Problem 50 - Thomas Calculus

Question

Let \(f(x)=2 x+3\) and let \(\epsilon>0\) be any positive number. Prove that \(|f(x)-f(0)|<\epsilon\) whenever \(|x-0|<\frac{\epsilon}{2} .\) Here the notation \(f(a)\) means the value of the expression \(2 x+3\) when \(x=a\) . (See Section \(1.3 . )\)

Step-by-Step Solution

Answer

Thus, \(|f(x) - f(0)| < \epsilon\) whenever \(|x - 0| < \frac{\epsilon}{2}\).

Step 1: Understanding the Function

Given the function \(f(x) = 2x + 3\), we need to calculate \(f(0)\). Substitute \(x = 0\) into \(f(x)\) to find \(f(0) = 2\times0 + 3 = 3\).

Step 2: Setup the Expression

We need to prove that \(|f(x) - f(0)| < \epsilon\). So, calculate \(|f(x) - 3| = |2x + 3 - 3| = |2x|\).

Step 3: Applying Given Condition

We are given that \(|x - 0| < \frac{\epsilon}{2}\). This implies that \(|x| < \frac{\epsilon}{2}\).

Step 4: Substituting \(|x|\) in \(|2x|\)

Since \(|x| < \frac{\epsilon}{2}\), substitute in \(|2x|\) to find \(|2x| = 2|x| < 2\times\frac{\epsilon}{2} = \epsilon\).

Step 5: Conclusion

We have shown that \(|f(x) - f(0)| = |2x| < \epsilon\) whenever \(|x| < \frac{\epsilon}{2}\), satisfying the requirement to prove that \(|f(x) - f(0)| < \epsilon\).

Key Concepts

Understanding Continuity

Continuity is a fundamental concept in calculus that ensures a function behaves predictably. A function is continuous at a point if the value of the function at that point equals the value it approaches as we get closer and closer from either side. This simply means there are no jumps or breaks in the graph of the function at that point.
When dealing with the epsilon-delta definition of continuity, it involves two main parameters: \( \epsilon \) and \( \delta \). The idea is to find a \( \delta \) for every \( \epsilon > 0 \), ensuring that if the distance between \( \text{x} \) and the point is less than \( \delta \), then the distance between \( \text{f(x)} \) and the function value at that point is less than \( \epsilon \). This means if you get close in \( \text{x} \), you will also get close in \( \text{f(x)} \).
In our exercise involving the function \( f(x) = 2x + 3 \), we demonstrate continuity by showing how the differences in \( f(x) \) and \( f(0) \) can be kept below any threshold \( \epsilon > 0 \) as long as the difference in \( x \) from zero is kept sufficiently small.

Linear Functions Characteristics

Linear functions are among the simplest and most fundamental types of functions. They are defined generally by an equation of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. The function \( f(x) = 2x + 3 \) is a perfect example of a linear function.
One of the key features of linear functions is that their graphs are straight lines. This property implies:

• They grow at a constant rate specified by the slope \( a \).
• There are no curves or changes in direction.
• They have exactly one slope, making calculations and predictions straightforward.

In the exercise, \( f(x) = 2x + 3 \) has slope \( 2 \) and y-intercept \( 3 \). This makes it easy to apply concepts like continuity, as linear functions tend to be all about consistent, predictable behavior. Understanding these properties helps in seeing why and how \( |f(x) - f(0)| \) can be made less than any \( \epsilon \).

Mastering Inequality Manipulation

Inequality manipulation is an essential skill in solving mathematical proofs and problems, especially those involving continuity, like epsilon-delta proofs. Here's how it plays out:

• Recognizing relationships: You begin by adjusting and transforming inequalities to reveal how one expression relates to another.
• Substitution: Often, you use given conditions to substitute into your inequalities, simplifying them further, as seen with substituting \( |x| < \frac{\epsilon}{2} \) into \( |2x| \).
• Logical reasoning: Ensuring each logical step follows clearly from the previous one is vital. This reasoning confirms that \( |f(x) - f(0)| < \epsilon \) holds.

In the provided exercise, moving from \( |2x| \) to \( 2|x| < \epsilon \) was crucial to concluding the proof. Proficiency in handling inequalities allows for such transformations, ultimately leading to the desired outcomes in proofs and mathematical arguments.