- #1
Simon Bridge said:You get it from the equation of the growth.
Simon Bridge said:Derived as part of the answer to part (a), used in part (b).
The whole question is about you ability to turn words into equations and vise versa.
Consider: what does "rate of population growth" mean?
Here is your question:
Q8. A territory will support a maximum population of ##P_0##. Let the ratio of the population ##P## to the maximum population be ##p##... the rate of change of this ratio is proportional to the product of ##p## and the difference between ##p## and ##1##.
(a) write down the differential equation in ##p##
(b) show that the growth of the population is greatest when ##P=\frac{1}{2}P_0##
(c) the population starts at ##\frac{1}{4}P_0## and reaches ##\frac{1}{2}P_0## in 20 years.
Find the time for the population to reach ##\frac{7}{8}P_0##.
(d) find the value of ther maximum growth of the population to 3dp.
Differentiation is a mathematical process used to find the rate of change of a function. It involves finding the derivative of a function, which represents the slope of the tangent line at any given point on the function's graph.
Differentiation is important because it allows us to analyze the behavior of functions and their rates of change. It is used in many fields such as physics, engineering, economics, and more to model real-world situations and make predictions.
Some common applications of differentiation include optimization, curve sketching, related rates problems, and finding maximum and minimum values of functions. It is also used in physics to calculate velocity and acceleration, and in economics to determine marginal cost and revenue.
There are three main types of differentiation: power rule, product rule, and chain rule. The power rule is used for functions with exponents, the product rule is used for the product of two or more functions, and the chain rule is used for compositions of functions.
Differentiation and integration are inverse operations of each other. The derivative of a function is the slope of its tangent line, while the integral of a function is the area under its curve. They are both used to solve problems involving rates of change and have many practical applications.