# Math Help - Derivatives/linear approximation problem

1. ## Derivatives/linear approximation problem

The population of a certain country grew from 2.1 million in 1970 to 3 million in 1995.

a) What was the average rate of change of the population over that period? (worked out to be 36 thousand people per year).

b)Suppose that the population grew exponentially. By what percentage did the population grow each year? What was the average rate of change from 1970 to 1975? What was the average rate of change from 1990 to 1995? Illustrate your answers using a sketch.

I worked out no. a), but I am having problems with number b) could some please help? This question appeared under a section in my note under the heading of derivatives and linear approximations, so I am thinking you have to use that to solve the problem.

2. You assume exponential growth. This means that the rate of change of the population is an exponenbtial, in pathematics terms, with P being the population

$\frac{dP}{dt}=Ae^{Rt}$. To solve for your two constants, you use the conditions given. Taking the zero time to be 1970 gives
$Ae^0=2.1$. This gives A. Then, after 25 years (1995), you have
$2.1e^{25R}=3$. Solve for R.

For the percentage change, consider that if the function is not thought of as continous, then it can be written discretely as

$P(t+1) = RP(t)$

Use the function to get the change between 1975 and 1970. Divide by 5 to get the average. etc.

3. $

\frac{dP}{dt}=Ae^{Rt}
$

Why do you have e and a compound exponent in the formula?

$

P(t+1) = RP(t)
$

How did you make this formula?

4. Sorry, that is a typo. That should be $P=Aexp(rt).$

So, you can either do this problem discretely, i.e. P(t+1)=RP(t), or you can approximate it to a continous function, which is the first way that I did it. Sorry, I am not explaining myself very well here..

An exponential function is of the form Aexp(Bt), only I have called my B r instead. However, a discrete exponential function is one that increases by a certain percentage each time unit, i.e. it satisfies $P(t+1)=C P(t)$. Writing it either way should work.

The e in my formula there is the exponential.