Problem 41 - Calculus

Question

give an example of: A function which is defined for all \(x\) and continuous everywhere except at \(x=15\)

Step-by-Step Solution

Answer

A piecewise function such as \( f(x) = \begin{cases} \sin(x), & \text{if } x \neq 15 \\ 0, & \text{if } x = 15 \end{cases} \)

Step 1: Identify Requirements

We need to create or choose a function that is defined for all values of \( x \) but has a discontinuity at \( x = 15 \).

Step 2: Identify the calculus concept

Determine which concept is being tested.

Step 3: Set up the problem

Write the problem in mathematical notation.

Step 4: Execute the computation

Apply calculus techniques step by step.

Step 5: State the result

Simplify and state the final answer.

Step 6: Conclude with the answer

\end{cases} \)

Key Concepts

Continuity

In mathematics, continuity is a property of a function where the function values change smoothly without any sudden jumps or interruptions. Formally, for a function \( f(x) \) to be continuous at a point \( x = a \), the following three conditions must be met:

• The function \( f(x) \) must be defined at \( x = a \).
• The limit of \( f(x) \) as \( x \) approaches \( a \) must exist.
• The value of the function at \( x = a \) must equal the limit as \( x \) approaches \( a \), i.e., \( f(a) = \lim_{x \to a} f(x) \).

When any of these conditions fail, we say the function is discontinuous at that point. In the original exercise, the function has a discontinuity at \( x = 15 \), meaning at this specific point, one or more of the above conditions aren't satisfied. Usually, continuity is considered across the entire domain unless noted otherwise.

Limit

In calculus, a limit helps us understand the behavior of a function as the input approaches a certain value. The notation \( \lim_{x \to a} f(x) = L \) means that as \( x \) gets closer and closer to \( a \), \( f(x) \) gets arbitrarily close to \( L \).
The concept of a limit is crucial in discussing the continuity of a function. If the limit of \( f(x) \) exists at a particular point, then we can begin to talk about whether the function is continuous at that point. If we cannot assign a definite value to \( f(x) \) as \( x \) nears \( a \), then the function might be discontinuous there.
In practical terms, if you are dealing with a function and want to check continuity at \( x = 15 \), you would evaluate \( \lim_{x \to 15} f(x) \). If this limit does not exist or does not match \( f(15) \), it indicates discontinuity at that point.

Mathematical Function

A mathematical function is an expression or rule that assigns each input exactly one output. Functions are usually denoted as \( f(x) \), where \( x \) represents the input and \( f(x) \) the output.
Functions can take many forms: linear, quadratic, exponential, logarithmic, trigonometric, and more. Depending on their structure, these functions can exhibit different properties such as continuity, discontinuity, limits, and behavior over their domain.

• For example, the function \( f(x) = \frac{x-15}{x-15} \) is defined for all \( x eq 15 \) and continuous everywhere except at \( x = 15 \). The discontinuity takes the form of a removable discontinuity because the factors \( x - 15 \) cancel each other out, yet a value for \( x = 15 \) is undefined.
• Understanding the properties of a function helps in predicting its behavior and resolving the requirements such as described in the original exercise.

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes both rational numbers (like 7 and -2) and irrational numbers (like \( \pi \) and \( \sqrt{2} \)).
The domain of a typical function, unless otherwise specified, will involve real numbers.

• Some functions may not be defined at certain real numbers, leading to potential gaps or jumps - indicative of discontinuity.
• In our context, the function is defined for all real numbers except at \( x = 15 \), where the discontinuity is specifically introduced.

Understanding how functions interact with real numbers is fundamental in studying calculus and analyzing functions for properties like continuity. It helps us comprehend both the domain and range within real-world scenarios and theoretical problems.