9:41LG
Specimen V
Reading an equation on a phone
The equation built one term at a time, with a graph you can scrub.
The question
How does a population grow when food and space are limited?
It cannot climb for ever. Calculus lets us write the brake straight into the growth, so the rule and its limit sit in one line.
Build it, one part at a time
dP/dt
Part 1 of 4
The solution, plotted live
The full working
Separate the variables, integrate, and the rule becomes a closed form for P at any time t. That line is very wide, so it stays whole and scrolls sideways rather than being crushed.
∫ dP / [ P (1 − P/K) ] = ∫ r dt ⟹ ln[ P / (K − P) ] = r t + C ⟹ P(t) = K / ( 1 + A e^(−r t) ), A = (K − P₀) / P₀
Turn your phone sideways for the whole line at once.