Understanding the problem; visualizing a solution
- Do you know the
current world population? What units did you use in this estimate?
- How much do you
think the population grew in the last 10 years?
- We will compare
the population growth in the last ten years to the growth
in 1950. How could your compare growth over different periods
of time? Compare your ideas with others in your group.
- Do you think it
would be useful to be able to predict future population? Why?
- What do you think
are some important factors that influence population growth?
- What do you think
are some of the important consequence of population growth
Learn or review mathematical concepts and skills needed to study
population change. See the menu at the left.
The table below gives world population for selected years.
U.S. Census Report
- What patterns
do you observe from this information?
how much is the world population growing each year? How did
you arrive at this answer?
- Make points out
of the data. Use a (t,P) coordinate system where t = 0 in
2000 and P is the world population, in billions. Plot the
points on a (t,P) coordinate system.
- Use exponential
regression to determine the exponential function P(t) which
best fits the data. Describe in your own words exactly what
- Graph the function
P(t). Estimate from the graph the rate of change of population
for 1950, for 2000 and for the current year. What units did
you use to describe the rate of change?
- Determine the
derivative P'(t). Evaluate the derivative for t = -50, t =
0 and t = current year. Compare your answers to the answer
of the previous question.
The growth rate is the rate of change described as a percent
of the population. That is,
7. Compute the growth rate for t = -50, t =
0 and t = current year -2000. What patterns do you notice?
The growth ratqe
of an exponential function is constant.
7. Compute the growth rate using this formula.
- What assumnptions
did you use in constructing the population model?
- Whatis the difference
between the rate of population change and the growth rate?
- What does it mean
if the growth rate is negative?
In this problem we will see what is the effect of a change in
growth rate. In particular we will suppose the growth rate drops
to 1% beginning in 2005.
- Use the function
P(t) to predict the population in 2005.
- Write an exponential
function E(t) with the following properties: the population
in 2005 should be the population you predicted in a) and the
growth rate should be 1%.
- What is a reasonable
domain for this function?
- Compare the graphs
of P(t) and E(t). Summarize what you see.