./A(c) The value of C(2,2) is

Solution: (b) C=[0,-1;-1,0] (c) C(2,2)=0

Multiple choice (Answers in bold):

Choice 1) 1 Choice 2) -1 Choice 3) 0 Choice 4) 2

(Question 2)Use the linspace function to create vectors identical to the colon notation t = 4:6:34 and select the correct answer(s)

Solution: t = linspace(4,34,6);


Choice 1) t = linspace(4,6,34); Choice 2) t = linspace(34,4,6); Choice 3) t = linspace(4,34,6); Choice 4) None

(Question 3)Use colon notation to create vectors identical to the following created with the linspace function The value of r = linspace (8,4.5,8); is identical to

Solution: r= [8, 7.5, 7, 6.5, 6, 5.5, 5, 4.5]


Choice 1) r = 8 : .5 : 4.5; Choice 2) r = 8 : 4.5 : 8; Choice 3) r = 8 : -0.5 : 4.5; Choice 4) r = 8 : 0.5 : 4.5;

(Question 4)The following matrix is entered in MATLAB:A=[3 2 1;0:0.5:1;linspace(6, 8, 3)]; (a) Write out the resulting matrix. (b) If we use colon notation to write a single-line MATLAB command to multiply the second row by the third column and assign the result to the variable C, the value of C would be same as

Solution: (a) A=[3, 2, 1;0, 0.5, 1; 6, 7, 8] (b) C = A(2,:)*A(:,3); C = 8.5;


Choice 1) C = A(:,3)*A(2,:); Choice 2) C = A(2,:)*A(:,3); Choice 3) C = a(2,:)*A(:,3); Choice 4) C = 8.5;

(Question 5)The density of freshwater can be computed as a function of temperature with the following cubic equation rho = 5.5289E-8 Tc^3 - 8.5016E-6 Tc^2 + 6.5622E-5 Tc + 0.99987 where rho = density (g/cm^3) and Tc = temperature (degree Celsius). Write a MATLAB code (a) to generate a vector of temperatures ranging from 32 degrees Fahrenheit to 93.2 degrees Fahrenheit using increments of 3.6 degrees Fahrenheit (b)to convert this vector to degrees Celsius (c)to compute a vector of densities based on the cubic formula (d)to create a plot of rho versus Tc. Recall that Tc = (5/9)*(Tf - 32)

(e)The value of rho(10) is closest to

Solution: a) Tf=32:3.6:93.2; b) Tc=5*(Tf-32)/9; c) rho=5.5289*1e-8*Tc.^3 - 8.5016*1e-6*Tc.^2 + 6.5622*1e-5*.Tc + 0.99987; d) plot(rho, Tc);

e) 0.9986

Multiple choice (Answers in bold):Choice 1) 1.0000 Choice 2) 2.0000 Choice 3) 0.9986 Choice 4) 0.9944

(Question 6)An example of computer hardware is Solution: An example of computer hardware is computer chips

Multiple choice (Answers in bold):Choice 1) computer chips

Choice 2) MS Word

Choice 3) A computer program with a series of instructions

Choice 4) All of the above

(Question 7)ASCII number for the character a=97, d=100, e=101, D=68, then what is the ASCII number for the character A? Solution: The ASCII number for the character A is 65

Multiple choice (Answers in bold):Choice 1) 97 Choice 2) 66 Choice 3) 65 Choice 4) 98

(Question 8)How many bits does 1101 have? Solution: 1101 has 4 bits


(Question 9)How many items can be represented by 3 bits binary number? Solution: 8 items can be represented by 3 bits


(Question 10)64 bits equals Solution: 64 bits equals 8 bytes

Multiple choice (Answers in bold):Choice 1) 1 byte Choice 2) 8 bytes Choice 3) 3 bytes Choice 4) None

(Question 11)How many bytes does a 1TB hard disk have Solution: A 1TB hard disk has more than 1000 billion bytes

Multiple choice (Answers in bold):Choice 1) 1 trillion Choice 2) greater than 1000 billion Choice 3) 1 billion Choice 4) less than 1 trillion

(Question 12)The value of the binary number 01001 in decimal is Solution: The value of the binary number 01001 in decimal is 9


(Question 13)The value of the binary number 111 in decimal is

Solution: The value of the binary number 111 in decimal is 1*2^2 + 1*2^1 + 1*2^0 = 7


Choice 1) 111 Choice 2) 5 Choice 3) 7 Choice 4) None

(Question 14)The value of the decimal number 10 in binary is

Solution: The value of the decimal number 10 in binary is 1010 which is equal to 01010


Choice 1) 10 Choice 2) 110 Choice 3) 1010 Choice 4) 01010

(Question 15)If b=1; what is value of a at the end of following operations b=b+1; a = b + 1;

Solution:

If b=1

b=b+1 b=1+1=2

a=b+1 b=2+1=3



(Question 16)If b=1; b=b+1; a =2*b + 1; What is the value of c if c=a+b?

Solution: b=1 b=b+1 = 1+1 = 2 a=2*b + 1 = 2*2 + 1 = 5 c=a+b = 5 + 2=7



(Question 17)If x=2; y= x^2 + 2*x + 10=?

Solution: y= x^2 + 2*x + 10 y= 2^2 + 2*2 +10 y= 4 + 4 + 10 y= 18



(Question 18)If f= inline('x^2 + 2*x + 10'); f(2) = ?

Solution: f = inline('x^2 + 2*x + 10') f = inline('2^2 + 2*2 + 10')=18



(Question 19)If f(x)= x^2 + 2*x ; g(y) = 2*y; then f(1) * g(2) = ?

Solution:

f(x)= x^2 + 2*x f(1)=1^2 + 2*1=3

g(y) = 2*y g(2)=2*2=4

f(1)*g(2)=3*4=12

Multiple choice (Answers in bold):Choice 1) 18 Choice 2) 12 Choice 3) 3 Choice 4) None

(Question 20)A MATLAB code to compute magnitude of a vector v is given by: mag_v=0; for i=1:length(v); mag_v=mag_v + v(i)^2; end; mag_v=sqrt(mag_v); If v=[3 4]; what is the value of mag_v? Solution: mag_v = 5


(Question 21)If a=3, the MATLAB output to the last line b=a*4 is identical to Solution:

a=3 b=a*4=3*4=12

Multiple choice (Answers in bold):Choice 1) 3+3+3+3 Choice 2) 3*3*3*3 Choice 3) 16 Choice 4) 12

(Question 22)Which one is an illegal variable name? Solution: 1dog is an illegal variable name

Multiple choice (Answers in bold):Choice 1) dog1 Choice 2) under_dog Choice 3) underdog Choice 4) 1dog

(Question 23)In MATLAB,floor(1.3) - round(1.4) gives an output of Solution: floor(1.3) - round(1.4) = 1 - 1 = 0

Multiple choice (Answers in bold):Choice 1) -1 Choice 2) 0 Choice 3) -.1 Choice 4) .1

(Question 24)To calculate the value of cosine of 90 degrees, the correct MATLAB expression is Solution: The correct MATLAB expression for the value of cosine of 90 degrees is cos(pi/2)

Multiple choice (Answers in bold):Choice 1) cos(Pi/2) Choice 2) cos(90) Choice 3) Cos(pi/2) Choice 4) cos(pi/2)

(Question 25)3^2^2 in MATLAB will give

Solution: ans = 3^2^2=81



(Question 26)Consider that xr is the only root for the function, f(x) = 0. If x is less than than xr, then the value |f(x) - f

(xr)|is

Solution: The value |f(x) - f(xr)|is always greater than 0


Choice 1) 0 Choice 2) Greater than 0 Choice 3) Less than 0 Choice 4) None

(Question 27)If xr1 and xr2 are the two roots for the function, f(x) = 0, then the value |f(xr1) - f(xr2)|is

Solution: The value |f(xr1) - f(xr2)|is 0


Choice 1) 0 Choice 2) greater than 0 Choice 3) less than 0 Choice 4) None

(Question 28)Which of the following are examples of bracketing methods for finding the root of a function

Solution: Bisection and False Position are examples of bracketing methods for finding the root of a function


Choice 1) Bisection Choice 2) Newton-Raphson method Choice 3) False position Choice 4) Fixed-point iteration

(Question 29)Which of the following are examples of open methods for finding the root of a function?

Solution: Newton-Raphson Method, Fixed-point Iteration, and Secant Methods are examples of open methods for

finding the root of a function


Choice 1)

Bisection

Choice 2) Newton-Raphson method

Choice 3) Fixed-point iteration

Choice 4) Secant methods

(Question 30)Which of the following must require two initial guess to find the roots of a function?

Solution: Bracketing Methods require two initial guess to find the roots of a function


Choice 1) Bracketing methods Choice 2) Newton-Raphson method Choice 3) Fixed-point iteration Choice 4) None

(Question 31)(a) Sketch the function f=inline('x^2 -4') (b) Using graphical method, what is the maximum number of

roots that this function has.

Solution: (a) x=-10:10; plot(x, f(x)); See the plot in the figure window. (b) Maximum number of roots = 2


Choice 1) 1 Choice 2) 2 Choice 3) 0 Choice 4) infinite

(Question 32)A MATLAB code is given as follows:

f=inline('x^3 - x^2 -18'); df=inline('3*x^2 - 2*x'); ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1;

xi1 = xi - f(xi)/df(xi); ea = 100*abs((xi1-xi)/xi1); disp([count xi1,xi, ea]); xi=xi1; end

The numerical scheme(s) used here is/are

Solution: The numerical scheme used here is Newton-Raphson


Choice 1) Bisection Choice 2) Fixed-Point Choice 3) Newton-Raphson Choice 4) Secant

(Question 33)

The MATLAB code f=inline('x^3 - x^2 -18'); df=inline('3*x^2 - 2*x'); ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1; xi1 = xi - f(xi)/df(xi); ea = 100*abs((xi1-xi)/xi1); disp([count xi1,xi, ea]); xi=xi1; end

finds the root for

Solution: The code finds the root for f(x)=x^3 -x^2 -18 =0 and f(x)=x*x*x -x^2 -18 =0


Choice 1) f(x)=x^3 -x^2 -18 =0 Choice 2) f(x)=3 *x^2 - 2*x =0 Choice 3) f(x)=x*x*x -x^2 -18 =0 Choice 4) None

(Question 34)Using the MATLAB code f=inline('x^3 - x^2 -18'); df=inline('3*x^2 - 2*x'); ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1; xi1 = xi - f(xi)/df(xi); ea = 100*abs((xi1-xi)/xi1); disp([count xi1,xi, ea]); xi=xi1; end

If ea is the error defined in percentage, the value of the error, ea, after 5th iteration (i.e., count=5) is

Solution: ea (iteration=1)=100.69 ea (iteration=2)=50.35 ea (iteration=3)=50.53 ea (iteration=4)=50.82 ea (iteration=5)=51.30


Choice 1) greater than 80% Choice 2) greater than 60% Choice 3) greater than 50% Choice 4) less than 1%

(Question 35)Using the MATLAB code

f=inline('x^3 - x^2 -18'); df=inline('3*x^2 - 2*x'); ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1; xi1 = xi - f(xi)/df(xi); ea = 100*abs((xi1-xi)/xi1); disp([count xi1,xi, ea]); xi=xi1; end

What is the value of the error, ea, after 22nd iteration (i.e., count=22) Solution: ea (iteration=22)=0.0048=.48%

Multiple choice (Answers in bold):Choice 1) 1% Choice 2) 0.5% Choice 3) less than 0.5% Choice 4) greater than 0.5%

(Question 36)Using the MATLAB code f=inline('x^3 - x^2 -18'); df=inline('3*x^2 - 2*x'); ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1; xi1 = xi - f(xi)/df(xi); ea = 100*abs((xi1-xi)/xi1); disp([count xi1,xi, ea]); xi=xi1; end

The root of the function is closest toSolution: xi=3.0000 xi1=3.0001


(Question 37)False-position method is a bracketing with bracket xl and xu where the new root is approximated asSolution: The new root is approximated as xr = xu - f(xu)(xu - xl)/(f(xu) - f(xl))

Multiple choice (Answers in bold):Choice 1) xr = 0.5*(xl+xu) Choice 2) xr = xu - f(xu)(xu - xl)/(f(xu) - f(xl)) Choice 3) xr=(xl+xu) Choice 4) None

(Question 38)Open methods differ from bracketing methods, in that open methods Solution: Open methods require a single or two starting values that do not necessarily bracket a root and can also diverge but when they converge they usually converges at much faster rate than the bracketing methods

Multiple choice (Answers in bold):Choice 1) require a single or two starting values that do not necessarily bracket a root

Choice 2) require two starting values that bracket a root

Choice 3) can also diverge but when they converge they usually converges at much faster rate than the bracketing methods

Choice 4) None

(Question 39)Newton-Raphson method hasSolution: Newton-Raphson method has quadratic convergence


Choice 1) linear convergence Choice 2) quadratic convergence Choice 3) no convergence Choice 4) None

(Question 40)Use the linspace function to create vectors identical to the colon notation

x = -4:2

Solution:

x = linspace(-4,2,7);


Choice 1) x = linspace(-4,2,7); Choice 2) x = linspace(-4,7,2); Choice 3) x = linspace(7,2,-4); Choice 4) None

(Question 41)(a) Use the linspace function to create a vector that is identical to the colon notation t=6:4:38;

(b) What is the value of t(8)?

Solution:

(a) t=linspace(6,38,9);

(b) t(8)=34


Choice 1) 34 Choice 2) 32 Choice 3) 30 Choice 4) None

(Question 42)(a)Use colon notation to create vectors identical to the following created with the linspace function v = linspace (-2,1.5,8); (b)The value of v is identical to

Solution: (a) v=-2:.5:1.5; (b) v=[-2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5]


Choice 1) v=[2, 1.5, 1, 0.5, 0, -

0.5, -1, -1.5]

Choice 2) v=[-2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5]

Choice 3) v=[-2, -1.5, -1, -0.5,

0.5, 1, 1.5]

Choice 4)

None

(Question 43)Which of the following MATLAB commands can be used to generate a vector z that starts from 0, ends at

50, and counts by 2.

Solution: z=[0:2:50]


Choice 1) z=[0:50] Choice 2) z=[2:0:50] Choice 3) z=[0:2:50] Choice 4) None

(Question 44)Which of the following MATLAB commands can be used to generate a vector w that starts at 0 counts by

squares to 81 (9^2), that is 0 1 4 9 16 ...

Solution: w=[0:9].^2


Choice 1) w=[0:9]^2 Choice 2) w=[0:9].^2 Choice 3) w=[0:9]. Choice 4) 4

(Question 45)(a) Write the MATLAB commands to generate a vector w that starts at 0 counts by 4th power to 256, that

is 0 1 16 81 ...

(b) Given the vector w above, what is the value of w(3)?

Solution: (a) w=(0:4).^4; (c)w(3) = 16



(Question 46)(a) Write the MATLAB commands to generate a vector w that starts at 0 counts by squares to 81 (9^2),

that is 0 1 4 9 16 ... (b) Given the vector w above, what is w(3)?

Solution: (a) w=(0:9).^2; (c)w(3) = 4


(Question 47)Given vectors x= [3,5,1,2] y=[5,6,10,8]

Write the output of the following MATLAB command: x.*y

Solution: ans = 15 30 10 16

Multiple choice (Answers in bold):Choice 1) 15 30 10 16 Choice 2) 10 15 30 16 Choice 3) 10 15 16 30 Choice 4) None


Write the output of the following MATLAB command: y'




Write the output of the following MATLAB command: y*x(2)+4



(Question 50)Using a for-loop in Matlab for the given vectors V=[1 5 10 7 2]; W=[1 2 1 -3 -1]; a) Write a program that calculates dot-product of the following two vectors b) What is the value of the dot-product?

Solution: V = [1 5 10 7 2]; W= [1 2 1 -3 -1]; n=length(V); dot_product=0; for i=1:n; dot_product = dot_product + V(i)*W(i); end

fprintf('dot product = %f \n',dot_product)

dot product = -2.000000

Multiple choice (Answers in bold):Choice 1) -2 Choice 2) -1 Choice 3) 0 Choice 4) 1

(Question 51)If b=5; b=b+1; a =2*b + 1; What is the value of c if c=a+b?

Solution:

b=5

b=b+1 b=5+1=6

a=2*b + 1 a=2*6 + 1=13

c=a+b c=13+6=19


(Question 52)If a=3; a=a^2; b=a+1; What is the value of c if c=a*b?

Solution:

a=3 a=a^2 a=3^2=9

b=a+1 b=9+1=10

c=a*b c=9*10=90


(Question 53)If c=2; c=4*c; c=c^2+1; b=2*c; What is the value of a if a=b+c

Solution:

c=2 c=4*c c=4*2=8 c=c^2+1 c=8^2+1 c=65

b=2*c b=2*65=130

a=b+c a=130+65

a=195


(Question 54)If a=2; a=a^2; What is the value of a+3?Solution: a=2 a=a^2 a=2^2 a=4

a+3=4+3=7


(Question 55)If a=1; b=a+5 What is the value of d if d=a*(b^2)?Solution: a=1 b=a+5 b=1+5=6

d=a*(b^2) d=1*(6^2)=36


(Question 56)If b=10; b=b^2; What is the value of b=b+2?Solution: b=10 b=b^2 b=10^2=100 b=b+2 b=100+2=102


(Question 57)If c=1; c=c^3; What is the value of c=c+9Solution: c=1 c=c^3 c=1^3=1 c=c+9 c=1+9=10


(Question 58)If a=4; a=a+1; b=a^2; What is the value of c if c=a*b?Solution: a=4 a=a+1 a=4+1=5 b=a^2 b=5^2=25 c=a*b c=5*25=125


(Question 59)If f=10; f=f+1; What is the value of f^2?Solution: f=10 f=f+1 f=10+1=11

f^2=11^2=121


(Question 60)If g=1; h=g+10; h=h^2; What is the value of f if f=g*h?Solution: g=1 h=g+10 h=1+10=11 h=h^2 h=11^2=121 f=g*h f=1*121=121


(Question 61)Exactly what will be displayed after the following MATLAB commands are types? a) >> x=5; >> x^3; >> y=8-x; b) >> q = 4:2:12; >> r = [7 8 4 ; 3 6 -5] ; >> sum(q) * r(2, 3) c) What is the value of sum(q) * y Solution: a) 3; b) -200; c) 120

Multiple choice (Answers in bold):Choice 1) -200 Choice 2) 120 Choice 3) 40 Choice 4) None

(Question 62)a) In the absence of air resistance, the Cartesian coordinates of a projectile launched with an initial velocity (v0) and angle (theta) can be computed with

x = v0 * cos(theta) * t; y = v0*sin(theta) * t - g * t * t /2; where g =9.81 m/s2. Develop a MATLAB script to generate a plot of the projectile's trajectory given that v0 =5 m/s and theta0=45 degree. b) The maximum height that the projectile will reach is closest to Solution: t=0:.1:2; v0=5; theta=45; g=9.81; t_radian=theta*pi/180; x=v0*cos(t_radian)*t; y=v0*sin(t_radian)*t - g*t.^2/2; plot(x,y); b) t=v0*sin(t_radian)/g; y=v0*sin(t_radian)*t - g*t.^2/2;

Multiple choice (Answers in bold):Choice 1) 0.3604 Choice 2) 0.6371 Choice 3) 1 Choice 4) 2

(Question 63)a) In the absence of air resistance, the Cartesian coordinates of a projectile launched with an initial velocity (v0) and angle (theta) can be computed with x = v0 * cos(theta) * t; y = v0*sin(theta) * t - g * t * t /2; where g =9.81 m/s2. Develop a MATLAB script to generate a plot of the projectile's trajectory given that v0 =5 m/s and theta0=45 degree. b) How long will it take for the projectile to reach the maximum heightSolution: t=0:.1:2; v0=5; theta=45; g=9.81; t_radian=theta*pi/180; x=v0*cos(t_radian)*t; y=v0*sin(t_radian)*t - g*t.^2/2; plot(x,y); b) t=v0*sin(t_radian)/g;


(Question 64)(a) Write a MATLAB code using for-loop to compute C=A*A where A=[1 3; 4 5]; (b) The value of C(2,2) isSolution: (a) A=[1 3; 4 5]; for i=1:2; for j=1:2; C(i,j)=0; for k=1:2; C(i,j) = C(i,j) + A(i,k)*A(k,j); end end end (b) 37


(Question 65)The "divide and average" method to find the square root of 2 is given by x = (x + 2/x)/2; (a) Write a MATLAB Script using for-loop to find the square root of 2 for 4 iterations with starting guess for x=1. What is the value (or closest value) of x after the 2nd iterationsSolution: xold=1; for i=1:4; xnew = (xold + 2/xold)/2; disp([xold xnew i]); xold=xnew; end

Multiple choice (Answers in bold):Choice 1) 1 Choice 2) 1.4167 Choice 3) 1.4142 Choice 4) 1.5

(Question 66)The "divide and average" method to find the square root of 2 is given by x = (x + 2/x)/2; (a) Write a MATLAB Script using while-loop to find the square root of 2 for 4 iterations with starting guess for x=1. What is the value (or closest value) of x after the 3rd iterations

Solution: xold=1; i=1;

while (i<=4) xnew = (xold + 2/xold)/2; disp([xold xnew i]); xold=xnew; i=i+1; end


(Question 67)The "divide and average" method to find the square root of 16 is given by x = (x + 16/x)/2; (a) Write a MATLAB Script using while-loop and if-break condition to find the square root of 16 for 4 iterations with starting guess for x=1. What is the value (or closest value) of x after the 3rd iterations

Solution: xold=1; i=1; while (1==1) xnew = (xold + 16/xold)/2; disp([xold xnew i]); if(i==4) break; end; xold=xnew; i=i+1; end

Multiple choice (Answers in bold):Choice 1) 8.5000 Choice 2) 4 Choice 3) 4.1367 Choice 4) 4.0023

(Question 68)The "divide and average" method to find the square root of 16 is given by x = (x + 16/x)/2; A MATLAB Script using while-loop and if-break condition to find the square root of 16 for 4 iterations with starting guess for x=1 is given by

xold=1; i=1; while (1==1) xnew = (xold +,16/xold)/2; disp([xold xnew i]); if(i==4) break; end; xold=xnew; i=i+1; end

(a) This code has a bug. Fix the bug and write the correct code. (b) What is the value (or closest value) of x after the 4th iterations

Solution: xold=1; i=1; while (1==1) xnew = (xold + 16/xold)/2; disp([xold xnew i]); if(i==4) break; end; xold=xnew; i=i+1; end


Choice 1) 8.5000 Choice 2) 4 Choice 3) 4.1367 Choice 4) 4.0023

(Question 69)(a) A MATLAB code to compute magnitude of a vector v is given by:

mag_v=0; v=[1 3 5]; for i=1:length(v); mag_v=mag_v + v(i)*2; end; mag_v=sqrt(mag_v);

This code however has a bug. Fix the bug and write the correct code. <b> If v=[1 3 5]; what is the correct value of mag_v?

Solution:

mag_v=0; v=[1 3 5]; for i=1:length(v); mag_v=mag_v + v(i)^2; end; mag_v=sqrt(mag_v);


Choice 1) 3 Choice 2) 5 Choice 3) 5.9161 Choice 4) 4.2426

(Question 70)(a) A MATLAB code to compute magnitude of a vector v is given by:

mag_v=0; v=[3 4];

for i=1:length(v);

mag_v=mag_v + v(i^2);

end;

mag_v=sqrt(mag_v);

This code however has a bug. Fix the bug and write the correct code. <b> If v=[3 4]; what is the correct value of mag_v?

Solution:

mag_v=0; v=[1 3 5];

for i=1:length(v);

mag_v=mag_v + v(i)^2;

end;

mag_v=sqrt(mag_v);

Multiple choice (Answers in bold):Choice 1) 3 Choice 2) 5 Choice 3) 5.9161 Choice 4) 4

(Question 71)(a) Sketch the function f=inline('x^2 -2') (b) Using graphical method, approximately find the value of x (root of the function) such that f(x)=0. (c) Write a MATLAB script for 4 iterations of Bi-section method with xL=0; and xU=2.0; (d) What is the closest value of xR after the 4th iteration?

Solution: (a)

xL = 0; xU = 2; fU = xU^2 - 2; fL=xL^2 - 2;

n=4;

for i=1:n

fR = xR^2 - 2; fL = xL^2 -2; fU = xU^2 - 2;

if(fR*fL > 0 ) xL = xR; else xU = xR; end

xR = xU - (fU) * (xL-xU)/(fL-fU);

disp([i xL xU xR])

end


(Question 72)A MATLAB code is given as follows:

f=inline('x^3 - x^2 -18'); df=inline('3*x^2 - 2*x');

ea=100;

count=0; xi=0.5;

while (ea>.01);

count = count + 1;

xi1 = xi - f(xi)/df(xi);

ea = 100*abs((xi1-xi)/xi1);

disp([count xi1,xi, ea]);

xi=xi1;

end

The numerical scheme(s) used here is/are

Solution: The numerical scheme used here is Newton-Raphson with xi=0.5. Note that x(i+1) = x(i) - f(x(i)) / f'(x(i));


Choice 1) Bisection with xL=0; xU=1;

Choice 2) Fixed-Point with xi=0.5

Choice 3) Newton-Raphson with xi=0.5

Choice 4) Secant with xi=0.5

(Question 73)

The MATLAB code

f=inline('x^3 - x^2 -18'); df=inline('3*x^2 - 2*x');ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1; xi1 = xi - f(xi)/df(xi); ea = 100*abs((xi1-xi)/xi1); disp([count xi1,xi, ea]); xi=xi1; end

finds the root forSolution: The code finds the root for f(x)=x^3 -x^2 -18 =0 and f(x)=x*x*x -x^2 -18 =0 using Newton-Raphson method

Multiple choice (Answers in bold):Choice 1) f(x)=x^3 -x^2 -18 =0 using secant method

Choice 2) f(x)=3 *x^2 - 2*x =0 using Newton Raphson method

Choice 3) f(x)=x*x*x -x^2 -18 =0 using Newton-Raphson Method

Choice 4) f(x)=x^3 -x^2 -18 =0 using false position method



If ea is the error defined in percentage, the value of the error, ea, after 5th iteration (i.e., count=5) is closest to Solution: ea (iteration=1)=100.69 ea (iteration=2)=50.35 ea (iteration=3)=50.53 ea (iteration=4)=50.82 ea (iteration=5)=51.30

Multiple choice (Answers in bold):Choice 1) greater than 80% Choice 2) 50.35 Choice 3) 51.30 Choice 4) 20%


f=inline('x^3 - x^2 -18'); df=inline('3*x^2 - 2*x');ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1; xi1 = xi - df(xi)/df(xi); ea = 100*abs((xi1-xi)/xi1);

disp([count xi1,xi, ea]); xi=xi1; end

(a) Fix the bug and write the correct code. (b)Using the correct code, what is the value of the error, ea, after 22nd iteration (i.e., count=22) closet toSolution:


ea (iteration=22)=0.0048=.48%

Multiple choice (Answers in bold):Choice 1) 10% Choice 2) 1.2% Choice 3) 0.48% Choice 4) 0.5%

(Question 76)(a) Find the bug and fix it in the the following Newton-Raphson code to find the square root of 3 and write the correct code.

f=inline('x^2 - 7'); df=inline('2*x');ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1; xi1 = xi1 - f(xi),/df(xi); ea = 100*abs((xi1-xi)/xi1); disp([count xi1,xi, ea]); xi=xi1; end

Using The root (xi1) of the function after 5th iteration (count=5) is closest toSolution:

f=inline('x^2 - 7'); df=inline('2*x');ea=100; count=0; xi=0.5; while (ea>.01); count = count + 1; xi1 = xi - f(xi)/df(xi); ea = 100*abs((xi1-xi)/xi1); disp([count xi1,xi, ea]); xi=xi1; end


(Question 77)False-position method is a bracketing with bracket xl and xu where the new root is approximated as xr = xu - f(xu)(xu - xl)/(f(xu) - f(xl)) (a) Write a Matlab script to compute the root for f(x) = x^3 - 27; after the 7th iterations. Use xL=0; xU=10; (b) What is the value of the root closest to after the 3rd iteration?

Solution:

xL = 0; xU = 10; fU = xU^3 - 27; fL=xL^3 - 27; xR = xU - (fU) * (xL-xU)/(fL-fU); n=7; for i=1:n fR = xR^3 - 27; fL = xL^3 -27; fU = xU^3 - 27; if(fR*fL > 0 ) xL = xR; else xU = xR; end xR = xU - (fU) * (xL-xU)/(fL-fU); disp([i xL xU xR])end

(b) 1.0355


(Question 78)False-position method is a bracketing with bracket xl and xu where the new root is approximated as xr = xu - f(xu)(xu - xl)/(f(xu) - f(xl)) (a) Matlab script with a bug to compute the root for f(x) = x^3 - 27; after the 7th iterations using xL=0; xU=10; is given by

xL = 0; xU = 10; fU = xU^3 - 27; fL=xL^3 - 27; xR = xU - (fU) * (xL-xU)/(fL-fU); n=7; for i=1:n fR = xR^3 - 27; fL = xL^3 -27; fU = xU^3 - 27; if(fR*fL < 0 ) xL = xR; else xU = xR; end xR = xU - (fU) * (xL-xU)/(fL-fU); disp([i xL xU xR])end

Fix the bug and write the correct code. (b) What is the value of the root closest to after the 3rd iteration? Solution:

xL = 0; xU = 10; fU = xU^3 - 27; fL=xL^3 - 27; xR = xU - (fU) * (xL-xU)/(fL-fU); n=7; for i=1:n fR = xR^3 - 27; fL = xL^3 -27; fU = xU^3 - 27; if(fR*fL > 0 ) xL = xR; else xU = xR; end xR = xU - (fU) * (xL-xU)/(fL-fU); disp([i xL xU xR])end

(b) 0.5327

Multiple choice (Answers in bold):Choice 1) 0.5 Choice 2) 1 Choice 3) 2 Choice 4) 3

(Question 79)Secant method is given by

df = (f(i+1) - f(i))/(x(i+1)-x(i)); x(i+2) = x(i+1) - f(i+1)/df;

Write a MATLAB Script using for for-loop to compute the root of f(x) = x^4 - 16 after 6-iterations; Use x(1) = 0; x(2)=1; What is the value of the root closest to after 2nd iteration (i.e., i=2)?

Solution:

x1 = 0; f1=x1^4-16; x2 = 1; f2=x2^4-16; n=3; for i=1:n df=(f2 - f1)/(x2-x1); x3 = x2 - f2/df; f3=x3^4-16; disp([i x1 x2 x3]) x1=x2; x2=x3; f1=f2; f2=f3; end




A MATLAB Script (with bug) using for for-loop to compute the root of f(x) = x^4 - 16 is given by

x1 = 0; f1=x1^4-16; x2 = 1; f2=x2^4-16; n=3; for i=1:n df=(f2 - f1)/(x2-x1); x3 = x2 - f2/df; f3=x3^4-16; disp([i x1 x2 x3])end

Fix the bug and write the correct code. What is the value of the root closest to after 15th iteration (i.e., n=15)? Solution:






x1 = 0; f1=x1^4-16; x2 = 1; f2=x2^4-16; n=3; for i=1:n df=(f2 - f1)/(x2*x1); x3 = x2 - f2/df; f3=x3^,4-16;

Printing All the question & existing

disp([i x1 x2 x3]) x1=x2; x2=x3; f1=f2; f2=f3; end

Fix the bug and write the correct code.

What is the value of the root closest to after 7th iteration (i.e., n=7)?

Solution:







x1 = 0; f1=x1^4-16; x2 = 1; f2=x2^4-16; n=3; for i=1:n df=(f2 - f1)*(x2-x1); x3 = x2 + x2/df; f3=x3^4-16; disp([i x1 x2 x3]) x1=x2; x2=x3; f1=f2; f2=f3; end

Fix the bug and write the correct code.

What is the value of the root closest to after 8th iteration (i.e., n=8)?

Solution:



Choice 1) 1.5 Choice 2) 2.5 Choice 3) 3 Choice 4) 4.5

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