The reaction R of the body to a dose Q of medication is...

The reaction $R$ of the body to a dose $Q$ of medication is often represented by the general function $$R(Q)=Q^{2}\left(\frac{k}{2}-\frac{Q}{3}\right)$$ where $k$ is a constant and $R$ is in millimeters of mercury $(\mathrm{mmHg})$ if the reaction is a change in blood pressure or in degrees Fahrenheit $\left(^{\circ} \mathrm{F}\right)$ if the reaction is a change in temperature. The rate of change $d R / d Q$ is defined to be the body's sensitivity to the medication. a) Find a formula for the sensitivity. b) Explain the meaning of your answer to part (a).

The reaction $R$ of the body to a dose $Q$ of medication is often represented by the general function $$R(Q)=Q^{2}\left(\frac{k}{2}-\frac{Q}{3}\right)$$
where $k$ is a constant and $R$ is in millimeters of mercury $(\mathrm{mmHg})$ if the reaction is a change in blood pressure or in degrees Fahrenheit $\left(^{\circ} \mathrm{F}\right)$ if the reaction is a change in temperature. The rate of change $d R / d Q$ is defined to be the body's sensitivity to the medication.
a) Find a formula for the sensitivity.
b) Explain the meaning of your answer to part (a).

Key Concepts

Differentiation

Differentiation is the process of computing the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. In the context of this problem, differentiating the reaction function R(Q) with respect to Q yields the sensitivity of the body's response to medication, showing how small changes in the dose Q affect the reaction R.

Product Rule

The product rule is a fundamental technique in calculus used to find the derivative of a product of two functions. Since the reaction function R(Q) is given as the product of Q² and (k/2 - Q/3), the product rule is required to differentiate it correctly by taking into account the derivative of each factor and their interactions.

Polynomial Functions

The reaction function and its derivative are both polynomial functions. Polynomials are important in calculus as they allow for straightforward computation of derivatives and provide clear, interpretable results that describe how the response changes with varying dosages. This helps in understanding the dose-response relationship in a comprehensible mathematical form.

Interpretation of Sensitivity

The derivative dR/dQ, referred to as the sensitivity, quantifies how the reaction changes in response to small variations in the medication dose. A higher sensitivity indicates that the reaction R (such as blood pressure or temperature change) responds strongly to changes in Q, meaning that minor adjustments in the dose can produce significant changes in the body's reaction, an insight that is crucial for dosaging decisions.

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