The population (in millions) of the United States (excluding...

The population (in millions) of the United States (excluding Alaska and Hawaii) $t$ years after 1800 is given by the function $f(t)$ in Fig. $18(\mathrm{a})$. The graphs of $f^{\prime}(t)$ and $f^{\prime \prime}(t)$ are shown in Figs. $18(\mathrm{~b})$ and $18(\mathrm{c})$. (a) What was the population in 1925 ? (b) Approximately when was the population 25 million? (c) How fast was the population growing in 1950 ? (d) When during the last 50 years was the population growing at the rate of $1.8$ million people per year? (e) In what year was the population growing at the greatest rate?

The population (in millions) of the United States (excluding Alaska and Hawaii) $t$ years after 1800 is given by the function $f(t)$ in Fig. $18(\mathrm{a})$. The graphs of $f^{\prime}(t)$ and $f^{\prime \prime}(t)$ are shown in Figs. $18(\mathrm{~b})$ and $18(\mathrm{c})$.
(a) What was the population in 1925 ?
(b) Approximately when was the population 25 million?
(c) How fast was the population growing in 1950 ?
(d) When during the last 50 years was the population growing at the rate of $1.8$ million people per year?
(e) In what year was the population growing at the greatest rate?

Key Concepts

Function Evaluation

Understanding that the population function f(t) represents the number of people (in millions) at time t and that evaluating this function at specific values of t (corresponding to years) provides the actual population values is crucial. This concept emphasizes the connection between real-world situations and abstract functions, and it underpins the process of reading and extracting data from graphs.

Interpretation of the First Derivative

The first derivative f'(t) indicates the rate at which the population is changing over time; it measures the instantaneous growth rate (in millions per year). Recognizing this is important for analyzing how quickly the population is increasing or decreasing at any given moment, and it helps in answering questions regarding the speed of population change.

Interpretation of the Second Derivative

The second derivative f''(t) provides information about the acceleration or deceleration of the population growth rate. This concept explains whether the increase in population is speeding up or slowing down, which is reflected in the curvature (or concavity) of the population graph. Understanding f''(t) is key to analyzing changes in the growth rate over time.

Graphical Analysis and Estimation

Using graphical methods to estimate key values of functions and their derivatives is a fundamental skill. This involves reading and interpreting intersections, scales, and approximate values from the graphs to answer questions such as when the population reached a certain level or when the rate of change was at a specific value.

Inverse Function Concepts

Determining the time at which the population reaches a specific value, such as 25 million, involves the concept of finding the inverse: identifying the input (year) for which the function output (population) matches the desired value. This concept connects the idea of reversing a function to obtain the necessary information about the independent variable.

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