Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use absolute values to write the following statement more compactly: Whenever \(x\) is within \(0.02\) of \(7, f(x)\) differs from 19 by no more than \(0.3\).

Step-by-Step Solution

Answer

The compact form using absolute values of the given statement is: Whenever \(\mid x - 7\mid \leq 0.02\), \(\mid f(x) - 19\mid \leq 0.3\).

Step 1 Understanding the given statement

The given statement can be broken into two parts: 'Whenever \(x\) is within \(0.02\) of \(7\)' and '\(f(x)\) differs from 19 by no more than \(0.3\)'. Essentially, the first part means that \(x\) falls between \(7 - 0.02\) and \(7 + 0.02\). Similarly, the second part indicates that \(f(x)\) falls between \(19 - 0.3\) and \(19 + 0.3\).

Step 2 Converting the statement into absolute value notation

\(\mid x - 7\mid\) is the absolute value representation of the first part, and it means that the value of \(x\) is within \(0.02\) of \(7\). \(\mid f(x) - 19\mid\) corresponds to the second part, i.e., \(f(x)\) is within \(0.3\) of \(19\).

Step 3 Writing the complete statement

Putting both parts together, we render the original statement more compactly as: Whenever \(\mid x - 7\mid \leq 0.02\), \(\mid f(x) - 19\mid \leq 0.3\). This means whenever \(x\) is within \(0.02\) of \(7\), \(f(x)\) is within \(0.3\) of \(19\).

Key Concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It's particularly useful in solving problems that involve changing conditions or rates of change. At the heart of calculus is the concept of the infinitesimally small and how these tiny changes impact a system as a whole.

When it comes to absolute value notation in calculus, it often represents the distance from a certain point on a number line, regardless of direction. This concept is crucial for understanding how variables behave in relation to a specific point, which is a common problem in calculus. For example, the absolute value tells us how a function approximately behaves near a certain point, which can be essential for solving limits and derivatives as they deal with changes near specific points.

Functions and Rates of Change

In the context of functions, the rate of change is an important concept that indicates how one quantity changes in relation to another. Mathematically speaking, this is typically represented by the derivative in calculus. The derivative of a function at a point is the slope of the tangent line at that point, which also translates as the instantaneous rate of change of the function.

The absolute value notation can greatly simplify the expression of a function's behavior near a target value or within a certain interval. In our exercise, absolute value is used to conveniently encapsulate the idea that the function's value, or output, remains close to a particular number, which is central to understanding how the function changes or maintains stability within that range.

Inequalities in Calculus

Inequalities are a fundamental part of calculus that show the relationship between two values when they are not strictly equal but instead one is lesser or greater than the other. They help to describe a range of possible values that satisfy certain conditions, adding a layer of versatility to solving calculus problems.

When using absolute value notation in inequalities, we can denote a range of numbers that are within a certain distance from a given value. This is exemplified in the given exercise, where the statement describes a condition on the proximity of the independent variable, \(x\), to a particular number, as well as the dependent variable, \(f(x)\), to a fixed value. Understanding how to write and interpret such inequalities is important for describing the behavior of functions within specified bounds, a skill that is essential when dealing with practical applications in calculus.