One hour after a dose of x milligrams of a particular drug...

One hour after a dose of $x$ milligrams of a particular drug is administered to a person, the change in body temperature $T(x)$, in degrees Fahrenheit, is given approximately by $$ T(x)=x^{2}\left(1-\frac{x}{9}\right) \quad 0 \leq x \leq 7 $$ The rate $T^{\prime}(x)$ at which $T$ changes with respect to the size of the dosage $x$ is called the sensitivity of the body to the dosage. (A) Use the product rule to find $T^{\prime}(x)$. (B) Find $T^{\prime}(1), T^{\prime}(3),$ and $T^{\prime}(6)$.

One hour after a dose of $x$ milligrams of a particular drug is administered to a person, the change in body temperature $T(x)$, in degrees Fahrenheit, is given approximately by
$$
T(x)=x^{2}\left(1-\frac{x}{9}\right) \quad 0 \leq x \leq 7
$$
The rate $T^{\prime}(x)$ at which $T$ changes with respect to the size of the dosage $x$ is called the sensitivity of the body to the dosage.
(A) Use the product rule to find $T^{\prime}(x)$.
(B) Find $T^{\prime}(1), T^{\prime}(3),$ and $T^{\prime}(6)$.

Key Concepts

Function Evaluation of Derivatives

Evaluating a derivative at specific points involves substituting particular input values into the derivative expression to determine the rate of change at those points. This process is essential in applied mathematics and real-world problems, where it is important to know the behavior of a system at certain values of the independent variable.

Sensitivity Analysis

Sensitivity analysis in the context of calculus refers to the study of how changes in the input of a function affect its output. The derivative is used as a measure of sensitivity, indicating how responsive a system is to changes. In applied problems, such as those involving dosages in medicine, the derivative helps quantify how a small change in dosage can lead to a corresponding change in a measured effect.

Differentiation

Differentiation is a fundamental concept in calculus that measures the rate at which a function changes with respect to changes in its input. It provides a way to calculate the instantaneous rate of change or slope of a function at any given point, which is essential for understanding how small variations in input affect the outcome.

Product Rule

The product rule is an important rule in differential calculus used to differentiate expressions that are products of two or more functions. It states that the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function. This rule is crucial for efficiently finding derivatives without the need to expand the product beforehand.

Transcript

Please give Ace some feedback