Problem 9 - Calculus Concepts

Question

The number of U.S. farms with milk cows can be modeled as $$ f(x)=45.183\left(0.831^{x}\right)+60 \text { thousand farms } $$ where \(x\) is the number of years since \(2000,\) based on data for years between 2001 and 2007 . (Source: Based on data from Statistical Abstract, 2007 and 2008\()\). a. Were the number of farms with milk cows increasing or decreasing between 2001 and \(2007 ?\) b. What is the concavity of the function on the interval \(1 \leq x \leq 7 ?\)

Step-by-Step Solution

Answer

a) Decreasing; b) Concave upward.

Step 1: Understanding the Decrease or Increase of Farms

The function given is an exponential decay function: \[f(x) = 45.183(0.831^x) + 60.\]Since the base of the exponent (0.831) is less than 1, it indicates that as \(x\) increases, \(45.183(0.831^x)\) will decrease towards zero. Hence, the number of U.S. farms with milk cows is decreasing over time between 2001 and 2007.

Step 2: Analyzing the Concavity of the Function

To determine the concavity, we need to find the second derivative of the function. The first derivative \[f'(x) = 45.183 \cdot 0.831^x \cdot \ln(0.831).\]Notice \(\ln(0.831) < 0\), so \(f'(x)\) is negative, affirming the decrease.The second derivative is \[f''(x) = 45.183 \cdot (\ln(0.831))^2 \cdot 0.831^x.\]Since \((\ln(0.831))^2\) is positive (as squaring eliminates negativity), and \(0.831^x\) is positive, \(f''(x)\) is positive. This implies the function is concave upward over the interval \(1 \leq x \leq 7\).

Key Concepts

First Derivative

In calculus, the first derivative of a function provides crucial information about the function's rate of change. For the given function, \(f(x) = 45.183(0.831^x) + 60\), we need to figure out how the number of U.S. farms with milk cows changes over time. When we differentiate \(f(x)\) with respect to \(x\), we obtain the first derivative, \(f'(x) = 45.183 \cdot 0.831^x \cdot \ln(0.831)\). The natural logarithm, \(\ln(0.831)\), is negative because it indicates the decrease rate in an exponential decay scenario.

• If \(f'(x)\) is negative, the function is decreasing.
• A negative \(f'(x)\) confirms that the milk cow farms are decreasing over the period from 2001 to 2007.

The first derivative gives us powerful insights into the behavior of the function as time progresses.

Concavity

Concavity is another essential concept that helps understand the shape of a graph. It describes whether a function is bending upwards or downwards. We determine concavity by inspecting the second derivative of a function. If the second derivative, \(f''(x)\), is positive, the function is concave upwards. Conversely, if \(f''(x)\) is negative, the function is concave downwards.
For our function, we know from the calculations that \(f''(x) = 45.183 \cdot (\ln(0.831))^2 \cdot 0.831^x\), which resulted in a positive number across the interval from 1 to 7. Therefore, we deduce that the graph of the function is bending upwards within this time frame.

• Knowing about concavity helps in understanding the general trend and acceleration of decrease.
• In this example, even as the number of farms decreases, the rate of this decrease is becoming less steep as time progresses.

Understanding concavity is crucial to interpreting how the change in the number of farms behaves over time.

Second Derivative

The second derivative of a function, \(f''(x)\), provides information about the concavity of the function by indicating the rate of change of the first derivative. For our function \(f(x)\), the second derivative is calculated as \(f''(x) = 45.183 \cdot (\ln(0.831))^2 \cdot 0.831^x\). This expression is essential because it indicates how the rate of change itself is changing.

• It tells us about the acceleration or deceleration of the function change.
• A positive \(f''(x)\) signifies concave upwards, showing the decrease in farms is slowing down.

Therefore, by analyzing the second derivative, it's clear that the rate at which farms are decreasing isn't constant; instead, the reduction slows down, indicating a more gradual decrease as the years go by. Understanding the second derivative helps us appreciate these nuances better.