The table gives the population of the world in the 20th...

The table gives the population of the world in the 20th century. (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing calculator or computer to find a cubic function (a third-degree polynomial) that models the data. (c) Use your model in part (b) to find a model for the rate of population growth in the 20th century. (d) Use part (c) to estimate the rates of growth in 1920 and $1980 .$ Compare with your estimates in part (a). (e) Estimate the rate of growth in $1985 .$

The table gives the population of the world in the 20th
century.
(a) Estimate the rate of population growth in 1920 and in
1980 by averaging the slopes of two secant lines.
(b) Use a graphing calculator or computer to find a cubic
function (a third-degree polynomial) that models the data.
(c) Use your model in part (b) to find a model for the rate of
population growth in the 20th century.
(d) Use part (c) to estimate the rates of growth in 1920 and
$1980 .$ Compare with your estimates in part (a).
(e) Estimate the rate of growth in $1985 .$

Key Concepts

Polynomial Interpolation and Cubic Functions

Polynomial interpolation involves constructing a polynomial that passes through a set of known data points. Choosing a cubic function, which is a third-degree polynomial, strikes a balance between accuracy and computational complexity, making it a common choice for modeling trends in data such as population growth across time.

Graphical and Numerical Tools

Graphing calculators and computer software are essential for visualizing data, performing polynomial interpolation, and calculating derivatives. These tools enable users to create models from data, estimate function values, and determine rates of change, thus bridging the gap between theoretical mathematics and practical application.

Population Growth Rate

This concept refers to the rate at which a population increases or decreases over a period. In mathematical modeling, it is often represented as the derivative of a population function with respect to time, providing insight into how population changes in each time interval relative to the previous one.

Secant Line and Averaging Slopes

A secant line connects two points on a curve and its slope gives an average rate of change between these points. Averaging the slopes of secant lines over multiple intervals is a method to estimate the instantaneous rate of change when dealing with discrete or tabulated data, serving as a practical numerical differentiation technique.

Derivative and Rate of Change in Modeling

The derivative represents the instantaneous rate of change of a function and is a fundamental concept in calculus. In the context of modeling growth, the derivative of a population function quantifies how quickly the population is increasing or decreasing at any specific point in time, allowing for precise estimations and comparisons between different time periods.

Transcript

00:01 In this question, we're given a population table of the population of the world in the 20th century, and we want to estimate the rate of population growth, find a cubic model for this data, and do a couple of things with a model.

00:19 So how do we do this? for 1920, let's establish a convention for this question first.

00:29 We're going to call t the number of years of the number of years since 1920.

00:39 This will make our calculations a little bit easier to manage.

00:45 So that means that t equals 20 corresponds to the year 1920.

00:54 And what we'll do is we will, what we do to find the average rate of growth by using cigant lines, we're going to take the average of two of them.

01:05 If we want to do so around the 20 mark, we can just take the average of the population in 1910 and 1920, and the population growth between 1920 and 1930.

01:19 So it'll be like this.

01:37 And this plus here will be a minus.

01:40 And all we do is just take the half of that.

01:53 So in this case, when we plug this in, we're going to plug in the values from the table and then subtract.

02:01 And when we do so, we're going to get that the population growth was around 16 million per year.

02:12 Similarly, if we want to do this for the year 1980, 80, we'll use t equals 80 for this, and we'll pretty much do the same thing.

02:23 We'll use p of 80 minus p of 70 over 80 minus 70.

02:29 And this up here should actually be a 20 minus 10 as well.

02:33 Let me correct that.

02:36 And we'll add this with p of 90 minus p of 80, divided by 90 minus 80.

02:44 And then we'll take the half of that...