Biology Arthur and Zettler ^20 created a mathematical model...

Biology Arthur and Zettler $^{20}$ created a mathematical model that showed that the percentage mortality $y$ of the red flour beetle was approximated by $y=f(x)=$ $9.1+86.3 e^{-0.64 x},$ where $x$ is the number of weeks after a malathion treatment. Use calculus to show that $f(x)$ is a decreasing function. What happens in the long-term? Sketch a graph. Check using your grapher.

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Key Concepts

Limits and Asymptotic Behavior

Evaluating the limit of a function as the independent variable approaches infinity reveals its long-term behavior. This concept helps determine steady states or asymptotes in models, showing whether the function levels off, continues to grow, or declines indefinitely.

Graphical Analysis

Graphical analysis involves sketching the function based on key features such as initial values, rate of change, and asymptotic behavior. This visualization aids in confirming analytical results and enhancing understanding of how the function behaves over time.

Exponential Decay

Exponential decay describes processes where a quantity decreases by a consistent percentage over uniform time intervals. In modeling scenarios, an exponential term such as e^(–kx) captures the rapid initial decline that slows over time, reflecting natural decay phenomena.

Differentiation and Monotonicity

Calculus is used to analyze whether functions are increasing or decreasing by calculating their derivatives. If the derivative of a function is negative over an interval, it signifies that the function is decreasing on that interval. This approach is vital for understanding the dynamic behavior of many real-world models.

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