represent a function in Matlab

I don't understand the graph of this exponential function, it is look like a two line instead of an exponential, this is the code and the figure:

xo = 400;

yo = 7000;

x = 260;

y = 6954;

a = 0.5, b =0.4;

K = b*xo-a*yo;

t= 0:0.5:50

y = K./(exp((K*t)-(K*(log(b*xo/yo)/(-(K)))))-a);

figure

plot (t,y)

Answer 1

[Steven Lord]

Let's look at your function symbolically.

xo = 400;
yo = 7000;
x  = 260;
y  = 6954;
a  = 0.5; b =0.4;
K = b*xo-a*yo;
% t= 0:0.5:50
syms t
 
y =  K./(exp((K*t)-(K*(log(b*xo/yo)/(-(K)))))-a);
vpa(y, 5)

ans = 

That exponential term is the exponential of a very large (in magnitude) negative number. It very quickly underflows to 0.

format longg
fun = @(t) exp(-3340*t-3.7785);
t = logspace(-9, 0, 10).';
results = table(t, fun(t), 'VariableNames', ["t", "fun(t)"])

results = 10x2 table
      t              fun(t)        
    ______    _____________________

     1e-09       0.0228568748085117
     1e-08       0.0228561877411816
     1e-07       0.0228493182036898
     1e-06       0.0227807362834381
     1e-05       0.0221061373691749
    0.0001       0.0163668063215704
     0.001     0.000809980811568495
      0.01     7.13809961741543e-17
       0.1    2.01679308890843e-147
         1                        0

If we evaluated that at a symbolic value of 50, what's the value?

vpa(fun(sym(50)), 10)

ans = 

1.51544597e-72529

When you're talking about numbers that small you're not quite at the probability of a monkey typing Hamlet first try, but maybe one of Shakespeare's shorter plays? For most other intents and purposes, that value is 0.

[Alvaro Mª Zumalacarregui Delgado]

thanks!