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Chapter 5: Problem 23

Biology Reed and Semtner \(^{57}\) showed that the proportional yield loss \(y\) of flue-cured tobacco was approximated by \(y=f(x)=0.259\left(1-e^{-0.000232 x}\right),\) where \(x\) is cumulative aphid-days in thousands. Use calculus to draw a graph. What happens to the proportional yield as the cumulative aphid- days becomes large without bound?

Short Answer

Expert verified
As cumulative aphid-days increase, the yield loss approaches 0.259 and stabilizes.

Step by step solution

01

Understand the function

The given function is \( y=f(x)=0.259\left(1-e^{-0.000232x}\right) \), where \(x\) is the cumulative aphid-days in thousands. This is an exponential growth function representing the yield loss as the cumulative aphid-days increase.
02

Identify the limit of the function as x approaches infinity

To understand what happens to the proportional yield loss as \( x \) becomes very large, we need to evaluate \( \lim_{x \to \infty} f(x) \). This can be computed as:\[ \lim_{x \to \infty} 0.259\left(1-e^{-0.000232x}\right) = 0.259 \times 1 - 0.259 \times \lim_{x \to \infty} e^{-0.000232x} \]Since \( \lim_{x \to \infty} e^{-0.000232x} = 0 \), the limit simplifies to \( 0.259 \).
03

Graph the function

To graph the function, we plot \( y = f(x) = 0.259\left(1-e^{-0.000232x}\right) \) for various values of \( x \). At \( x = 0 \), \( f(x) = 0 \). As \( x \) increases, \( f(x) \) approaches \( 0.259 \). The graph will show an asymptotic behavior approaching \( y = 0.259 \) as \( x \) becomes very large, reflecting the yield loss stabilization.
04

Interpret the results

As the cumulative aphid-days \( x \) becomes large without bound, the proportional yield loss \( y \) approaches a maximum value of \( 0.259 \). This means that even with very high aphid presence, the maximum yield loss stabilizes at this value and does not increase further.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the realm of calculus, these functions are vital to model growth or decay processes. The function given in the exercise, \( y=f(x)=0.259\left(1-e^{-0.000232x}\right) \), is an exponential function. Here, \( e^{-0.000232x} \) represents exponential decay. As the variable \( x \) increases, the term \( e^{-ax} \) (where \( a \) is a positive constant) approaches zero.

  • The exponential decay component means that initially, changes in \( x \) cause significant changes in \( y \), but these changes diminish as \( x \) gets larger.
  • In biological applications, such as this tobacco yield model, exponential functions can help predict outcomes under continuous changes like aphid populations.
Understanding these functions is crucial in analyzing not just biological phenomena but a wide array of scientific fields, from physics to finance.
Limit of a Function
The concept of limits is foundational in calculus. Limits help describe the behavior of functions as they approach certain points or infinity. In this exercise, we calculate the limit as \( x \) approaches infinity for the exponential function modeling yield loss.

When we compute \( \lim_{x \to \infty} f(x) \), we are essentially determining how \( f(x) \) behaves when \( x \) becomes very large. For the function \( y=f(x)=0.259\left(1-e^{-0.000232x}\right) \):
  • The limit \( \lim_{x \to \infty} e^{-0.000232x} \) is zero, because \( e^{-ax} \) approaches zero as \( x \) becomes infinitely large.
  • This means \( y \) simplifies to a constant value, here it鈥檚 0.259.
By understanding limits in exponential functions, we can predict long-term behavior, which is invaluable for scientists and mathematicians alike.
Graphing Functions
Graphing functions allows us to visualize how a function changes over its domain. In the context of the exponential function \( y=f(x)=0.259\left(1-e^{-0.000232x}\right) \), the graph helps us see the stabilization of yield loss over time.

To graph this function:
  • Start with plotting basic points, such as \( x = 0 \) to yield \( f(x) = 0 \). As \( x \) increases, the output \( y \) approaches 0.259.
  • Recognize the asymptotic behavior: initially rapid change slowing and approaching a steady value.
  • The horizontal line \( y = 0.259 \) represents an asymptote, giving visual insight into the function's limit.
Graphing makes it easier to observe trends, such as the settling point of this biological model, showing us how even with infinite aphid presence, the yield loss caps at 0.259. This graphical interpretation makes complex data accessible and interpretable.

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