Human hands covered with cotton fabrics impregnated with the...

Human hands covered with cotton fabrics impregnated with the insect repellent DEPA were inserted for 5 minutes into a test chamber containing 200 female mosquitoes. The function $f(x)=26.48-14.09 \ln x$ gives the number of mosquito bites received when the concentration was $x$ percent. $[$ Note: The answers to parts $(\mathrm{b})-(\mathrm{e})$ can be obtained either algebraically or from the graphs. You might consider trying both methods.] (Source: Journal of Medical Entomology.) (a) Graph $f(x)$ and $f^{\prime}(x)$ for $0<x \leq 6$. (b) How many bites were received when the concentration was $3.25 \%$ ? (c) What concentration resulted in 15 bites? (d) At what rate is the number of bites changing with respect to concentration of DEPA when $x=2.75 ?$ (e) For what concentration does the rate of change of bites with respect to concentration equal $-10$ bites per percentage increase in concentration?

Human hands covered with cotton fabrics impregnated with the insect repellent DEPA were inserted for 5 minutes into a test chamber containing 200 female mosquitoes. The function $f(x)=26.48-14.09 \ln x$ gives the number of mosquito bites received when the concentration was $x$ percent. $[$ Note: The answers to parts $(\mathrm{b})-(\mathrm{e})$ can be obtained either algebraically or from the graphs. You might consider trying both methods.] (Source:
Journal of Medical Entomology.)
(a) Graph $f(x)$ and $f^{\prime}(x)$ for $0<x \leq 6$.
(b) How many bites were received when the concentration was $3.25 \%$ ?
(c) What concentration resulted in 15 bites?
(d) At what rate is the number of bites changing with respect to concentration of DEPA when $x=2.75 ?$
(e) For what concentration does the rate of change of bites with respect to concentration equal $-10$ bites per percentage increase in concentration?

Key Concepts

Interpretation of the Derivative as Rate of Change

The derivative of a function represents its instantaneous rate of change with respect to the independent variable. This interpretation is valuable in applied contexts, where it allows one to quantify how a small change in the input, such as concentration, influences the output, like the number of occurrences or events. Understanding this concept is critical for making informed decisions based on the model’s predictions.

Solving Equations Involving Logarithmic Expressions

Solving equations that include logarithmic expressions often requires isolating the logarithmic term and then exponentiating to eliminate the logarithm. This skill is useful in inverse problems where one needs to determine the input value that corresponds to a specific output, such as finding the concentration for a desired response.

Evaluating Functions at Specific Values

Evaluating a function at a given point involves substituting the input value into the function to compute the output. This technique is essential in determining precise numerical predictions in models and verifying the accuracy of more general functional behavior, such as the number of events or measurements predicted by the function.

Logarithmic Functions

Logarithmic functions are functions that involve the logarithm of the variable, which transform multiplicative relationships into additive ones. They often have domains restricted to positive values and exhibit a slow rate of increase at large values of the variable. Understanding the properties of logarithmic functions is essential for modeling phenomena that change at a diminishing rate as the input increases.

Graphing Functions and Their Derivatives

Graphing a function alongside its derivative provides insight into both the function’s values and the rate at which those values change. The derivative graph helps identify intervals of increase or decrease, and points where the function attains local maxima or minima. This combined graphical analysis is a powerful tool in understanding overall function behavior and verifying algebraic work.

Differentiation of Logarithmic Functions

The process of differentiating logarithmic functions, particularly using the derivative of the natural logarithm ln(x) which is 1/x, is a critical technique in calculus. This derivative is used to calculate how rapidly the output of the logarithmic function is changing with respect to changes in its input, which is fundamental in many applied problems.

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