Problem 27 Each of Exercises $15-30$ give... [FREE SOLUTION]

Chapter 2: Problem 27

Each of Exercises $15-30$ gives a function $f(x)$ and numbers $L, c,$ and $\varepsilon>0 .$ In each case, find an open interval about $c$ on which the inequality $|f(x)-L|<\varepsilon$ holds. Then give a value for $\delta>0$ such that for all $x$ satisfying $0<|x-c|<\delta$ the inequality $|f(x)-L|<\varepsilon$ holds. $$f(x)=m x, \quad m>0, \quad L=2 m, \quad c=2, \quad \varepsilon=0.03$$

Short Answer

Expert verified

The open interval is $ (2 - \frac{0.03}{m}, 2 + \frac{0.03}{m}) $ and $ \delta = \frac{0.03}{m} $.

Step by step solution

Understand the Problem

We need to find an interval around the given point $c = 2$ where the function $f(x) = mx$ is within $\varepsilon = 0.03$ of the limit $L = 2m$. Afterward, find $\delta > 0$ such that for all $x$ satisfying $0 < |x-2| < \delta$, $|f(x)-L| < \varepsilon$ holds.

Express the Condition

Express the inequality $|f(x) - L| < \varepsilon$. Substitute the given values: $|mx - 2m| < 0.03$. This simplifies to $m|x - 2| < 0.03$.

Solve for Boundaries of x

To find the interval around $c = 2$, solve the inequality $|x - 2| < \frac{0.03}{m}$. This tells us that $x$ must lie within $2 - \frac{0.03}{m} < x < 2 + \frac{0.03}{m}$.

Determine \\delta

From the inequality $|x - 2| < \frac{0.03}{m}$, we need to ensure this condition holds. Thus, choose $\delta = \frac{0.03}{m}$ so that $0 < |x - 2| < \delta$ implies $|f(x) - L| < \varepsilon$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Epsilon-Delta Definition

The epsilon-delta definition is a foundational concept in calculus used to rigorously define limits. It provides a precise way of showing how a function behaves as the input approaches a particular point. In this context, we are concerned with establishing that the function $ f(x) = mx $ gets arbitrarily close to a limit $ L = 2m $ as $ x $ becomes close to $ c = 2 $.Here's how it works:

The goal is to ensure the function is within an $ \varepsilon $ distance (often called "tolerance") from $ L $.
$ \delta $ is the maximum distance $ x $ is allowed to be from $ c $ to guarantee that $ |f(x) - L| < \varepsilon $.

By setting $ \delta = \frac{0.03}{m} $, we show that if $ x $ is within $ \delta $ of $ c $, then the function value stays within the desired $ \varepsilon $ range of $ L $.

Continuity

Continuity ensures that a function behaves predictably without any jumps or breaks within a particular interval. For linear functions like $ f(x) = mx $, which compose of straight lines through the origin, continuity means that there are no sudden changes in value for tiny changes in $ x $.In the epsilon-delta framework, a function $ f $ is continuous at a point $ c $ if, for every $ \varepsilon > 0 $, there exists a $ \delta > 0 $ so that whenever $ 0 < |x - c| < \delta $, it holds that $ |f(x) - f(c)| < \varepsilon $.For our exercise,

We prove continuity by showing that choosing $ \delta = \frac{0.03}{m} $ keeps $ f(x) = mx $ within $ 0.03 $ of its limit $ L = 2m $.

This makes $ f(x) $ not just continuous at $ x = 2 $, but over any open interval surrounding $ x = 2 $.

Linear Functions

Linear functions are among the simplest and most intuitive types of functions in mathematics. They have the form $ f(x) = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. In the provided exercise, the function $ f(x) = mx $ is a special type of linear function where the line passes through the origin (i.e., $ b = 0 $).Key properties of linear functions include:

Constant Rate of Change: The slope $ m $ determines how steep the line is, and it remains consistent across the entire function.
Graph Representation: These functions graph as straight lines with no curvature.
Easy Predictability: For any change in $ x $, the change in $ f(x) $ is precisely $ m \cdot $ (change in $ x $).

In solving the problem, understanding this linear nature helps in simplifying the inequality $ |mx - 2m| < 0.03 $ to $ m|x - 2| < 0.03 $.

Inequalities in Calculus

Inequalities are a powerful tool in calculus used to describe ranges and limits effectively. They play a crucial role in concepts like continuity, differentiability, and integration. In our context, inequalities help us confine the behavior of a function within specified bounds.Here's a breakdown of their use in solving the problem:

The inequality $ |mx - 2m| < 0.03 $ describes how close $ f(x) $ should be to the limit $ L = 2m $.
We manipulated this to $ |x - 2| < \frac{0.03}{m} $ to find an interval around $ x = 2 $ where the function behaves as desired.
This conversion from a strict inequality involving $ f(x) $ to one involving $ x $ allows us to determine the required $ \delta $.

By strategically solving inequalities, we've bound $ x $ within precise limits to preserve the close proximity ($ \varepsilon $) of $ f(x) $ to $ L $, a central pursuit in calculus for validating accuracy and control over functions.

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Short Answer

Step by step solution

Understand the Problem

Express the Condition

Solve for Boundaries of x

Determine \\delta

Key Concepts

Epsilon-Delta Definition

Continuity

Linear Functions

Inequalities in Calculus

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