Drug sensitivity. One hour after x milligrams of a...

Drug sensitivity. One hour after $x$ milligrams of a particular drug are given to a person, the change in body temperature $T$ (in degrees Fahrenheit) is given by $$ T=x^{2}\left(1-\frac{x}{9}\right) \quad 0 \leq x \leq 6 $$ Approximate the changes in body temperature produced by the following changes in drug dosages: (A) From 2 to 2.1 milligrams (B) From 3 to 3.1 milligrams (C) From 4 to 4.1 milligrams

Drug sensitivity. One hour after $x$ milligrams of a particular drug are given to a person, the change in body temperature $T$ (in degrees Fahrenheit) is given by $$ T=x^{2}\left(1-\frac{x}{9}\right) \quad 0 \leq x \leq 6 $$ Approximate the changes in body temperature produced by the following changes in drug dosages:
(A) From 2 to 2.1 milligrams
(B) From 3 to 3.1 milligrams
(C) From 4 to 4.1 milligrams

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Key Concepts

Derivative

The derivative of a function represents its instantaneous rate of change with respect to its independent variable. It provides a measure of how much the output value of a function is changing at a specific input value. In the context of drug sensitivity, the derivative indicates the rate at which the body temperature changes in response to small changes in the dosage.

Linear Approximation

Linear approximation, also known as differential approximation, uses the derivative to estimate the change in a function for small changes in the input. This method involves calculating the derivative at a particular point and multiplying it by the small change in the variable to approximate the corresponding change in the function. This technique is particularly useful when dealing with small modifications in drug dosage to predict their effects on body temperature.

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