In Exercise 55 in the section on Polynomial and Rational...

In Exercise 55 in the section on Polynomial and Rational Functions, we gave the function defined by $$ A(x)=0.003631 x^{3}-0.03746 x^{2}+0.1012 x+0.009 $$ as the approximate blood alcohol concentration in a $170-1 \mathrm{b}$ woman $x$ hours after drinking 2 $\mathrm{oz}$ of alcohol on an empty stomach, for $x$ in the interval $[0,5] .$ Source: Medicolegal Aspects of Alcohol Determination in Biological Specimens. a. Approximate the change in alcohol level from 1 to 1.2 hours. b. Approximate the change in alcohol level from 3 to 3.2 hours.

In Exercise 55 in the section on Polynomial and Rational Functions, we gave the function defined by
$$
A(x)=0.003631 x^{3}-0.03746 x^{2}+0.1012 x+0.009
$$
as the approximate blood alcohol concentration in a $170-1 \mathrm{b}$ woman $x$ hours after drinking 2 $\mathrm{oz}$ of alcohol on an empty stomach, for $x$ in the interval $[0,5] .$ Source: Medicolegal Aspects of Alcohol Determination in Biological Specimens.
a. Approximate the change in alcohol level from 1 to 1.2 hours.
b. Approximate the change in alcohol level from 3 to 3.2 hours.

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Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They are smooth and continuous, making them ideal for modeling various real-world phenomena through approximations and predictions without abrupt changes.

Change in Function Values

The change in function values refers to the difference between the function's output at two distinct points. This concept is crucial when determining how a quantity modeled by a function varies over a small interval. It involves calculating the difference f(x + ?x) – f(x), which provides insight into how a modeled quantity increases or decreases over a specific period.

Differential Approximation

Differential approximation, or linear approximation, leverages the derivative of a function to estimate changes in its values for small increments. When the increment ?x is small, the change in the function can be approximated by f'(x)·?x. This method is particularly useful for quickly estimating variations without computing the exact values of the function at both endpoints.

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