MATLAB: Represent a function in Matlab

exponential function

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)

Best Answer

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.
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