Problem 11 - Applied Calculus

Question

An agricultural worker in Uganda is planting clover to increase the number of bees making their home in the region. There are 100 bees in the region naturally, and for every acre put under clover, 20 more bees are found in the region. (a) Draw a graph of the total number, \(N(x)\), of bees as a function of \(x\), the number of acres devoted to clover. (b) Explain, both geometrically and algebraically, the shape of the graph of: (i) The marginal rate of increase of the number of bees with acres of clover, \(N^{\prime}(x)\). (ii) The average number of bees per acre of clover, \(N(x) / x\)

Step-by-Step Solution

Answer

(a) N(x) = 100 + 20x is a straight line. (b)(i) N'(x) = 20, a horizontal line. (b)(ii) N(x)/x = 20 + 100/x, a hyperbola.

Step 1: Understanding the problem

We have a formula for the number of bees, which is given by the problem. Naturally, there are 100 bees, and for every acre of clover, there are 20 additional bees.

Step 2: Defining the function N(x)

The total number of bees, \( N(x) \), can be expressed as \( N(x) = 100 + 20x \), where \( x \) is the number of acres of clover planted.

Step 3: Sketching the Graph of N(x)

Plot the graph of \( N(x) = 100 + 20x \). It's a straight line with a y-intercept at 100 and a slope of 20, indicating that for every increase of 1 in \( x \), \( N(x) \) increases by 20.

Step 4: Calculating the Marginal Rate, N'(x)

The marginal rate of increase is the derivative of \( N(x) \). Since \( N(x) = 100 + 20x \), the derivative is \( N'(x) = 20 \). This constant derivative indicates a constant rate of increase.

Step 5: Graph Description for N'(x)

Geometrically, the graph of \( N'(x) \) would be a horizontal line at \( y = 20 \). Algebraically, it's because the slope of \( N(x) \) is constant and equal to 20.

Step 6: Calculating the Average Number of Bees per Acre, N(x)/x

The average number of bees per acre is given by \( N(x)/x = \frac{100 + 20x}{x} = \frac{100}{x} + 20 \).

Step 7: Graph Description for N(x)/x

Geometrically, this is a hyperbola. As \( x \) increases, \( \frac{100}{x} \) decreases, meaning the average number per acre approaches 20. Algebraically, for large \( x \), \( \frac{100}{x} \) becomes negligible and the average approaches 20.

Key Concepts

Marginal Analysis

In this exercise, marginal analysis plays a significant role in understanding how the number of bees changes with each additional acre of clover. Marginal analysis focuses on the derivative of a function, representing how a small change in one variable affects another.

For this problem, the number of bees, denoted by the function \( N(x) = 100 + 20x \), shows how the number of bees increases with each acre of clover planted. The derivative, \( N'(x) \), captures the marginal rate of change. Here, \( N'(x) = 20 \) tells us that each additional acre of clover adds 20 more bees.

• Constant Derivative: The derivative \( N'(x) = 20 \) is constant, meaning the rate of increase is fixed.
• Geometric Interpretation: The graph is a horizontal line at \( y = 20 \), illustrating no change in the rate of increase.
• Algebraic Representation: This constant derivative results from the linear relationship in \( N(x) \).

Therefore, marginal analysis provides insight through both geometric and algebraic lenses, showing the impact of additional acres planted.

Average Rate of Change

The average rate of change is another vital concept, helping to understand how the total number of bees per acre changes as more land is devoted to clover. The average rate of change is given by \( \frac{N(x)}{x} \), or how many bees per acre are obtained over a specified number of acres.

When calculating \( N(x)/x = \frac{100 + 20x}{x} = \frac{100}{x} + 20 \), it becomes clear how the average number of bees per acre evolves.

• Initial High Value: At smaller \( x \), the term \( \frac{100}{x} \) contributes significantly, making the average look high.
• Decreasing Effect: As \( x \) increases, \( \frac{100}{x} \) becomes negligible, driving the average toward 20.
• Graph Shape: The function graphically resembles a hyperbola, constantly approaching the horizontal asymptote \( y = 20 \).

Ultimately, this metric shows how effectively land is being utilized as more clover is planted, spotlighting how the average levels off over time.

Graphing Calculus Functions

Graphing is an essential skill in calculus, allowing visual interpretation of functions. Here, two main graphs are considered: the graph of \( N(x) \) and the graphs of its derived functions.

For the function \( N(x) = 100 + 20x \), the graph is straightforward:

• Straight Line: The graph forms a straight line because it is a linear function.
• Understanding the Slope: The slope, 20, signifies for every acre of clover, 20 more bees appear.

Graphing derived functions helps to further elucidate the behavior of \( N(x) \):

• Marginal Rate \( N'(x) \): A horizontal line at 20, reinforcing the constant rate of bee increase.
• Average Rate \( \frac{N(x)}{x} \): Appears hyperbolic with a horizontal asymptote at \( y = 20 \), indicating that the average stabilizes to 20.

By graphing these functions, complex ideas become more accessible, providing a clearer picture of the relationship between acres of clover and the bee population.