Question

The amount of construction loans of peer banks as a percentage of its capital from the beginning of 2015 (t = 0) through to the beginning of 2018 is approximated by the function g(t) = -0.656t +5.693t3- 16.789t2 +36.08t +51.9, 0st15 (c) When did the construction loans of peer banks as a 100%? percentage of capital first exceed (d) What was the highest amount of construction loans period from the beginning of 2015 to the beginning 2008? Solve this question using differentiation. as a share of capital over the 100 over the interval 0 t5. Label or trace over its plot with a (e) Plot both functions, f and g and the line y f and g. Plot the function f in red the functions as highlighter.

Answer 1

Matlab code for plotting clear all close all syms t tfunction ro peecage ap t.2+30.08.*tt51.9; - 16.78 g(t)-0.656.*t.^4+5.693. t.' %finding the ercentage first crossing 100 gg (t)-0.656.*t."4+5.693.*t.^3-16.789.*t.^2+30.08.*t+51.9-100; SHere we willbe using Newton method for finding percentage q1 (t)diff(gg,t); $1st Derivative of this function sinitial guess) хX-1; Loop for all intial guesses n eps; %error limit for close itteration for i-1:1000 x2 double(xx-(gg(xx). /gl(xx))); %Newton Raphson Formula cc abs (gg(x2)) erri) cc; хх-x2; SError if cc<n break end end root xx; fprintf\n\ tUsing Newton method it will reach 100 percentage at t-f n',root) finding root for 1st derivative zero g_1(t) diffg,t); fprintf\n\ tFor finding maxima 1st derivative should be zero\n' fprintf\n\tlst derivative of the function f(t)=') disp (vpa (g 1,6)) g_2(t)diff (g_1,t); хх-1; %Loop for all intial guesses n 10^-6; error limit for close itteration for i-1:1000 %1st Derivative of this function g %initial guess] x2 double(xx-(g_1(xx). /g_2 (xx) )); Newton Raphson Formula cc double(abs (g_1(x2)) ) erri) cc; хх3x2; if cc<=n break SError 1

end end root-xx; fprintf'\n\tUs ing Newton method maxima will occured at t f\n',root) Splotting all functions tt1nspace ( 0,5,1000) ; fl =g_1(tt); y 100*ones (size (tt) ) ; hold on plot(tt,gl, 'r','linewidth',2) plot(tt,fl, 'b', 'linewidth',2) plot(tt,y, 'k', linewidth',2) legend'g(t)','f(t)','y') title Bank percentage of capitals') xlabelt') ylabel(t) and g(t)) box on 88888888s888888883888 End of Code %888888888888888 Using Newton method it will reach 100 percentage at t -3.990076 For finding maxima 1st derivati ve should be zero 1st derivative of the function f(t) =- 2.624 *t ^3 33.578*t 30.08 17. 079*t^2 - symbolic function inputs: t Using Newton method maxima will occured at t=4.0464 93 2

Bank percentage of capitals 120 100 80 60 40 20 F -20 40 4.5 3 3.5 25 15 05 1 t Published with MATLAB R2018a 6 pue a)

%%Matlab code for plotting
clear all
close all

syms t
%function for percentage of capital
g(t)=-0.656.*t.^4+5.693.*t.^3-16.789.*t.^2+30.08.*t+51.9;

%finding the percentage first crossing 100
gg(t)=-0.656.*t.^4+5.693.*t.^3-16.789.*t.^2+30.08.*t+51.9-100;

%Here we willbe using Newton method for finding percentage

g1(t) =diff(gg,t);   %1st Derivative of this function
xx=1;                %initial guess]
%Loop for all intial guesses
    n=eps; %error limit for close itteration
    for i=1:1000
        x2=double(xx-(gg(xx)./g1(xx))); %Newton Raphson Formula
        cc=abs(gg(x2));                 %Error
        err(i)=cc;
        xx=x2;
        if cc<=n
            break
        end
      
    end
    root=xx;
  
fprintf('\n\tUsing Newton method it will reach 100 percentage at t=%f\n',root)

%finding root for 1st derivative zero
g_1(t)=diff(g,t);
fprintf('\n\tFor finding maxima 1st derivative should be zero\n')
fprintf('\n\t1st derivative of the function f(t)=')
disp(vpa(g_1,6))

g_2(t) =diff(g_1,t);   %1st Derivative of this function g
xx=1;                 %initial guess]
%Loop for all intial guesses
    n=10^-6; %error limit for close itteration
    for i=1:1000
        x2=double(xx-(g_1(xx)./g_2(xx))); %Newton Raphson Formula
        cc=double(abs(g_1(x2)));                 %Error
        err(i)=cc;
        xx=x2;
        if cc<=n
            break
        end
      
    end
    root=xx;
  
fprintf('\n\tUsing Newton method maxima will occured at t=%f\n',root)

%plotting all functions
tt=linspace(0,5,1000);
g1=g(tt);
f1=g_1(tt);
y=100*ones(size(tt));

hold on
plot(tt,g1,'r','linewidth',2)
plot(tt,f1,'b','linewidth',2)
plot(tt,y,'k','linewidth',2)

legend('g(t)','f(t)','y')
title('Bank percentage of capitals')
xlabel('t')
ylabel('f(t) and g(t)')

box on


%%%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%