Q1

In [41]:

ezplot('tan(x)',[0,10])
hold on;
g = @(x) 4*x./(4-x.^2);

lb = 3*pi/2+0.1;
ub = 5*pi/2-0.1;
hold on;
ezplot(g, [0,10])
plot([lb,ub],[0,0], 'ro')
plot([lb,ub],[0,0], 'r.')

In [39]:

format long
% Solve (4-x.*x)*tan(x) - 4x = 0
% tan(x) disc at each (2n+1)(pi/2)
% Guess lower, upper bracket = 3pi/2+0.1, 5pi/2 -0.1

hx = @(x) (4-x.^2).*tan(x)-4*x;
ezplot(hx, [0, 10])
hold on;
plot([lb,ub],[0,0], 'r.')
plot([lb,ub],[0,0], 'ro')

In [26]:

hx = @(x) (4-x.^2).*tan(x)-4*x;
TOL = 1e-7;
a = 3*pi/2+0.1;
b = 5*pi/2-0.1;
for n = 1:100
	c = (a+b)/2
	if abs((b-a)/2) <= TOL
		disp(n)
		break
	elseif sign(hx(c)) == sign(hx(a))
		a = c;
	else
		b = c;
	end
	
end

c =  6.28318530717959
c =  5.54778714378214
c =  5.91548622548086
c =  5.73163668463150
c =  5.63971191420682
c =  5.59374952899448
c =  5.61673072160065
c =  5.60524012529756
c =  5.59949482714602
c =  5.59662217807025
c =  5.59805850260814
c =  5.59734034033919
c =  5.59698125920472
c =  5.59680171863749
c =  5.59671194835387
c =  5.59675683349568
c =  5.59677927606658
c =  5.59676805478113
c =  5.59677366542386
c =  5.59677086010249
c =  5.59677226276317
c =  5.59677156143283
c =  5.59677191209800
c =  5.59677208743059
c =  5.59677217509688
 25

In [27]:

r = 6.4e6;
v = 5e3;
f = v*c/r;
T = 1/f

T =  228.703252509621

In [32]:

ezplot(hx, [0, 10])
hold on;
plot(c,0,'r.')
plot(c,0,'ro')

Q2

In [33]:

format longE;

In [34]:

function [x] = newt(x0,n)
f = @(y) y^2 -3;
df = @(y) 2*y;

if n == 0
	x = x0;
else
	temp = newt(x0,n-1);
	x = temp - f(temp)/df(temp);
end
end

In [36]:

TOL = 1e-10;
err = TOL + 1;
n = 0;
while err > TOL
	n += 1
	err = newt(2,n) - sqrt(3)
end

n =    1.00000000000000E+00
err =    1.79491924311228E-02
n =    2.00000000000000E+00
err =    9.20495739800131E-05
n =    3.00000000000000E+00
err =    2.44585041109247E-09
n =    4.00000000000000E+00
err =    0.00000000000000E+00

Q3

In [37]:

u = 16;
g = 9.8;
time = @(th) 20/(u*cosd(th));
y = @(th) u*time(th)*sind(th)-(g/2)*time(th)^2 -4;

th1 = fzero(y, 55)
th2 = fzero(y, 50)

th1 =    6.00350373511171E+01
th2 =    4.12748951229031E+01

In [38]:

th = th1;
t = time(th);
min(norm([u*t*cosd(th), u*t*sind(th)- (g/2)*t.^2])-[20,4])
th = th2;
t = time(th);
min(norm([u*t*cosd(th), u*t*sind(th)- (g/2)*t.^2])-[20,4])

ans =    3.96078054371142E-01
ans =    3.96078054371142E-01

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