areas highlighted in orange I don't understand what it means. can anyone explain in simple terms
Simple example is radioactive decay, every xxx years (half life), the activity decays by 1/2. The gradient or rate must slow down as the activity decreases
Obviously, you can exponential growth as well. Bank interest. The more money you have, the greater the amount is credited to your account. The interest rate (exponent) stays the same.
y = Ae^(kt)
dy/dt = kAe^(kt) = ky
as y increases or decreases, the rate of change increases or decreases.
and no it's as level not GCSE. nothing to worry about
dy/dt = ky
which you can solve by splitting it up as
(1/y) dy = k dt
integrate to get
ln(y) - ln(A) = kt
take inverse logs (exponential)
y = Ae^kt
A is the initial value y(0). So they're basically the same thing saying:
* y(t) is an exponential
* the derivative (rate of change) is proportional to its value.
not really sure about this but if dy/dx gives the slope of some function of x say x^2 or x+2 or e^x, and y gives the value of that function, so y=x^2 or y=x+2 or y=e^x, then we are after a function where the slope of the function is proportional to the value of the function.
This can be written as dy/dx=ky, where k is the constant of proportionality.
This equation is talking about some function of x with the property that the value of that function is proportional to the value of the slope. So if you double the value of the function, you double the slope. And im not sure but exponentials might have this property which might explain the paragraph.
If y=e^x then dy/dx = e^x
This is a standard differential, that the function is the same as the differential equation.
Does that make sense?