An Engineers Guide to Matlab Solutions

1Solution to Exercises

Chapter 1

1.1 E1 = 1+5^3+3^3-153

E2 = 1+6^4+3^4+4^4-1634

E3 = 5^6+4^6+8^6+8^6+3^6+4^6-548834

E4 = 1+factorial(7)-71^2

E5 = factorial(1)+factorial(4)+factorial(5)+factorial(6)+factorial(7)+factorial(8)-215^2

1.2 pie = pi - (100-(2125^3+214^3+30^3+37^2)/82^5)^(1/4)

1.3 format long g

I1 = 53453/log(53453)

I2 = 613*exp(1)/37-35/991

format short

1.4 x = (49-27*sqrt(5)+3*sqrt(6)*sqrt(93-49*sqrt(5)))^(1/3)

R = 0.5*sqrt((8*2^(2/3)-16*x+2^(1/3)*x^2)/(8*2^(2/3)-10*x+2^(1/3)*x^2))

Answers: x = 2.5375 + 1.9080i R = 0.8161 - 0.0000i

1.5 E1 = cot(pi/5)-sqrt(25+10*sqrt(5))/5

E2 = sin(pi/15)-sqrt(7-sqrt(5)-sqrt(30-6*sqrt(5)))/4

E3 = pi-16*atan(1/5)+4*atan(1/239)

1.6 g = sqrt(17-sqrt(17));

b = sqrt(17+sqrt(17));

a = sqrt(34+6*sqrt(17)+sqrt(2)*(sqrt(17)-1)*g-8*b*sqrt(2));

E1 = sin(pi/17)-sqrt(2)/8*sqrt(g^2-sqrt(2)*(a+g))

1.7 a = 1; b = 2;

E1 = exp(a)-sinh(a)-cosh(a)

E2 = sinh(2*b)/(cosh(a+b)*cosh(a-b))-tanh(a+b)+tanh(a-b)

1.8 L = 1.5; h = 1; r = 1.6;

V = L*(r^2*acos((r-h)/r)-(r-h)*sqrt(2*r*h-h^2))

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2Answer: V = 3.2209

1.9 g = 60*pi/180; a = 35*pi/180; n = 4/3;

D = a-g+asin(n*sin(g-asin(sin(a)/n)))

Answer: D = 0.4203

1.10 k = 7; n =12;

Cnk = factorial(n)/factorial(k)/factorial(n-k)

Answer: Cnk = 792

1.11 r = 2.5;

I = (pi/8-8/9/pi)*r^4

Answer: I = 4.2874

1.12 c = 5;

K = (4*c-1)/(4*c+4)+0.615/c

Answer: K = 0.9147

1.13 B = 0.6;

K = 3/(1-B)^3*(0.5-2*B+B*(1.5-log(B)))

Answer: K = 23.7420

1.14 R = 30; r = 12; S = 50;

theta = asin((R-r)/S);

L = 2*S*cos(theta)+pi*(R+r)+2*theta*(R-r)

Answer: L = 238.4998

1.15 Fn = 250; f = 0.35; r = 0.4; theta = 60*pi/180;

T = (4*f*r*Fn*sin(theta/2))/(theta+sin(theta))

Answer: T = 36.5875


31.16 A = 1.7; B = 1.2;

D = 1.265*((A*B)^3/(A+B))^(1/5)

Answer: D = 1.5682

1.17 n = 6;

M = 1/sin(pi/n);

C = (1+M^2)/4/M;

alpha = acos(sqrt(C^2+2)-C);

aG = M*(1-M^2)*sin(alpha)/(1+M^2-2*M*cos(alpha))^2

Answer: aG = -1.3496

1.18 L = 3000; d = 45; V = 1600;

Deltap = 0.03*L/d^1.24*(V/1000)^1.84

Answer: Deltap = 1.9048

1.19 E1 = 206e9; E2 = 206e9; nu1 = 0.3; nu2 = 0.3;

d1 = 0.038; d2 = 0.070; F = 450; z = 0.00025;

a = (0.375*F*((1-nu1^2)/E1+(1-nu2^2)/E2)/(1/d1+1/d2))^(1/3);

pmax = 1.5*F/pi/a^2;

za = z/a;

sxsy = -pmax*((1-za*atan(1/za))*(1-nu1)-0.5/(1+za^2))

sz = -pmax/(1+za^2)

Answers: sxsy = 2.0779e+008 sz = -1.2421e+009

1.20 E1 = 206e9; E2 = 206e9; nu1 = 0.3; nu2 = 0.3;

d1 = 0.038; d2 = 0.070; F = 450; z = 0.000025; L = 0.05;

b = sqrt(2*F/pi/L*((1-nu1^2)/E1+(1-nu2^2)/E2 )/(1/d1+1/d2));

pmax = 2*F/pi/b/L;

zb = z/b;

sx = -pmax*2*nu2*(sqrt(1+zb^2)-zb)

sy = -pmax*((2-1/(1+zb^2))*sqrt(1+zb^2)-2*zb)

sz = -pmax/sqrt((1+zb^2))

Answers: sx = -5.0360e+007 sy = -3.5432e+007 sz = -1.3243e+008

1.21


4e = 0.8;

NL = pi*e*sqrt(pi^2*(1-e^2)+16*e^2)/(1-e^2)^2

Answer: NL = 72.0220

1.22 h = 0.03; d0 = 0.006;

d1 = 0.016; E = 206e9;

d2 = d1+h*tan(pi/6);

k = pi*E*d0*tan(pi/6)/log((d2-d0)*(d1+d0)/(d2+d0)/(d1-d0))

Answer: k = 5.2831e+009

1.23 alpha = 2e-5; E = 206e9; nu = 0.3;

Ta = 260; Tb = 150; a = 0.006; b = 0.012; r = 0.010;

c1 = alpha*E*(Ta-Tb)/2/(1-nu)/log(b/a);

c2 = a^2/(b^2-a^2)*log(b/a);

br = b/r;

sr = c1*(c2*(br^2-1)-log(br))

st = c1*(1-c2*(br^2+1)-log(br))

T = Tb+(Ta-Tb)*log(br)/log(b/a)

Answers: sr = -3.7670e+007 st = 1.1859e+008 T = 178.9338

1.24 k = 1.4; pepo = 0.3;

psi = sqrt(k/(k-1))*sqrt(pepo^(2/k)-pepo^((k+1)/k))

Answer: psi = 0.4271

1.25 x = 0.45;

K = 1.2/x/(sqrt(16*x^2+1)+log(sqrt(16*x^2+1)+4*x)/4/x)^(2/3)

Answer: K = 1.3394

1.26 c = sqrt(8)/9801;

n = 0;

oneopi0 = c*(1103+26390*n)*factorial(4*n)/factorial(n)^4/396^(4*n);

n = 1;

oneopi1 = c*(1103+26390*n)*factorial(4*n)/factorial(n)^4/396^(4*n);

format long e

pin0 = 1/oneopi0

diffpino = pin0-pi

pin0n1 = 1/(oneopi0+oneopi1)

diffpin0n1 = pin0n1-pi


5format short

Answers: pin0 = 3.141592730013306e+000 diffpino = 7.642351240733092e-008

pin0n1 = 3.141592653589794e+000 diffpin0n1 = 4.440892098500626e-016

1.27 r = 10; rc = 3; k = 1.4;

eta = 1-1/r^(k-1)*((rc^k-1)/k/(rc-1))

Answer: eta = 0.4803

1.28 M = 2; k = 1.4;

AA = 1/M*(2/(k+1)*(1+(k-1)/2*M^2))^((k+1)/(2*(k-1)))

Answer: AA = 1.6875

1.29 n = 5; ep = 0.1;

w = 0.5

T = 1/sqrt(1+(ep*cos(n*acos(w))).^2)

w = 1

T = 1/sqrt(1+(ep*cos(n*acos(w))).^2)

w = 1.5

T = 1/sqrt(1+(ep*cosh(n*acosh(w))).^2)

Answers: w = 0.5000 T = 0.9988 w = 1 T = 0.9950 w = 1.5000 T = 0.1605

1.30 m = 1; n = 2; nu = 0.3; hR = 0.05; Rl = 0.1;

Lm = pi/4*Rl*(4*m+1);

Om = (1-nu^2)*Lm^4/(Lm^2+n^2+1.78*n^2*Lm^2);

Om = Om+hR^2/12*(Lm^4+n^4+1.78*m^2*Lm^2)

Answer: Om = 7.5159e-003

1.31 V = 2.9; D = 0.3; ep = 0.00025; nu = 1.012e-6;

Re = V*D/nu;

f = ((64/Re)^8+9.5*(log(ep/3.7/D+5.74/Re^0.9)-(2500/Re)^6)^-16)^(1/8)

Answer: f = 0.0193

1.32 n = 3; th = pi/7;

a = cos(th)+(n^2*cos(2*th)+sin(th)^4)/(n^2-sin(th)^2)^(3/2)


6Answer: a = 1.1168

1.33 [Note: These solutions are obtained with a large amount of user interaction, almost on a line

by line basis.]

(a) x = a*sin(t);

y = a*cos(t)^2*(2+cos(t))/(3+sin(t)^2);

xp = diff(x,t,1);

xpp = diff(x,t,2);

yp = diff(y,t,1);

ypp = diff(y,t,2);

[Nnum Nden] = numden(xp*ypp-yp*xpp);

Nden = factor(Nden)

Nnum = simple(Nnum)

[Dnum Dden] = numden(simple(xp^2+yp^2))

Partial answers: Nden = (sin(t)^2 + 3)^3 Nnum = -a^2*cos(t)^2*(cos(t) + 2)^3*(9*cos(t) - 6)

Dnum = a^2*(cos(2*t) + 1)*(9*cos(2*t) - 80*cos(t) + 73) Dden = 4*(cos(t) - 2)^4

At this point, we place these results in standard notation and make a few trigonometric substitutions as follows:

( ) ( )( )

( ) ( )( )( )

( ) ( )( )

( ) ( ) ( )

( )( ) ( )

3 ' 232 2 4

3 3 / 22 2

3 62 2

3 3 / 2 3 / 22 2 2

32 2

cos cos 2 9cos( ) 6 4(cos( ) 2)

sin 3 cos(2 ) 1 9cos(2 ) 80cos( ) 73

24 cos cos 2 3cos( ) 2 cos( ) 2

4 cos 2 cos 9cos(2 ) 80cos( ) 73

24 cos cos 2 cos( ) 2 3cos( ) 2

a t t t t

t a t t t

a t t t t

t a t t t

a t t t t

! +

=+ + +

+ =

+

# + % &=( )

( ) ( )

( ) ( )( )

( ) ( )

( )( )

( )

3

3 3 / 23 / 2 3 3 2

3 3

3 3 / 22

3

3 / 2

cos( ) 2

2 cos 4 cos 9cos(2 ) 80cos( ) 73

4 cos( ) 3cos( ) 2 cos( ) 26 2

cos 4 cos 9cos(2 ) 80cos( ) 73

sec 3cos( ) 2 cos( ) 26 2

9cos(2 ) 80cos( ) 73

t

a t t t t

t t t

a t t t t

t t t

a t t

+

=

+

=

+

(b) syms a t

x = 3*t*a/(1+t^3);

y = x*t;

xp = diff(x,t,1);

xpp = diff(x,t,2);

yp = diff(y,t,1);

ypp = diff(y,t,2);

kn = simple(xp*ypp-yp*xpp);

kd = simple(xp^2+yp^2);

k = simple(kn/kd^(3/2))

Answer: k = (2*a^2)/(3*(t^3 + 1)^2*((a^2*(t^8 + 4*t^6 - 4*t^5 - 4*t^3 + 4*t^2 + 1))/(t^3 + 1)^4)^(3/2))


7(c) syms a t

x = a*(t-tanh(t));

y = a*sech(t);

xp = diff(x,t,1);

xpp = diff(x,t,2);

yp = diff(y,t,1);

ypp = diff(y,t,2);

kn = simple(xp*ypp-yp*xpp);

kn = subs(kn, cosh(t)^2 - 1, sinh(t)^2);

kd = collect(simple(xp^2+yp^2), a);

kd = subs(kd,1 - 1/cosh(t)^2, sinh(t)^2/cosh(t)^2);

kd = subs(kd, (1/cosh(t)^2)-1, -sinh(t)^2/cosh(t)^2);

k = simple(kn/kd^(3/2))

Answer: k = (a^2*sinh(t)^2)/(cosh(t)^3*((a^2*sinh(t)^2)/cosh(t)^2)^(3/2))

1.34

(a) z = vpa('cos(pi*cos(pi*cos(log(pi+20))))+1', 40)

Answer: 0.000000000000000000000000000000000039321609261272240194541

(b) z = vpa('exp(pi*sqrt(163))', 35)

Answer: z = 262537412640768743.99999999999925007

1.35 syms x

tay1 = taylor(x*(7*x+1)/(1-x)^3, 5, x, 0)

tay2 = taylor((1+x-sqrt(1-6*x+x^2))/4,5,x,0)

tay3 = taylor(2*x/(1-x)^3,5,x,0)

tay4 = taylor(x*(x^2+4*x+1)/(1-x)^3,5,x,0)

Answers: tay1 = 52*x^4 + 27*x^3 + 10*x^2 + x

tay2 = 11*x^4 + 3*x^3 + x^2 + x

tay3 = 20*x^4 + 12*x^3 + 6*x^2 + 2*x

tay4 = 37*x^4 + 19*x^3 + 7*x^2 + x

1.36 syms x n e a t

L1 = limit((x^e-1)/e, e, 0)

L2 = limit((1-sin(2*x))^(1/x), x, 0)

L3 = limit(((exp(t)-1)/t)^(-a), t, 0)

L4 = limit(log(x^n)/(1-x^2), x, 1)

Answers: L1 = log(x) L2 = 1/exp(2) L3 = 1 L4 = -n/2


81.37 syms s t

rD = roots(sym2poly(s^4+0.282*s^3+4.573*s^2+0.4792*s+2.889));

rN = roots(sym2poly(0.1*s^3+0.0282*s^2-0.0427*s+0.0076));

Xt = vpa(ilaplace(((s-rN(1))*(s-rN(2))*(s-rN(3)))/((s-rD(1))*(s-rD(2))*(s-rD(3))*(s-rD(4))), s, t),5)

Answer: Xt = (1.3974*(cos(1.9456*t)+0.050033*sin(1.9456*t)))/exp(0.097401*t)-

(0.39736*(cos(0.87145*t)+0.049799*sin(0.87145*t)))/exp(0.043599*t)



Chapter 2

2.1 n = 0:7;

a = 2*n-1;

b = 2*n+1;

aplusb = a+b

aminusb = a-b

atimesb = a'*b

detatimesb = det(atimesb)

atimesbtranspose = a*b'

Answers: aplusb = 0 4 8 12 16 20 24 28

aminusb = -2 -2 -2 -2 -2 -2 -2 -2

atimesb =

-1 -3 -5 -7 -9 -11 -13 -15

1 3 5 7 9 11 13 15

3 9 15 21 27 33 39 45

5 15 25 35 45 55 65 75

7 21 35 49 63 77 91 105

9 27 45 63 81 99 117 135

11 33 55 77 99 121 143 165

13 39 65 91 117 143 169 195

detatimesb = 0 atimesbtranspose = 552

2.2 x = sort( [17 -3 -47 5 29 -37 51 -7 19], 'descend');

xs = [x(find(sort(x, 'descend')0))]

Answer: w = -3 -7 -37 -47 51 29 19 17 5

2.3 y = [0, -0.2, 0.4, -0.6, 0.8, -1.0, -1.2, -1.4, 1.6];

sy = sin(y);

% (a)

mx = max(sy(find(sy

22.5 z = magic(5)

z(:,2) = z(:,2)/sqrt(3);

z(5,:) = z(5,:)+z(3,:);

z(:,1) = z(:,1).*z(:,4);

q = z-diag(diag(z))+diag([2 2 2 2 2]);

format long g

dq = diag(q*q')

format short

qmax = max(max(q))

qmin = min(min(q))

Answers: z =

17 24 1 8 15

23 5 7 14 16

4 6 13 20 22

10 12 19 21 3

11 18 25 2 9

dq = [486 104189 7300 44522 111024] qmax = 330 qmin = 1

2.6 w = magic(2);

wa = repmat(w, 2, 2)

wb = repmat(w, 3, 1)

wc = [repmat(w, 3, 1), repmat(w', 3, 1)]

wad = [w, w; w, w]

wbd = [w; w; w]

wcd = [w, w'; w, w'; w, w']

Answers: wa = % or wad

1 3 1 3

4 2 4 2

1 3 1 3

4 2 4 2

wb = % or wbd

1 3

4 2

1 3

4 2

1 3

4 2

wc = % or wcd

1 3 1 4

4 2 3 2

1 3 1 4

4 2 3 2

1 3 1 4

4 2 3 2

2.7


3(a)

x = magic(3); t = x(1,:); x(1:2,:) = x(2:3,:); x(3,:) = t; x

Answers: x =

3 5 7

4 9 2

8 1 6

(b)

x = magic(3) t = x(:,3); x(:,2:3) = x(:,1:2); x(:,1) = t; x

Answers: x =

6 8 1

7 3 5

2 4 9

2.8 x = magic(5);

xf = fliplr(x);

xf = fliplr(xf-diag(diag(xf)))

Answer: xf =

17 24 1 8 0

23 5 7 0 16

4 6 0 20 22

10 0 19 21 3

0 18 25 2 9

2.9 a = 1; b = 1.5; e = 0.3;

phi = linspace(0, 2*pi);

s = a*cos(phi)+sqrt(b^2-(a*sin(phi)-e).^2);

plot(phi*180/pi, s)

Answer:


40 50 100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

2.10 n = 1:25;

a = n*pi/sqrt(19);

P = 100*(1+2*cumsum((sin(a)./a).^2))/4.3589;

plot(n(2:25), P(2:25))

Answer:

0 5 10 15 20 2582

84

86

88

90

92

94

96

98

100

2.11 a = sqrt(2.8);

n = 1:100; % (1x100)

x = 1:0.5:5; % (1x9)

exact = a./sqrt(a^2+x.^2).*sin(pi*sqrt(a^2+x.^2))/sin(pi*a);

e = 100*(prod(1-(1./(n.^2-a^2))'*x.^2)-exact)./exact % prod works on columns

Answer: e = 1.0001 2.2643 4.0610 6.4176 9.3707 12.9670 17.2642 22.3330 28.2588

2.12 x = [72, 82, 97, 103, 113, 117, 126, 127, 127, 139, 154, 159, 199, 207];

beta = 3.644;

delta = (sum(x.^beta)/length(x))^(1/beta)

Answer: delta = 144.2741


52.13 b = 2;

phi = linspace(0, pi/2, 10);

the = linspace(0, 2*pi, 24);

[p, t] = meshgrid(phi, the);

mesh(b*sin(p).*cos(t), b*sin(p).*sin(t), b*cos(p))

Answer:

-2-1.5

-1-0.5

00.5

11.5

2

-2

-1

0

1

20

0.5

1

1.5

2

2.14 x = linspace(0.1, 1, 5);

n = -25:25;

y = pi*x*sqrt(2);

exact = 2*pi^4./y.^3.*(sinh(y)+sin(y))./(cosh(y)-cos(y));

[xx, nn] = meshgrid(x, n);

apx = sum(1./(nn.^4+xx.^4));

format long g

comp = [apx' exact']

format short

Answer: comp =

10002.1644054971 10002.1644456719

91.7751642291914 91.7752044040698

12.9244097924593 12.9244499673344

4.40350416999114 4.40354434485533

2.15691498449604 2.15695515933427

2.15 k = 0:25; n = 2;

x = linspace(1, 6, 6);

[xx, kk] = meshgrid(x, k);

Jn = sum((-1).^kk.*(xx/2).^(2*kk+n)./factorial(kk)./gamma(kk+1+n));

format long g

Comp = [Jn' besselj(n, x)']

format short

Answer: Comp =

0.1149034849319 0.114903484931901

0.352834028615638 0.352834028615638


6 0.486091260585891 0.486091260585891

0.364128145852073 0.364128145852073

0.0465651162777518 0.0465651162777523

-0.242873209960186 -0.242873209960185

2.16 n = 7; k = 1:(2*n-1);

ser = sum(cos(k*pi/n))

Answer: ser = -1.0000

2.17 n = 6; k = 1:n;

N1 = n*(2*n^2+3*n+1)/6

S1 = sum(k.^2)

N2 = n*(6*n^4+15*n^3+10*n^2-1)/30

S2 = sum(k.^4)

N3 = n^2*(2*n^4+6*n^3+5*n^2-1)/12

S3 = sum(k.^5)

Answers: N1 = 91 S1 = 91 N2 = 2275 S2 = 2275 N3 = 12201 S3 = 12201

2.18 N = 10; n = 0:N;

PieDiff = sum((4./(8*n+1)-2./(8*n+4)-1./(8*n+5)-1./(8*n+6)).*(1/16).^n) -pi

Answer: PieDiff = 0

2.19 k=1:300;

dn=0.5*cumsum(1./k);

On = find(dn>1);

d1 = On(1)

tw = find(dn>2);

d2 = tw(1)

th = find(dn>3);

d3 = th(1)

Answer: d1 = 4 d2 = 31 d3 = 227

2.20 N = 10; k = 0:N;

e = sum(1./factorial(k))-exp(1)

Jo = sum((-1).^k./factorial(k).^2)-besselj(0,2)

co = sum((-1).^k./factorial(2*k))-cos(1)

Answers: e = -2.7313e-008 Jo = 6.3838e-016 co = -1.1102e-016


72.21 K = 41; k = 1:2:K; y = pi/3; x = 0.75;

SKdiff = sum(exp(-k*x).*sin(k*y)./k)-0.5*atan(sin(y)/sinh(x))

Answer: SKdiff = -2.2204e-016

2.22 K = 14; k = 0:K; a = 3; x = -2;

axdiff = sum((x*log(a)).^k./factorial(k))-a^x

Answer: axdiff = 9.0230e-008

2.23 N = 500; k = 1:N; z = 10;

c = pi*coth(pi*z)/(2*z)-1/(2*z^2)

S1 = cumsum(1./(k.^2+z^2));

K = find(abs(S1-c)

8A = [1 2 2; 2 1 2; 2 2 1];

T = A*A-4*A-5*eye(3)

Answer: T =

0 0 0

0 0 0

0 0 0

2.27 A = [1 -1; 2 -1]; B = [1 1; 4 -1];

T = (A+B)^2-A^2-B^2

Answer: T =

0 0

0 0

2.28 A = [7 -2 1; -2 10 -2; 1 -2 7];

P = [1/sqrt(2) 1/sqrt(3) 1/sqrt(6); 0 1/sqrt(3) -2/sqrt(6); -1/sqrt(2) 1/sqrt(3) 1/sqrt(6)];

L = eig(A)

LL = diag(inv(P)*A*P)

Answers: L =

6.0000

6.0000

12.0000

LL =

6.0000

6.0000

12.0000

2.29 c = pi/180;

th1 = 30*c; th2 = 30*c; th3 = 30*c;

a1 = 1; a2 = 2; a3= 3;

a01 = [cos(th1) -sin(th1) 0 a1*cos(th1); sin(th1) cos(th1) 0 a1*sin(th1);0 0 1 0; 0 0 0 1];



t = a01*a12*a23;

thx3 = atan2(t(2,1), t(1,1))/c

thy3 = atan2(t(2,2), t(1,2))/c

qx = t(1,4)

qy = t(2,4)

Answers: thx3 = 90.0000 thy3 = 180 qx = 1.8660 qy = 5.2321

2.30 c = pi/180;

th1 = 15*c; th2 = 105*c; th3 = 145*c;


9th4 = 0; th5 = 215*c; th6 = 55*c;

a2 = 0.431; a3 = 0.019; r4 = a2; r3 = 0.125;

T12 = [cos(th1) 0 sin(th1) 0 ; sin(th1) 0 -cos(th1) 0 ; 0 1 0 0; 0 0 0 1];

T23 = [cos(th2) -sin(th2) 0 a2*cos(th2); sin(th2) cos(th2) 0 a2*sin(th2); 0 0 1 0; 0 0 0 1];

T34 = [cos(th3) 0 sin(th3) a3*cos(th3); sin(th3) 0 -cos(th3) a3*sin(th3); 0 1 0 r3; 0 0 0 1];

T45 = [cos(th4) 0 sin(th4) 0; sin(th4) 0 -cos(th4) 0; 0 1 0 r4; 0 0 0 1];

T56 = [cos(th5) 0 sin(th5) 0; sin(th5) 0 -cos(th5) 0; 0 1 0 0; 0 0 0 1];

T67 = [cos(th6) 0 sin(th6) 0; sin(th6) 0 -cos(th6) 0; 0 1 0 0; 0 0 0 1];

T17 =T12*T23*T34*T45*T56*T67

Answer: T17 =

0.2418 -0.5540 0.7966 -0.4729

0.9128 -0.1485 -0.3804 -0.2561

0.3290 0.8192 0.4698 0.5459

0 0 0 1.0000

2.31 x = [17 31 5;6 5 4;19 28 9;12 11 10];

H = diag(x*inv(x'*x)*x')'

Answer: H = 0.7294 0.9041 0.4477 0.9188

2.32 % square

n = 1:2:401;

tau = linspace(-0.5, 0.5, 200);

plot(tau, 4/pi*(1./n)*sin(2*pi*n'*tau))

% sawtooth

n = 1:200;

tau = linspace(-1, 1, 200);

plot(tau, 0.5+1/pi*(1./n)*sin(2*pi*n'*tau))

% sawtooth

n = 1:200;


plot(tau, 0.5-1/pi*(1./n)*sin(2*pi*n'*tau))

% triangular

n = 1:200;


plot(tau, 0.5*pi-4/pi*((1./(2*n-1).^2)*cos(pi*(2*n'-1)*tau)))

% recitified sine wave

n = 1:200;


plot(tau, 2/pi+4/pi*((1./(1-4*n.^2))*cos(2*pi*n'*tau)))

% half sine wave

n = 2:2:106;


plot(tau, 1/pi+.5*sin(pi*tau)-2/pi*(1./(n.^2-1)*cos(pi*n'*tau)))

% exponential

n = 1:250;

tau = linspace(0, 4*pi, 350);

plot(tau, (exp(2*pi)-1)/pi*(0.5+((1./(1+n.^2))*cos(n'*tau)-(n./(1+n.^2))*sin(n'*tau))))

% trapezoidal

n = 1:2:105;


10


plot(tau, 64*((sin(n*pi/4)./(pi*n).^2)*sin(pi*n'*tau)))

2.33 a = sqrt(3); n = 1:25;

t = (10:10:80)*pi/180;

c = pi/sinh(pi*a)/2;

b = 1./(n.^2+a^2);

e1 = 100*(b*cos(n'*t)./(c*cosh(a*(pi-t))/a-0.5/a^2)-1)

e2 = 100*(n.*b*sin(n'*t)./(c*sinh(a*(pi-t)))-1)

Answers: e1 = -1.2435 0.8565 0.8728 -1.9417 -0.9579 -8.1206 0.7239 1.1661

e2 = 8.0538 10.4192 -8.9135 -5.4994 12.9734 -0.5090 -17.2259 11.2961

2.34 n = (1:2:51)*pi; a = 2;

eta = 0:1/14:1; xi = 0:1/14:1;

temp = meshgrid(1./(n.*sinh(n*a)), eta);

z = 4*(temp.*sinh(a*n'*eta)')*sin(n'*xi);

mesh(xi, eta, z)

Answer:

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

1.2

1.4

2.35 n = (1:50)*pi; eta = 0:0.05:1;

tau = 0:0.05:2; a = 0.25;

temp = meshgrid(sin(a*n)./n.^2, eta);

z = 2*pi/a/(1-a)*temp.*sin(n'*eta)'*cos(n'*tau);

mesh(tau, eta, z)

Answers:


11

0

0.5

1

1.5

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4

-2

0

2

4

2.36 A = [1 3 4; 31 67 9; 7 5 9];

B = [11 34 6; 7 13 8; 43 10 53];

DADB = det(A)*det(B)

DAB = det(A*B)

Answer: DADB = -3832062 DAB = -3832062

2.37 syms x y

x1 = -1; y1 = 1; x2 = 0; y2 = 0; x3 = 1; y3 = 1;

p = det([x^2+y^2 x y 1; x1^2+y1^2 x1 y1 1; x2^2+y2^2 x2 y2 1; x3^2+y3^2 x3 y3 1]);

p = factor(p)

Answer: p = 2*(x^2 + y^2 - 2*y)

2.38 syms x y

x1 = -1; y1 = 1; x2 = 1; y2 = 1; x3 = 2; y3 = 2;

p = det([x^2 x y 1; x1^2 x1 y1 1; x2^2 x2 y2 1; x3^2 x3 y3 1]);

p = factor(p)

Answer: p = 2*(x^2 - 3*y + 2)

2.39 A = [16, 32, 33, 13; 5, 11, 10, 8; 9, 7, 6, 12; 34, 14, 15, 1];

c = [91, 16, 5, 43];

supw = A\c' % or supw = linsolve(A, c')

determin = det(A)

invA = inv(A)

Answers: supw = [-0.1258 -8.7133 11.2875 -0.0500] determin = 7680

invA =

-0.0177 -0.0023 0.0180 0.0333

-0.3344 1.2352 -0.4695 0.1000


12

0.3500 -1.1375 0.3875 -0.1000

0.0333 -0.1500 0.1500 -0.0333

2.40 a = 0.005; b = 0.0064; c = 0.008;

E1 = 2.1e9; E2 = 0.21e9;

nu1 = 0.4; nu2 = 0.4; Uo = 0.00025;

Matrix = [1, a^2, 0, 0; 1, b^2, -1, -b^2; ...

-(1+nu1), (1-nu1)*b^2, (1+nu2)*E1/E2, -(1-nu2)*b^2*E1/E2;...

0, 0, -(1+nu2), (1-nu2)*c^2];

y = [0, 0, 0, -Uo*E2*c]';

AB = Matrix\y;

hoop1 = -AB(1)/b^2+AB(2)

hoop2 = -AB(3)/b^2+AB(4)

% check

urc = -(1+nu2)*AB(3)/E2/c+(1-nu2)*c*AB(4)/E2

Answers: hoop1 = -6.3013e+007 hoop2 = -1.1790e+007 urc = -2.5000e-004

2.41 syms i1 i2 i3 R V1 V2 V3

Q1 = 'V1-6*R*i1+4*R*(i2-i1)=0,';

Q2 = 'V2+2*R*(i3-i2)-3*R*i2-4*R*(i2-i1)=0,';

Q3 = '-V3-R*i3-2*R*(i3-i2y)=0';

[i1 i2 i3] = solve([Q1 Q2 Q3])

Answer: i1 = 1/182/R*(23*V1 + 12*V2 - 8*V3)

i2 = 1/91/R*(6*V1 + 15*V2 - 10*V3)

i3 = 1/91/R*(4*V1 + 10*V2 - 37*V3)

2.42 Either

syms s t V1 V2 V3 U real

A = [2*s+t -s 0; -s 2*s+t -s; 0 -s s+t];

B = [s*U 0 0]';

V=inv(A)*B

or

syms s t V1 V2 V3 U real

[V1 V2 V3] = solve((2*s+t)*V1-s*V2-s*U, -s*V1+(2*s+t)*V2-s*V3, -s*V2+(s+t)*V3, V1, V2, V3);

V1 = factor(V1)

V2 = factor(V2)

V3 = factor(V3)

gives

Answer: V1 = (U*s*(s^2 + 3*s*t + t^2))/(s^3 + 6*s^2*t + 5*s*t^2 + t^3)

V2 = (U*s^2*(s + t))/(s^3 + 6*s^2*t + 5*s*t^2 + t^3)

V3 = (U*s^3)/(s^3 + 6*s^2*t + 5*s*t^2 + t^3)



Chapter 3

3.1 ft = input('Enter the value of length in feet: ');

disp([num2str(ft) ' ft = ' num2str(ft*0.3048) ' m'])

Answer: Enter the value of length in feet: 11.4

11.4 ft = 3.4747 m

3.2 acre = input('Enter the number of acres: ');

disp([num2str(acre) ' acres = ' num2str(acre*43560*0.092903) ' sq. m'])

Answer: Enter the number of acres: 2.4

2.4 acres = 9712.4512 sq. m

3.3 Decimal = input('Enter a positive integer < 4.5x10^15: ');

disp(['The binary representation of ' num2str(Decimal) ' is ' num2str(dec2bin(Decimal))])

Answer: Enter a positive integer < 4.5x10^15: 37

The binary representation of 37 is 100101

3.4 R = input('Enter the real part of a complex number: ');

I = input('Enter the imaginary part of a complex number: ');

z = complex(R,I);

disp(['The magnitude and phase of ' num2str(z) ' is'])

disp(['Magnitude = ' num2str(abs(z)) ' Phase angle = ' num2str(angle(z)*180/pi) ' degrees'])

Answer: Enter the real part of a complex number: -7

Enter the imaginary part of a complex number: 13

The magnitude and phase of -7+13i is

Magnitude = 14.7648 Phase angle = 118.3008 degrees

3.5 n = 0:15;

f = (((1+sqrt(5))/2).^n-((1-sqrt(5))/2).^n)/sqrt(5);

disp([repmat('F', length(n),1) num2str(n',2) repmat(' = ', length(n),1) num2str(f')])

% or

fprintf(1, 'F%2.0f = %3.0f \n', [n;f])

Answers: F 0 = 0

F 1 = 1

F 2 = 1

F 3 = 2

F15 = 610


23.6 n = 9:-1:2; L = length(n);

A = [repmat('cos(pi/', L, 1) int2str(n') repmat(') = ', L, 1) num2str(cos(pi./n'),'%1.5f')]

B = [repmat(' pi/', L, 1) int2str(n') repmat(' = ', L, 1) num2str(180./n','%2.1f') repmat(' degrees',L,1)]

disp([A B])

fprintf(1,'cos(pi/%1.0f) = %1.5f pi/%1.0f = %2.1f degrees\n', [n; cos(pi./n); n; 180./n])

Answer: cos(pi/9) = 0.93969 pi/9 = 20.0 degrees

cos(pi/8) = 0.92388 pi/8 = 22.5 degrees

cos(pi/7) = 0.90097 pi/7 = 25.7 degrees

cos(pi/6) = 0.86603 pi/6 = 30.0 degrees

cos(pi/5) = 0.80902 pi/5 = 36.0 degrees

cos(pi/4) = 0.70711 pi/4 = 45.0 degrees

cos(pi/3) = 0.50000 pi/3 = 60.0 degrees

cos(pi/2) = 0.00000 pi/2 = 90.0 degrees

3.7 Mon ={'January', 'February', 'March', 'April', 'May', 'June', 'July', 'August', ...

'September', 'October', 'November', 'December'};

Srt = char(sort(Mon));

disp([Srt(1:6,:) repmat(' ',6,1) Srt(7:12,:)])

Answer: April June

August March

December May

February November

January October

July September

3.8 N = input('Enter an integer < 12: ');

n = 1:N;

disp(' ')

disp([repmat('For n = ', N, 1) int2str(n') repmat(', ', N, 1) int2str(n') repmat('! = ', N, 1) int2str(factorial(n)')])

disp(' ')

disp(['The sum of these ' int2str(N) ' factorials = ' int2str(sum(factorial(n)))])

Answer: Enter an integer < 12: 4

For n = 1, 1! = 1

For n = 2, 2! = 2

For n = 3, 3! = 6

For n = 4, 4! = 24

The sum of these 4 factorials = 33



Chapter 4

4.1

day = char('Sunday','Monday','Tuesday','Wednesday','Thursday','Friday','Saturday');

nonleapyear = [31 28 31 30 31 30 31 31 30 31 30 31];

leapyear = [31 29 31 30 31 30 31 31 30 31 30 31];

datee = input('Enter month, day and year in the form xx/xx/xxxx for 2010, 2011, or 2012: ','s');

f = findstr(datee,'/');

monthno = str2num(datee(1:f(1)-1)); dayno = str2num(datee(f(1)+1:f(2)-1));

yearno = str2num(datee(f(2)+1:f(2)+4));

if yearno == 2010

offse t= 5;

nodaypermonth = nonleapyear;

elseif yearno == 2011

offset = 6;

nodaypermonth = nonleapyear;

elseif yearno == 2012

offset = 7;

nodaypermonth = leapyear; else

error('Not programmed for this year')

end

if monthno>1

dayofyear = sum(nodaypermonth(1:monthno-1))+dayno;

else

dayofyear = dayno;

end

dayofweek = offset+dayofyear-fix(dayofyear/7)*7;

if dayofweek>7

dayofweek = dayofweek-7;

end disp(['The date ' datee(1:f(2)+4) ' is the ' num2str(dayofyear) ' day of the year and falls on a '

deblank(day(dayofweek,: )) '.'])

Answer: Enter month, day and year in the form xx/xx/xxxx for 2010, 2011, or 2012: 08/31/2011

The date 08/31/2011 is the 243 day of the year and falls on a Wednesday.

4.2

dat = [45 38 47 41 35 43]; L = length(dat);

vvar = zeros(L,1);

for n = 2:L

vvar(n) = var(dat(1:n));

end

disp([repmat('var(n=', L-1, 1) int2str((2:L)') repmat('): ', L-1,1) num2str(vvar(2:L),4)])

Answer: var(n=2): 24.5

var(n=3): 22.33

var(n=4): 16.25

var(n=5): 24.2

var(n=6): 19.9


24.3 m = 6; N = 100; Lend = (N-m+1);

a = 5*(1+rand(N,1));

A = zeros(Lend,1);

Avg(1) = mean(a(1:m));

for k = 2:Lend

Avg(k) = Avg(k-1)+(a(m+k-1)-a(k-1))/m; end

plot(1:Lend, Avg, 'ks-')

Answer:

0 10 20 30 40 50 60 70 80 90 1006

6.5

7

7.5

8

8.5

4.4

xold = 1.5; fx = 1; cnt = 0;

while fx>1e-8

xnew = xold-(cos(xold)+sin(xold))/(-sin(xold)+cos(xold));

fx = abs(cos(xnew)+sin(xnew));

cnt = cnt+1;

xold = xnew;

end

disp(['At x = ' num2str(xold,8) ', f(x) = ' num2str(fx) ' after ' int2str(cnt) ' iterations'])

Answer: At x = 2.3561945, f(x) = 1.1102e-016 after 4 iterations

4.5

ir = 0.045; P = 250000; A = 25000;

disp('Principal Annuity Interest No. Remain')

disp(' ($) ($/yr) (%) years ($)') R = P-A; n = 1;

while R>A

n = n+1;

R = R*(1+ir)-A;

end

disp([num2str(P,6) blanks(6) num2str(A,5) blanks(8) num2str(ir*100,3) blanks(9) num2str(n,3) blanks(6)

num2str(R,5)])

Answer: Principal Annuity Interest No. Remain


3 ($) ($/yr) (%) years ($)

250000 25000 4.5 12 19112

4.6

alp = pi/4; tol = 1e-5;

ao = 1; bo = cos(alp); co = sin(alp);

while abs(co)>tol

an = (ao+bo)/2;

bn = sqrt(ao*bo);

co = (ao-bo)/2;

ao = an;

bo = bn;

end

format long e Kalp = pi/ao/2

K = ellipke(sin(alp)^2)

format short

Answers: Kalp = 1.854074677301372e+000 K = 1.854074677301372e+000

4.7

h = [1 3 6 -7 -45 12 17 9];

a = 3; b=13; n = zeros(length(h),1);

for k = 1:length(h)

if h(k)=b

n(k) = 0;

else

n(k) = 1;

end

end

disp(['n = ' int2str(n')])

Answer: n = 0 0 1 0 0 1 0 1

4.8

N = input('Enter a positive integer < 10: ');

h = zeros(N,N);

for m = 1:N

for n = 1:N

if (n+m-1)

44.9 num = 1.5;

while (num < 1)||(num> 19)||(rem(num,1)~=0)

num = input(' Enter any integer from 1 to 19: ');

end

4.10

x = zeros(1000,1);

x(1) = 3; cnt = 1; while x(cnt)~=1

cnt = cnt+1;

if rem(x(cnt-1), 2) > 0

x(cnt) = 3*x(cnt-1)+1;

else

x(cnt) = x(cnt-1)/2;

end

end

disp(['x = ' num2str(x(1:cnt)')])

Answer: x = 3 10 5 16 8 4 2 1

4.11

x(1) = 0; n = 201;

x = zeros(n,1);

for k = 2:n

x(k) = x(k-1)^2+0.25;

end

plot(0:5:n-1, x(1:5:n), 'ks')

m = 1; xw(1) = 0;

xw = zeros(201, 1);

while m

50 20 40 60 80 100 120 140 160 180 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

4.12

a = 7; x = [3 100];

for k=1:2

xold = x(k); test = 1; cnt = 0;

while test>1e-6

xnew = 0.5*(xold+a/xold);

test = abs(xnew-xold);

xold = xnew; cnt = cnt+1;

end

disp(['For xo = ' int2str(x(k)) ' the number of iterations = ' num2str(cnt)])

end

Answers: For xo = 3 the number of iterations = 4

For xo = 100 the number of iterations = 10

4.13

for k = 1:3

switch k

case 1

p=[1 2 3 4]; s = [10 20 30 40];

case 2

p = [11 12 13 14]; s = [101 102];

case 3

p = [43 54 55]; s = [77 66 88 44 33];

end

n = length(p); m = length(s);

d = n-m;

if d==0

h = p+s

elseif d

6 h = 11 12 114 116

h = 77 66 131 98 88

4.14

a = [1.45, 2.75, 3.2, 4]; N = 200;

for k = 1:length(a)

xold = 0.1; cnt = 1; test = 1;

while (test>1e-4)

cnt = cnt+1;

xnew = xold*a(k)*(1-xold);

test = abs((xnew-xold)/xnew);

xold = xnew;

if cnt>N

break end

end

disp(['For a = ' num2str(a(k)) ', x(n=' int2str(cnt) ') = ' num2str(xold)])

end

Answers: For a = 1.45, x(n=19) = 0.31031

For a = 2.75, x(n=31) = 0.63634

For a = 3.2, x(n=201) = 0.51304

For a = 4, x(n=201) = 0.08737

4.15

Nu = 10;

for k = 1:2

test = -1;

while test < 0

NM(k) = input(['Enter a positive integer

7for k = 1:3

xon = [];

eon = [];

switch k

case 1

xo=[1 7 8 6 5 7 3 5 4]; eo=[2 6 10 4 3 6 1 2 3];

case 2

xo=[7 11 13 6];

eo=[6 10 15 7];

case 3

xo=[3 14 20 25 14 6 2 0 1 0];

eo=[4 12 19 19 14 8 4 2 1 1];

end

cutoff = 5; n = 1; nend = length(eo); cnt = 1;

while n


Chapter 5

5.1 Dod = [6 3 2 1.5 1.2 1.1 1.07 1.05 1.03 1.01];

c = [.88 .89 .91 .94 .97 .95 .98 .98 .98 .92];

a = [.33 .31 .29 .26 .22 .24 .21 .20 .18 .17];

Kspline = spline(Dod, c, Dod).*((Dod-1)/2).^-spline(Dod, a, Dod);

Kpoly = polyval(polyfit(Dod, c, 5), Dod).*((Dod-1)/2).^-polyval(polyfit(Dod, a, 5), Dod);

Kt = c.*((Dod-1)/2).^-a;

disp(' Kt Kspline Kpoly')

disp([Kt' Kspline' Kpoly'])

Answers: Kt Kspline Kpoly

0.6504 0.6504 0.6504

0.8900 0.8900 0.8900

1.1126 1.1126 1.1135

1.3479 1.3479 1.3369

1.6098 1.6098 1.6970

1.9497 1.9497 1.8546

1.9814 1.9814 1.9221

2.0495 2.0495 1.9886

2.0870 2.0870 2.1037

2.2645 2.2645 2.4324

5.2 function Exercise5_2

N = 2^10; wn = 2*pi*[5, 9, 9.4, 20];

dT = 2*pi/wn(end)/4; T = N*dT; df = 1/T;

t = (0:N-1)*T/N;

NHSignal = foft(t, N, wn);

whamm = 0.54-0.46*cos(2*pi*t/T);

k1 = sum(whamm.*NHSignal)/sum(whamm);

k2 = sqrt(N/sum(whamm.^2));

HSignal = whamm.*(NHSignal-k1)*k2;

subplot(2,1,1)

ft = fft(NHSignal, N);

Sig = 2*abs(ft(1:N/2))/N;

plot((0:N/2-1)*df, Sig)

temp = [1 find(diff(Sig)

2freqpeak = temp(temp1+1)*df;

disp([' Peaks occur at ' num2str(freqpeak) ' Hz'])

function NHSignal = foft(t, N, wn)

x = [0.1, 0.04, 0.04, 0.03];

Ho = [1 1.3 1.3 1.8];

NHSignal = 0;

for k = 1:length(x)

NHSignal = NHSignal+Ho(k)*Hw(t,wn(k),x(k));

end

function hw = Hw(t, wn, x)

hw = exp(-x*wn*t).*sin(sqrt(1-x^2)*wn*t);

Answers: Average power = 0.07529

Peaks occur at 4.84375 9.14063 20.0781 Hz

Average power = 0.0013038

Peaks occur at 4.92188 9.0625 9.45313 20.0781 Hz

0 5 10 15 20 25 30 35 400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 5 10 15 20 25 30 35 400

0.002

0.004

0.006

0.008

0.01

5.3 sx = 100; sy = -60; sz = 80; txy = -40; tyz = 50; tzx = 70;

c2 = sx+sy+sz;

c1 = txy^2+tyz^2+tzx^2-sx*sy-sy*sz-sz*sx;

c0 = sx*sy*sz+2*txy*tyz*tzx-sx*tyz^2-sy*tzx^2-sz*txy^2;

ps = sort(roots([1 -c2 -c1 -c0]), 'descend')

t12 = 0.5*(ps(1)-ps(2))

t23 = 0.5*(ps(2)-ps(3))

t13 = 0.5*(ps(1)-ps(3))

Answers: ps = 160.7444 54.8980 -95.6424 t12 = 52.9232 t23 = 75.2702 t13 = 128.1934

In Exercises 5.4 to 5.12 we have used the following function:

function Rt = FindZeros(FunName, Nroot, x, w)

f = feval(FunName, x, w);

indx = find(f(1:end-1).*f(2:end)

3Rt = zeros(Nroot, 1);

for k = 1:Nroot

Rt(k) = fzero(FunName, [x(indx(k)), x(indx(k)+1)], [], w);

end

5.4 Ex4 = @(x) (sin(x)-x.*cos(x))

Rt = FindZeros(Ex4, 5, linspace(0,10*pi,100), [])

Answers: Rt = 4.4934 7.7253 10.9041 14.0662 17.2208

5.5 Ex5 = @(x,p) (2*cos(x)-(x/p-p./x).*sin(x));

p = [0.1 1];

for k = 1:2

Rt = FindZeros(Ex5, 5, linspace(0,10*pi,100), p(k))'

end

Answers: Rt = 0.4435 3.2040 6.3149 9.4460 12.5823

Rt = 1.3065 3.6732 6.5846 9.6317 12.7232

5.6 Ex6 = @(x,b) (besselj(0,x).*bessely(0,b*x)-besselj(0,b*x).*bessely(0,x));

Rt = FindZeros(Ex6, 5, linspace(0,10*pi,100), 2)'

Answers: Rt = 3.1230 6.2734 9.4182 12.5614 15.7040

5.7 Ex7 = @(x, m) (m*x.*(cos(x).*sinh(x)-sin(x).*cosh(x))+cos(x).*cosh(x)+1);

m = [0, 0.2, 1];

for k = 1:3

Rt = FindZeros(Ex7, 5, linspace(0,10*pi,100), m(k));

disp(['m = ' num2str(Rt')])

end

Answers: m = 0: 1.8751 4.69409 7.85476 10.9955 14.1372

m = 0.2: 1.6164 4.26706 7.31837 10.4016 13.5067

m = 1: 1.24792 4.03114 7.13413 10.2566 13.3878

5.8 Ex8 = @(x,w) (cos(x).*tanh(x)-sin(x));

Rt = FindZeros(Ex8, 5, linspace(0,10*pi,100), [])'

Answers: Rt = 3.9266 7.0686 10.2102 13.3518 16.4934

5.9 Ex9 = @(x,m) (besselj(m,x).*besseli(m+1,x)+besseli(m,x).*besselj(m+1,x));


4for m = 0:2

Rt = FindZeros(Ex9, 3, linspace(0,10*pi,100), m);

disp(['m = ' int2str(m) ': ' num2str(Rt')])

end

Answers: m = 0: 3.1962 6.3064 9.4395

m = 1: 4.6109 7.79927 10.9581

m = 2: 5.90568 9.19688 12.4022

5.10 Ex10 = @(x,L2) (sin(x)-(x-4*x.^3/L2).*cos(x));

L2 = [2, 4, 8]*pi^2;

for k = 1:3

Rt = FindZeros(Ex10, 1, linspace(0,2*pi,100), L2(k));

disp(['(Lambda/pi)^2 = ' int2str(L2(k)/pi^2) ': ' num2str(Rt')])

end

Answers: (Lambda/pi)^2 = 2: 2.5185

(Lambda/pi)^2 = 4: 3.1416

(Lambda/pi)^2 = 8: 3.8834

5.11 channel = inline('(1+c(1)*x).^2.*(x.^2-x.^3)-c(2)','x','c');

c0 = [0.4, 7]; c1 = [0.2, 4];

for k = 1:length(c0)

r = roots([-c0(k)^2 c0(k)^2-2*c0(k) 2*c0(k)-1 1 0 -c1(k)]);

val = r(find((real(r)>0)&(real(r)

5x = [72 82 97 103 113 117 126 127 127 139 154 159 199 207];

beta = fzero(weibullbeta, [2 4], [], x)

Answer: beta = 3.6437

5.14 bearingstress = inline('x.*log(sqrt(x.^2-1)+x)-sqrt(x.^2-1)-c*x', 'x', 'c');

r = fzero(bearingstress, [1 4], [], 0.5)

Answer: r = 2.1155

5.15 jeffcooper = inline('x.^4-2.^x', 'x','w');

Rt = FindZeros(jeffcooper, 3, linspace(-2, 20, 100), [])

Answers: Rt = -0.8613 1.2396 16.0000


format long e

Zi = zstanford(1,1)

Zii = zstanford(1/0.3,1)

Ziii = zstanford(2.5,0.5)

format short

% (b)

p = 0.6; t = 1/1.05; trecip = 1.05;

r = fzero(@pvrtr, 5, [], p, t);

z = zstanford(r,t);

disp(['p = ' num2str(p) ' 1/tau = ' num2str(trecip) ' r = ' num2str(r) ' Z = ' num2str(z)])

p = 2.18; t = 1/1.2; trecip = 1.2;

r = fzero(@pvrtr, 5, [], p, t);

z = zstanford(r, t);

disp(['p = ' num2str(p) ' 1/tau = ' num2str(trecip) ' r = ' num2str(r) ' Z = ' num2str(z)])

% (c)

p = 0.6; r = 1/1.4; rrecip = 1.4;

t = fzero(@pvrtt, 0.8, [], p, r);

z = zstanford(r,t);

disp(['p = ' num2str(p) ' 1/r = ' num2str(rrecip) ' t = ' num2str(t) ' Z = ' num2str(z)])

p = 2.18; r = 1/0.6; rrecip = 0.6;

t = fzero(@pvrtt, 0.8, [], p, r);

z = zstanford(r,t);

disp(['p = ' num2str(p) ' 1/r = ' num2str(rrecip) ' t = ' num2str(t) ' Z = ' num2str(z)])

function z = zstanford(rr, tt)

a = [0.062432384 0.12721477 -0.93633233 0.70184411 -0.35160896 0.056450032 ...

0.0299561469907 -0.0318174367647 -0.0168211055516 1.6020406008 ...

-0.001099967407467 -0.000727155024312992 -0.0045245465261 0.001304687241 ...

-0.00022216512840926 -0.0019814053565598 5.9757397292e-5 -3.6413534970217e-6 ...

8.413648453856e-6 -9.828688588219e-9 -1.57683056810249 0.040072898890756 ...

-0.08451944938128 -0.003409313119283 -0.0019512704990109 4.938999109783e-5 ...

-4.9326461293046e-5 8.856665723816e-7 5.347880295527e-8 -5.934205591923e-8 ...


6 -9.068133269285e-9 1.6182240726495e-9 -3.320447939146e-10];

g = 0.0588; k = 1:12; r = rr.^k; t = tt.^k(1:6);

z1 = r(1)*sum(a(1:6).*t(1:6)/t(1))+r(2)*sum(a(7:10).*t(1:4)/t(1))+r(3)*sum(a(11:13).*t(1:3)/t(1));

z2 = r(4)*a(14)*t(1)+r(5)*(a(15)*t(2)+a(16)*t(3))+r(6)*a(17)*t(2)+r(7)*(a(18)*t(1)+a(19)*t(3)) ...

+r(8)*a(20)*t(3);

z31 = r(2)*(a(21)*t(3)+a(22)*t(4))+r(4)*(a(23)*t(3)+a(24)*t(5))+r(6)*(a(25)*t(3)+a(26)*t(4));

z32 = r(8)*(a(27)*t(3)+a(28)*t(5))+r(10)*(a(29)*t(3)+a(30)*t(4));

z33 = r(12)*(a(31)*t(3)+a(32)*t(4)+a(33)*t(5));

z = 1+z1+z2+(z31+z32+z33)*exp(-g*r(2));

function s = pvrtr(r, p, t)

s = zstanford(r,t)-p*t/r;

function s = pvrtt(t, p, r)

s = zstanford(r,t)-p*t/r;

Answers: Zi = 7.024239692767296e-001

Zii = 2.999999998525347e-001

Ziii = 9.922185392802401e-001

p = 0.6 1/tau = 1.05 r = 0.71309 Z = 0.80134

p = 2.18 1/tau = 1.2 r = 3.3567 Z = 0.54121

p = 0.6 1/r = 1.4 t = 0.95319 Z = 0.80068

p = 2.18 1/r = 0.6 t = 0.65049 Z = 0.85084


re = 100000; dk = [200, 200000];

for n =1:2

lambda = fzero(@pipefrictioncoeff, [.008 .08], [], re, dk(n));

disp(['Re = ' num2str(re) ' d/k = ' num2str(dk(n)) ': lambda = ' num2str(lambda)])

end

function x = pipefrictioncoeff(el, re, dk)

if (dk>100000)||(dk==0)

x = el-(2*log10(re*sqrt(el)/2.51))^-2;

else

x = el-(2*log10(2.51/re/sqrt(el)+0.27/dk))^-2;

end

Answers: Re = 100000 d/k = 200: lambda = 0.031298

Re = 100000 d/k = 200000: lambda = 0.01799

5.18 rts = roots([10 0 0 -75 0 -190 21]);

rr = rts(find(imag(rts)==0))'

Answer: rr = 2.2323 0.1100

5.19


7ro = roots([3*pi*sqrt(5) -60 0 0 0 0 0 0 0 20]);

posroot = ro(find(real(ro)>1))

Answer: posroot = 2.8468


tto = fzero(@Tto, [0.2 0.9])

function t = Tto(x)

t = Thq(exp(-x*pi^2))-pi/2/sqrt(5);

function qq = Thq(q)

k = 0:10;

qq = 2*q^(1/4).*sum((-1).^k./(2*k+1).*q.^(k.*(k+1)));

Answer: tto = 0.4240

5.21 fx =@(x) (x-(1+1./x).^x);

r = fzero(fx, [0 10])

Anwser: r = 2.2932

5.22 are = inline('sin(x)-0.5*abs(sin(2*x))', 'x');

x = linspace(0, pi);

aret = trapz(x, are(x))

areq = quadl(are, 0, pi)

Answers: aret = 0.9999 areq = 1.0000

5.23 bearing = inline('(1-(1-cos(x))/2/epsilon).^c.*cos(m*x)', 'x', 'c', 'm', 'epsilon');

m = 1; epsilon = 0.6;

c = 1.5; a = acos(1-2*epsilon);

Im = quadl(bearing, -a, a, [], [], c, m, epsilon)/2/pi

Answer: Im = 0.2416

5.24 Ee = inline('C1./lam.^5./(exp(C2./(lam*T))-1)', 'lam', 'C1', 'C2', 'T');

C1 = 3.742e8;

C2 = 1.439e4;

sig = 5.667e-8;

T = [300 400 500];

for k=1:length(T)

App = quadl(Ee, 1, 200, [], [], C1, C2, T(k));


8 exact = sig*T(k)^4;

err = 100*(exact/App-1);

disp(['T = ' num2str(T(k)) ' exact = ' num2str(exact) ' Approx. = ' num2str(App) ' %error = ' num2str(err)])

end

Answers: T = 300 exact = 459.027 Approx. = 458.7468 %error = 0.061087

T = 400 exact = 1450.752 Approx. = 1450.3993 %error = 0.024315

T = 500 exact = 3541.875 Approx. = 3541.4894 %error = 0.010889

5.25 arg = @(x,y) (cos(x-y).*exp(-x.*y/pi^2));

Z = dblquad(arg, 0, pi/2, pi/4, pi)

Answer: Z = 0.8449

5.26 arg = @(x) (1./sqrt((6-x.^2)));

Z = quadl(arg, 2, sqrt(6))-acot(sqrt(2))

Answer: Z = 1.3216e-006


Om = pi*[0.5969, 1.4942, 2.5002, 3.5:9.5];

Cn = -T(Om)./Q(Om);

N = zeros(10,1);

for n=1:10

N(n) = quadl(@W, 0, 1, [], [], Om(n), Cn(n));

end

disp(['N = ' num2str(N',4)])

function w = W(x, Om, Cn)

w = (Cn*T(Om*x)+S(Om*x)).^2;

function q = Q(x)

q = 0.5*(cosh(x)+cos(x));

function s = S(x)

s = 0.5*(cosh(x)-cos(x));

function t = T(x)

t = 0.5*(sinh(x)+sin(x));

Answers: N = 0.6133 0.3546 0.3137 0.2955 0.2854 0.2789 0.2745 0.2712 0.2687 0.2668


g = 9.8; alpha = 45*pi/180;

v0 = 180; cd = 0.007;


9[x, y] = ode45(@projectile, [0, 300], [v0*cos(alpha) v0*sin(alpha) 0 0]', [], v0, cd, g);

N = length(x);

indx = find(y(:,3)>0);

height = @(v,x,y) (-spline(x,y,v));

opt = optimset('display','off');

xmax = fminbnd(height, 0, 1000, opt, x, y(:,3));

ymax = -height(xmax, x, y(:,3));

disp(['ymax = ' num2str(ymax) ' m at x = ' num2str(xmax) ' m'])

xe = interp1(y(indx(end)-5:N, 3), x(indx(end)-5:N), 0);

te = interp1(y(indx(end)-5:N, 3), y(indx(end)-5:N, 4), 0);

disp(['Travel distance = ' num2str(xe) ' m and time of travel = ' num2str(te) ' s'])

function p=projectile(x, z, v0, cd, g)

v = sqrt(z(1)^2+z(2)^2);

if v

10

subplot(2,1,1)

plot(t, th(:,1))

subplot(2,1,2)

plot(th(:,1), th(:,2))

function v = IP(t, y, b, a)

v = [y(2); -a*y(2)+sin(y(1))-b*sin(y(1))*(1-1/sqrt(5-4*cos(y(1))))];

Answer:

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1

-0.5

0

0.5

1


p = 0.375; q = 7.43e-4;

Z = [10 50]; zer = zeros(length(Z),1);

for k = 1:2

[t, y] = ode45(@ReservoirOsc, [0 500], [Z(k) 0], [], p, q);

ind = min(find(y(:,1)

11

Answer: Minimum xi = 0.71873


e = 0.16; g = 0.4; w = 0.97;

[t, y] = ode45(@tunus, [0, 60], [-1,1], [], e, w, g);

plot(y(:,1), y(:,2),'k-')

function d = tunus(t, z, e, w, g)

c = w*t+9*pi/8;

L = 1+e*sin(c)^7;

Lp = 7*e*w*sin(c)^6*cos(c);

d = [z(2); -((2*Lp/L+g*L)*z(2)+sin(z(1))/L)];

Answer:

-1 -0.5 0 0.5 1 1.5-1

-0.5

0

0.5

1

1.5


a = 0.5; b = 0.6; k = 1;

[t, z] = ode45(@Ray, [0 30],[ 0 0.1], [], a, b, k);

plot(z(:,1), z(:,2), 'k-')

function d = Ray(t, y, a, b, k)

d = [y(2); -k*y(1)+a*y(2)-b*y(2)^3];

Answer:


12

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5


uo = 0.05; c = 0.02; r = 1;

w = 0.2; p = 2.01;

pe = fzero(@Static, [0.1, 0.2], [], p);

opt = odeset('RelTol', 0.00001);

[t, ph] = ode45(@Mecha, [0, 800], [pe, 0], opt, uo, c, r, w, p);

plot(t, ph(:,1), 'k-')

hold on

plot(t, uo*sin(w*t)+2*cos(ph(:,1))-2*cos(pe), 'b-')

function g = Mecha(t, y, uo, c, r, w, p)

den = 1/3+(1+2*r*p)*sin(y(1))^2;

g = [y(2); (-(1+2*r*p)*y(2)^2*sin(y(1))*cos(y(1))-(1+r*p)*uo*w^2*sin(y(1))*sin(w*t)...

-2*c*y(2)-2*y(1)+p*sin(y(1)))/den];

function d = Static(h, p)

d = 2*h-p*sin(h);

Answer:

0 100 200 300 400 500 600 700 800-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3


solinit = bvpinit(linspace(0,1,5),[1, 1, 1, -10, 0.91]);

sol = bvp4c(@Exode, @Exbc, solinit);


13

xx = linspace(0, 1, 50);

yy = deval(sol, xx);

plot(xx,yy(1,:))

hold on

plot(xx, yy(2,:))

plot(xx, yy(3,:))

function res = Exbc(ya,yb)

% y(1) = u, y(2) = v, y(3) = w, y(4) = z, y(5) = y.

res = [ ya(1) - 1; ya(2) - 1; ya(3) - 1; ya(4) + 10; yb(3) - yb(5)];

function dydx = Exode(x,y)

% y(1) = u, y(2) = v, y(3) = w, y(4) = z, y(5) = y.

dydx = [ 0.5*y(1)*(y(3) - y(1))/y(2)

-0.5*(y(3) - y(1))

(0.9 - 1000*(y(3) - y(5)) - 0.5*y(3)*(y(3) - y(1)))/y(4)

0.5*(y(3) - y(1))

100*(y(3) - y(5)) ];

Answer:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.8

1

1.2

1.4

1.6

1.8

2

2.2


LLo = 1.2;


beta = fzero(@cablebeta, 2, opt, LLo);

solinit = bvpinit(linspace(0,1,5), [0.1 0.1]);

sol = bvp4c(@inextcable, @BC, solinit, [], beta);

z = deval(sol, 0);

disp(['beta = ' num2str(beta) ' and the slope at eta = 0 is ' num2str(z(2,1))])

function a = cablebeta(beta, LLo)

solinit = bvpinit(linspace(0,1,5), [0.1 0.1]);

sol = bvp4c(@inextcable, @BC, solinit, [], beta);

eta = linspace(0, 1, 100);

zz = deval(sol, eta);

a = LLo-trapz(eta, sqrt(1+zz(2,:).^2));

function c = inextcable(t, z, beta)

c = [z(2); beta*sqrt(1+z(2)^2)];


14

function r = BC(za,zb,beta)

r = [za(1); zb(1)];

Answer: beta = 2.1297 and the slope at eta = 0 is -1.2778


qo = 1;

solinit = bvpinit(linspace(0, 1, 10), [0.5, 0.5, 0.5, 0.5]);

beamsol = bvp4c(@BeamODEqo, @CantBC, solinit, [], qo);


y = deval(beamsol, eta);

subplot(2,2,1)

plot(eta, y(1,:),'k-')

subplot(2,2,2)


subplot(2,2,3)


subplot(2,2,4)


function dydx = BeamODEqo(x, y, qo)

dydx = [y(2); y(3); y(4); qo.*(x>0.5)];

function bc = CantBC(y0, y1, qo)

bc = [y0(1); y0(2); y1(3); y1(4)];

Answer:

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

5.39 function Exerxise5_39

Pbar = 8; alph = pi/2;

sw = 0; tol = 0.001;

gl = 157; gh = 158;

qq = fzero(@Shoot, [gl, gh]*pi/180, [], Pbar, alph, sw, tol)*180/pi;

sw = 1;

[r, x] = NLBeam(qq*pi/180, Pbar, alph, sw, tol);

disp(['Pbar = ' num2str(Pbar) ' phi(0) = ' num2str(qq) ' (deg) x(0) = '...

num2str(x(1)), ' y(0) = ' num2str(x(2))])


15

function z = Shoot(q, Pbar, alph, sw, tol)

z = q-NLBeam(q, Pbar, alph, sw, tol);

function [res, dis]= NLBeam(y0, Pbar, alph, sw, tol)

solinit = bvpinit(linspace(0,1,5), [2 1]);

opt = bvpset('RelTol', tol);

solu = bvp4c(@Odes, @BC, solinit, opt, Pbar, alph, y0);

r = deval(solu(1,end), 1);

res = r(1);

dis = [999, 999];

if sw==1

xx = linspace(0,1,50);

s = deval(solu,xx);

dis = [trapz(xx, cos(s(1,:))), trapz(xx, sin(s(1,:)))] ;

end

function der = Odes(x, y, Pbar, alph, y0)

der = [y(2); -Pbar*sin(y(1)+alph-y0)];

function der = BC(ya, yb, Pbar ,alph, y0)

der = [ya(1); yb(2)];

Answer: Pbar = 8 phi(0) = 157.938 (deg) x(0) = -0.073844 y(0) = 0.63353


a = 0.8;

zinfi = fzero(@Eqn, [1.1 4], [], a)

function slop = Eqn(zinf, a)

zin = [zinf 1.05*zinf]; d(1:2) = 0;

for k = 1:2

solinit = bvpinit(linspace(0, zin(k), 5), [0.1, 0.1]);

exmp = bvp4c(@odeQ3, @bcQ3,solinit, [], a);

et = linspace(0, zin(k), 100);

wnew = deval(exmp, et);

slop = wnew(1, end)-0.05;

d(k) = interp1(et, wnew(1,:), 1.1);

end

slop = (d(2)-d(1))/0.05-0.03;

function de = odeQ3(t, y, a)

de = [y(2); -2*t*y(2)/sqrt(1-a*y(1))];

function b = bcQ3(ya, yb, a)

b = [ya(1)-1; yb(1)];

Answer: zinfi = 1.9501


v = 0.3;


16

solinit = bvpinit(linspace(0.2,1,5), [0.1, 0.1, 0.1, 0.1]);

plat = bvp4c(@Plate, @bcPlate, solinit, [], v);

rr = linspace(0.2, 1, 50);

w = deval(plat, rr);

subplot(2,2,1)

plot(rr, w(1,:), 'k')

subplot(2,2,2)

plot(rr, w(2,:), 'k')

subplot(2,2,3)

plot(rr, w(3,:)+v*w(2,:)./rr, 'k')

subplot(2,2,4)

plot(rr, w(4,:)+w(3,:)./rr-w(2,:)./rr.^2, 'k')

function s = Plate(r, y, v)

s = [y(2); y(3); y(4); -2/r*y(4)+y(3)/r^2-y(2)/r^3+1];

function d = bcPlate(yi, yo, v)

d = [yi(1); yi(2); yo(3)+v*yo(2); yo(4)+yo(3)-yo(2)];

Answer:

0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8 1-2.5

-2

-1.5

-1

-0.5

0

0.5


S = 0.4;

solinit = bvpinit(linspace(0, 1, 10), [0.5, 0.5, 0.5, 0.5]);

beamsol = bvp4c(@BeamODE, @CantileverBC, solinit, [], S);


y = deval(beamsol, eta);

subplot(2,2,1)


subplot(2,2,2)


subplot(2,2,3)


subplot(2,2,4)


function dydx = BeamODE(x, y, S)

dydx = [y(2); y(3); y(4); 1-S*y(3)];


17

function bc = CantileverBC(y0, y1, S)

bc = [y0(1); y0(2); y1(3); y1(4)+S*y1(2)];

Answer:

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0


Omguess = pi/2;

gbs = 3.12; Ro = 0.06;

solinit = bvpinit(linspace(0,1,10), @TimoBeamInit, Omguess);

beamsol = bvp4c(@TimoBeamODE, @TimoBeamBC, solinit, [], gbs, Ro);

Omega = beamsol.parameters

function yinit = TimoBeamInit(x)

yinit = [sin(x*pi/2); cos(x*pi/2); -sin(x*pi/2); -cos(x*pi/2)];

function dydx = TimoBeamODE(x, y, Om, gbs, Ro)

dydx = [y(2); y(4)-gbs*Ro^2*Om^4*y(1); y(4); -(y(2)+(gbs*Ro^4*Om^4-1)*y(3))/(gbs*Ro^2)];

function bc = TimoBeamBC(y0, y1, Om, gbs, Ro)

bc = [y0(1); y0(4)-0.05; y0(3); y1(2)-y1(3); y1(4)];

Answer: Omega = 1.8444


% Part (a)

wormleadmin = @(lambda, b) (b./sin(lambda)+1./cos(lambda));

bt = [0.02, 0.05, 0.08, 0.11, 0.15, 0.18, 0.23, 0.30];

opt = optimset('display','off'); z = zeros(length(bt),1);

for j = 1:length(bt)

z(j )= fminbnd(wormleadmin, pi/180, 40*pi/180, opt, bt(j))*180/pi;

end

disp(' beta Kmin')

disp([bt' z])

% Part (b)

wormlead = @(lambda, b, k) (k-wormleadmin(lambda,b));

lambda1 = fzero(wormlead, [10 25]*pi/180, opt, 0.16, 1.5)*180/pi


18

lambda2 = fzero(wormlead, [25 40]*pi/180, opt, 0.16, 1.5)*180/pi

Answers: beta Kmin

0.0200 15.1866

0.0500 20.2231

0.0800 23.3098

0.1100 25.6001

0.1500 27.9830

0.1800 29.4501

0.2300 31.4950

0.3000 33.7997

lambda1 = 22.5505 lambda2 = 35.0343

5.45 gasflowtank = inline('-sqrt(k/(k-1))*sqrt(pepo.^(2/k)-pepo.^((k+1)/k))','pepo','k');


k = 1.4;

maxpratio = fminbnd(gasflowtank, 0, 1, opt, k);

maxexact = (2/(k+1))^(k/(k-1));

pepo = linspace(0, 1, 200);

[minarray, index] = min(gasflowtank(pepo,k));

numpratio = pepo(index);

disp(['max ratio exact = ' num2str(maxexact) ' max ratio w/fminbnd = '

num2str(maxpratio) ' max ratio w/min = ' num2str(numpratio)])

Answers: max ratio exact = 0.52828 max ratio w/fminbnd = 0.5283 max ratio w/min = 0.52764

5.46 function Exercsie5_46

opt = optimset('Display', 'off');

sol = fsolve(@functExamp, [1 1 1], opt)

disp(['x = ' num2str(sol(1)) ' y = ' num2str(sol(2)) ' z = ' num2str(sol(3))])

function f = functExamp(x)

f = [sin(x(1))+x(2)^2+log(x(3))-7;

3*x(1)+2^x(2)+1-x(3)^3;

x(1)+x(2)+x(3)-5];

Answers: x = 0.59905 y = 2.3959 z = 2.005

5.47 fs = inline('[b-z(1)*(1-cos(z(2)));a-z(1)*(z(2)-sin(z(2)))]', 'z', 'a', 'b');

a = 1; b = 3;

opt = optimset('Display','off');

r = fsolve(fs, [5 pi/4], opt, a, b);

disp(['k = ' num2str(r(1)) ' theta = ' num2str(r(2)*180/pi) ' degrees'])

fz = inline('b*(th-sin(th))-a*(1-cos(th))', 'th', 'a', 'b');

th = fzero(fz, [pi/4], opt, a, b);

disp(['k = ' num2str(a/(th-sin(th))) ' theta = ' num2str(th*180/pi) ' degrees'])

Answers: k = 6.9189 theta = 55.4999 degrees

k = 6.9189 theta = 55.4999 degrees


19

5.48 QQ = inline('[T1^4-T(1)^4-T(3)/sig; T(1)^4-T(2)^4-T(3)/sig; T(2)^4-T2^4-T(3)/sig]', 'T', 'T1', 'T2', 'sig');

sig = 5.667e-8; T2 = 293; T1 = 373;

opt = optimset('display', 'off');

TQ = fsolve(QQ, [ 300 300 300], opt, T1, T2, sig);

disp('From fsolve:')

disp(['TA = ' num2str(TQ(1)) ' TB = ' num2str(TQ(2)) ' Q = ' num2str(TQ(3))])

% (b)

V = [1, 0, 1/sig; 1, -1, -1/sig; 0, 1, -1/sig] \[T1^4, 0, T2^4]';

disp(' ')

disp('From matrix inversion:')

disp(['TA = ' num2str(V(1)^(1/4)) ' TB = ' num2str(V(2)^(1/4)) ' Q = ' num2str(V(3))])

Answers: From fsolve:

TA = 352.052 TB = 326.5116 Q = 226.4312

From matrix inversion:

TA = 352.052 TB = 326.5116 Q = 226.4312

5.49 syms x

dfx = diff(exp(sin(x)), x);

xext = solve(dfx, x)

dfxx = limit(diff(dfx,x), x, xext)

Answers: xext = 1/2*PI dfxx = -exp(1)

5.50 syms x b

Intg = int((2*x+5)/(x^2+4*x+5), x, 0, b)

AofB = matlabFunction(Intg, 'vars', [b])

A = AofB(linspace(0, 4*pi, 10))'

Answers: A =

0

1.0964

1.8252

2.3653

2.7927

3.1459

3.4467

3.7084

3.9401

4.1478


[AlphMax fAlphaMax] = fminbnd(@(x) (-Integ(x)), 0, 1);


20

disp(['alphaMax = ' num2str(AlphMax) ' and f(alphaMax) = ' num2str(-fAlphaMax)])

[AlphMin fAlphaMin] = fminbnd(@Integ, 0.5, 5);

disp(['alphaMin = ' num2str(AlphMin) ' and f(alphaMin) = ' num2str(fAlphaMin)])

function In = Integ(alph)

In = (2+sin(10*alph)).*quadl(@Arg, 0, 1.8, [], [], alph);

function A = Arg(x, alph)

A = x.^alph.*sin(alph./(2-x));

Answers: alphaMax = 0.78153 and f(alphaMax) = 3.2284

alphaMin = 3.8882 and f(alphaMin) = -0.99339


ka = pi/4; v = 0.3; b = 0.05/sqrt(12);

N = 10; rot = zeros(N+1,3);

for meth = 1:3

switch meth

case 1

[C1 C2 C3 C4] = Coef;

for n = 0:N

a11 = ka^2+(1-v)/2*n.^2;

a12 = (1+v)*n*ka/2;

a13 = v*ka;

a23 = n;

a22 = (1-v)/2*ka^2+n.^2;

a33 = 1+b^2*(ka^2+n.^2).^2;

C11 = C1(a11,a12,a13, a22,a23,a33);

C22 = C2(a11,a12,a13, a22,a23,a33);

C33 = C3(a11,a12,a13, a22,a23,a33);

C44 = C4(a11,a12,a13, a22,a23,a33);

rot(n+1,:) = sqrt(roots([C44 C33 C22 C11]));

end

case 2

for n = 0:N

A = MatrixElements(n, ka, b, v);

rot(n+1,:) = sqrt(eig(A));

end

case 3

Nroot = 3;

x = logspace(-2, 1.2, 100);

for n = 0:N

rot(n+1,:) = FindZero(@deter, Nroot, x, [n,ka,b,v]);

end

end

figure(meth)

for m=1:3

semilogy(0:10, rot(:,m), 'ks-' )

hold on

end

axis tight

disp(' ')

disp(' n L(1,n) L(2,n) L(3,n) ')


21

disp([(0:10)' rot])

end

function [C1 C2 C3 C4] = Coef

syms La a11 a12 a13 a21 a22 a23 a31 a32 a33

A = [-La+a11, a12, a13;

a12, -La+a22, a23;

a13, a23, -La+a33];

DetA = collect(simple(det(A)), La);

C = coeffs(DetA, La);

C1 = matlabFunction(C(1), 'vars', [a11, a12, a13, a22, a23, a33]);




function D = deter(La, w)

n = w(1); ka = w(2); b = w(3); v = w(4);

A = MatrixElements(n, ka, b, v);

A(1,1) = A(1,1)-La^2;

A(2,2) = A(2,2)-La^2;

A(3,3) = A(3,3)-La^2;

D = det(A);

function Rt=FindZero(FunName, Nroot, x, w)

cnt = 0; Rt = []; n = 3;

D = zeros(length(x),1);

% Following change required because 'deter' cannot

% deal with an array of values for x

for k = 1:length(x)

D(k) = feval(FunName, x(k), w);

end

for k = 1:length(x)

if sign(D(n))*sign(D(n-1))>0

n=n+1;

else

cnt = cnt+1;

rte = fzero(FunName,[x(n-1),x(n)],[],w);

Rt = [Rt rte];

n = n+2;

end

if n>length(x)-1||cnt==Nroot

break

end

end

function A = MatrixElements(n,ka,b,v)

a11 = ka^2+(1-v)/2*n.^2;

a12 = (1+v)*n*ka/2;

a13 = v*ka;

a23 = n;

a22 = (1-v)/2*ka^2+n.^2;

a33 = 1+b^2*(ka^2+n.^2).^2;

A = [a11, a12, a13; a12, a22, a23; a13, a23, a33];

Answers: n L(1,n) L(2,n) L(3,n)

0 1.0546 0.7105 0.4646


22

1.0000 1.5258 0.8883 0.2574

2.0000 2.3405 1.3203 0.1271

3.0000 3.2468 1.8546 0.1433

4.0000 4.1920 2.4212 0.2348

5.0000 5.1566 2.9996 0.3630

6.0000 6.1320 3.5831 0.5213

7.0000 7.1141 4.1693 0.7088

8.0000 8.1004 4.7570 0.9253

9.0000 9.0897 5.3457 1.1706

10.0000 10.0811 5.9351 1.4449

0 1 2 3 4 5 6 7 8 9 10

100

101



Chapter 6

6.1 % Cycloid

phi = linspace(-pi, 3*pi, 200);

ra = [0.5, 1, 1.5];

for k = 1:length(ra)

subplot(3,1,k)

plot(ra(k)*phi-sin(phi), ra(k)-cos(phi))

axis equal

title(['r_a = ' num2str(ra(k))])

axis equal off

end

Answer:

ra = 0.5

ra = 1

ra = 1.5

% Leminscate

phi = linspace(-pi/4, pi/4, 101);

plot(cos(phi).*sqrt(2*cos(2*phi)), sin(phi).*sqrt(2*cos(2*phi)))

axis equal off

Answer:

% Archimedean spiral

phi = linspace(0, 6*pi, 200);

plot(phi.*cos(phi), phi.*sin(phi))

axis equal off

Answer:


2

%Logarithmic spiral

phi = linspace(0, 6*pi, 200); k = 0.1;

plot(exp(k*phi).*cos(phi), exp(k*phi).*sin(phi))

axis equal off

Answer:

% Cardioid


plot(2*cos(phi)-cos(2*phi), 2*sin(phi)-sin(2*phi))

axis equal off

Answer:

% Astroid


plot(4*cos(phi).^3, 4*sin(phi).^3)

axis equal off

Answer:


3

% Epicycloid

en = [2*pi, 4*pi]; r = [3, 2.5]; a = [0.5 2];

for k = 1:2

subplot(1,2,k)

phi = linspace(0, en(k), 200);

rr = r(k); ar = a(k);

plot((rr+1)*cos(phi)-ar*cos(phi*(rr+1)), (rr+1)*sin(phi)-ar*sin(phi*(rr+1)))

axis equal off

end

Answer:

% Hypocycloid


rr = 3; ar = [0.5, 1, 2];

for k = 1:length(ar)

subplot(1,3,k)

plot((rr-1)*cos(phi)+ar(k)*cos(phi*(rr-1)), (rr-1)*sin(phi)-ar(k)*sin(phi*(rr-1)))

axis equal off

end

Answer:

% Eight curve

ph = linspace(0, 2*pi, 200); a = 2;

plot(a*sin(ph), a*sin(ph).*cos(ph), 'k-')

axis equal off


4Answer:

% Butterfly

x = linspace(-1, 1, 201); a = 2;

plot(x, (x.^4-x.^6).^(1/6), 'k-', x, -(x.^4-x.^6).^(1/6), 'k-')

axis equal off

Answer:

% Dumbbell

x = linspace(-1, 1, 200); a = 2;

plot(x, a*sqrt((x.^4-x.^6)), 'k-', x, -a*sqrt((x.^4-x.^6)), 'k-')

axis equal off

Answer:

% Bicuspid

a = 1; x = linspace(-a, a, 200);

rt = @(x,a) sqrt((a^2-x.^2).*(x-a).^2);

plot(x, sqrt(a^2+rt(x,a)), 'k-', x, -sqrt(a^2+rt(x,a)), 'k-')

hold on

A = @(x,a) (a^2-rt(x,a));

xl = fzero(A, -a/2, [], a)

x = linspace(-a, xl, 50);

plot(x, sqrt(a^2-rt(x,a)), 'k-', x, -sqrt(a^2-rt(x,a)), 'k-')

x= linspace(0, a, 50);

plot(x, sqrt(a^2-rt(x,a)

An Engineers Guide to Matlab Solutions

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