using limit in differentiation

Hi,

I am calculating differentiation of a function v(t) wrt y which is also function of t i.e y(t) as follows:

syms t

x = 0.7071*t;

y = 0.7071*t - 4.9050*t^2;

v = 0.7071 - 9.81*t;

dv_dy = diff(v,t)/diff(y,t);

t = 0:0.001:1;

res = double(vpa(subs(dv_dy,{t}))); % result

The problem here is the differentiation approaches infinity when the denominator (diff(y,t)) of the term dv_dy approaches zero and it makes my result discontinuos (see fig). But it should be continous. How to correct it?

[Dyuman Joshi]

"But it should be continous."

Why? Could you show the mathematics to back your claim?

syms t
y = 0.7071*t - 4.9050*t^2;
v = 0.7071 - 9.81*t;
dv_dy = diff(v,t)/diff(y,t)

dv_dy = 

limit(dv_dy, t, 7071/98100, 'left')

ans = 

limit(dv_dy, t, 7071/98100, 'right')

ans = 

∞

fplot(dv_dy, [0 1])
ylim([-14e3 2e3])

Answer 1

[Saurabh]

Hi Mukesh,

I tried to reproduce the steps at my end and found out that the expression given by dv_dy will be discontinous at some point of time t, which is very close to 0.0721.

So, the possible work around solution which i can suggest is that you can exclude values of t which are very close to value 0.0721.

For reference I am attaching a code snippet:

syms t
x = 0.7071*t;
y = 0.7071*t - 4.9050*t^2;
v = 0.7071 - 9.81*t;
dv_dy = diff(v,t)/diff(y,t);
t = 0:0.001:1;
lower_bound = 0;
upper_bound = 0.1;
% Create a logical index that is true for values outside the range
index = ~(t >= lower_bound & t <= upper_bound);
% disp(size(t_values))
t = t(index);
% disp(size(t_values))

res = double(vpa(subs(dv_dy,{t}))); % result
plot(t, res);

This is the resultant graph I am getting after executing the code.

I hope this was helpful!!