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Sorry, I don't know how to write this besides just plain text, but it shouldn't be too hard to read:
lim k->0 (e^kt-a)/k = at
It seems strange to me, and I have no idea how to derive it. Here's how I got it. Starting with:
dv/dt = a+kv
Seperate variables and integrate to get:
dv/(a+kv) = dt
(1/k)*ln(a+kv) = t+c (I'll set c=0 from here on)
a+kv = e^(kt)
v = (e^kt-a)/k
which is the left side of the limit. But as k goes to 0 in the original equation:
dv/dt = a
v = at
Is this right? If so, is there a better way to derive it? If not, where did I go wrong? (I know there are multiple divide by zeros, but I don't think that's the problem because k varies continuously, and the original curve could get closer and closer to a straight line as k is smaller and smaller but still greater than 0.)
lim k->0 (e^kt-a)/k = at
It seems strange to me, and I have no idea how to derive it. Here's how I got it. Starting with:
dv/dt = a+kv
Seperate variables and integrate to get:
dv/(a+kv) = dt
(1/k)*ln(a+kv) = t+c (I'll set c=0 from here on)
a+kv = e^(kt)
v = (e^kt-a)/k
which is the left side of the limit. But as k goes to 0 in the original equation:
dv/dt = a
v = at
Is this right? If so, is there a better way to derive it? If not, where did I go wrong? (I know there are multiple divide by zeros, but I don't think that's the problem because k varies continuously, and the original curve could get closer and closer to a straight line as k is smaller and smaller but still greater than 0.)