Problem 7

Question

Between 1940 and 1995 the size of the average farm in America increased from 174 acres to 469 acres. (Facts from the World Almanac and Book of Facts 1997.) Suppose that $A(t)$ gives the average number of acres of an American farm $t$ years after 1940 . $A(t)$ is an increasing function. (a) What are the units of $\frac{d A}{d t}$ ? (b) Average farm size increased much more dramatically in the 50 s than in the 80 s. Which is larger, $A(12)$ or $A(43)$ ? Which do you think is larger, $A^{\prime}(12)$ or $A^{\prime}(43)$ ?

Step-by-Step Solution

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Answer

The units of $ \frac{d A}{d t} $ are acres per year. $A(43)$ is larger than $A(12)$ indicating that the average farm size was larger in 1983 than in 1952. However, $A'(12)$ is larger than $A'(43)$ indicating that the farm size increased more quickly in 1952 than in 1983.

1Step 1: Understanding $ \frac{d A}{d t} $

$\frac{d A}{d t}$ represents the rate of change of the average farm size with respect to time. The units of $A(t)$ are acres, representing the size of the farm, and the unit of $t$ is years, marking the time elapsed after 1940. Hence the units of $ \frac{d A}{d t} $ are acres per year which represents the rate at which the average farm size is increasing per year.

2Step 2: Comparing $ A(12) $ and $ A(43) $

The year 1940 + 12 = 1952 falls within the 1950s, while the year 1940 + 43 = 1983 is within the 1980s. The average farm size increased more dramatically in the 1950s than in the 1980s, it is given. Since $ A(t) $ is an increasing function, $ A(43) $ will be larger than $ A(12) $, representing a larger average farm size in 1983 than in 1952.

3Step 3: Comparing $ A'(12) $ and $ A'(43) $

Since $A'(t)$ gives the rate of change of $A(t)$, a larger $A'(t)$ means the function $A(t)$ is increasing more rapidly at that time. Given that farm size increased more dramatically in the 1950s than the 1980s, $A'(12)$, representing the rate of change in farm size in 1952 (in the 1950s), should be larger than $A'(43)$, representing the rate of change in 1983 (in the 1980s).

Key Concepts

Rate of ChangeDifferentiationFunction Analysis

Rate of Change

When we talk about the **rate of change** of a function, we're discussing how fast the output of the function changes as the input changes. In simpler terms, it shows us the speed at which something is happening. For example, in this exercise, the rate at which the size of an average farm increases over time is important. The notation $ \frac{dA}{dt} $ represents this rate of change for the average farm size function $ A(t) $. Here:

$ A(t) $ is the number of acres after $ t $ years.
$ t $ is the number of years since 1940.

Thus, $ \frac{dA}{dt} $ is in the units of acres per year. This unit indicates how many acres the average farm size increases each year. To understand whether the changes are rapid or slow, one must look at both the sign and the magnitude of this rate of change. A larger magnitude indicates faster changes, while the sign tells you if the output is increasing or decreasing.

Differentiation

**Differentiation** is a fundamental operation in calculus. It involves calculating the derivative of a function. A derivative illustrates how a function changes at a specific point. In this exercise, the derivative $ A'(t) $ gives the rate at which the average farm size increases at any year $ t $ since 1940. Differentiation helps us find the slope of the tangent line to the curve at a particular point, thereby revealing how steep the curve of the function is. In practical terms:

High derivatives mean rapid changes.
Low derivatives indicate slower changes.

For this problem, comparing $ A'(12) $ and $ A'(43) $ helps us discern how much more quickly or slowly farm sizes were increasing in those respective years. The derivative value is higher when the growth is more striking. In the case of farm sizes, the derivative value is likely higher in the 1950s (such as $ A'(12) $) than in the 1980s ($ A'(43) $), consistent with historical data indicating more dramatic growth during the former decade.

Function Analysis

**Function analysis** involves examining how a function behaves in different scenarios. Looking at the function $ A(t) $ for this exercise, we analyze both its behavior over time and its rate of change.First, consider what it means for $ A(t) $ to be an increasing function. This indicates that as $ t $ increases, $ A(t) $ gets larger, meaning average farm size always goes up over the years. Function analysis also involves identifying periods of rapid change in farm size by evaluating the derivative at different points. Considering our specific instance:

In the 1950s, growth was rapid, indicating a higher derivative value for those years.
The 1980s experienced slower growth, indicating a smaller derivative.

Understanding these changes provides insight into historical patterns and trends, helping build a broader understanding of how functions like $ A(t) $ move and adapt over time. The analysis of $ A(t) $, accompanied by its derivative $ A'(t) $, offers a full picture of the dynamics in play.

Problem 6

Problem 7

Other exercises in this chapter

Problem 6

In Problems 5 through 8 , estimate $f^{\prime}(c)$ by calculating the difference quotient $\frac{f(c+h)-f(c)}{h}$ for successively smaller values of \(|h| .

View solution

Problem 6

Let $g(x)=\frac{x}{2 x+5}$. Using the limit de nition of derivative, nd $g^{\prime}(x)$.

View solution

Problem 7

In Problems 5 through 8 , estimate $f^{\prime}(c)$ by calculating the difference quotient $\frac{f(c+h)-f(c)}{h}$ for successively smaller values of \(|h| .

View solution

Problem 7

For Problems 7 through 13, find $f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)$, and $f^{\prime}(-1) .$ $$ f(x)=3 x+5 $$

View solution