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

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.

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.

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.

Transcript

00:01 One hour after a dose of x milligrams of a particular drug is administered to a person, the change in body temperature, t of x, in degrees fahrenheit, is approximately given by t of x equals x squared times the quantity of 1 minus x over 9, for x values from 0 through 7.

00:20 The rate t prime of x at which t changes with respect to the size of the dosage x is called the sensitivity of the body to the dosage.

00:32 In part a, we're going to use the product rule to find t prime of x.

00:37 So i'm using the product rule because t of x is the product of two functions of x.

00:45 The product rule says i start by writing down my first factor, x squared, times the derivative of my second factor.

00:53 The derivative of 1 minus x over 9 is negative 1 9.

01:01 So that we add our second factor, 1 minus x over 9, and then i multiply by the derivative of that first factor.

01:12 The derivative of x squared is 2 x.

01:17 Now if we were to distribute, to simplify this a little bit, my t prime of x is negative 1 9 times x squared plus 2 x minus 2 x squared minus 2 x squared, i should say, over 9.

01:38 Now i do have a couple of like terms.

01:41 Namely, we've got negative 1 9 x squared and negative 2 9 x squared.

01:46 They combine to make negative 3 9 x squared or negative 1 3 x squared.

01:52 So my t prime of x is negative 1 3 x squared plus 2 x...

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