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

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

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.

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.

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.

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