Problem 23 - Calculus for Biology and Medicine

Question

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\tan (2 \pi x) $$

Step-by-Step Solution

Answer

\( f(x) \) is continuous for all \( x \in \mathbb{R} \) except \( x = \frac{1}{4} + \frac{n}{2} \), where \( n \in \mathbb{Z} \).

Step 1: Understanding Function Definition

The function given here is the tangent function, \( f(x) = \tan(2\pi x) \). A tangent function, \( \tan(\theta) \), is undefined when \( \theta = \frac{\pi}{2} + n\pi \) for any integer \( n \), since tangent represents the ratio of \( \sin(\theta) \) and \( \cos(\theta) \) and cosine is zero at these points.

Step 2: Determine Points of Discontinuity

To find when \( f(x) \) is both defined and continuous, we need to ensure that \( 2\pi x eq \frac{\pi}{2} + n\pi \). This translates to \( x eq \frac{1}{4} + \frac{n}{2} \) for \( n \in \mathbb{Z} \).

Step 3: General Solution for x

The values of \( x \) for which \( f(x) \) is both defined and continuous are all real numbers except \( x = \frac{1}{4} + \frac{n}{2} \), where \( n \) is any integer.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), plays a pivotal role in trigonometry and calculus. It represents the ratio of the sine and cosine functions, specifically \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This function is periodic with a period of \( \pi \), meaning it repeats its values over intervals of \( \pi \).A key characteristic to remember about the tangent function is that it is undefined whenever the cosine function is zero, as division by zero is undefined. This occurs at angles \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is any integer. These points are where vertical asymptotes occur in the graph of the tangent function. If you imagine the unit circle, these are the points where the line that defines the tangent function becomes parallel to the vertical axis, shooting up to infinity.In the context of the function \( f(x) = \tan(2\pi x) \), the value of \( \theta \) becomes \( 2\pi x \). Thus, the tangent function behaves in its usual manner, but its undefined points are determined by these specific transformations.

Discontinuity Points

Discontinuity points in functions are specific values where a function is not continuous. For the tangent function, discontinuity occurs at points where the function value shoots to infinity, corresponding to the undefined values we discussed earlier.To find discontinuity in \( f(x) = \tan(2\pi x) \), consider where the tangent function is undefined: where \( \cos(2\pi x) = 0 \). These are the points \( 2\pi x = \frac{\pi}{2} + n\pi \). Simplifying this equation gives \( x = \frac{1}{4} + \frac{n}{2} \).Thus, in terms of real numbers, the function will be discontinuous at these \( x \) values:

• \( x = \frac{1}{4} \)
• \( x = \frac{1}{4} + \frac{1}{2} \)
• \( x = \frac{1}{4} + 1 \)
• and so on, each corresponding to different integer values of \( n \).

Understanding these discontinuity points helps determine when this function can be both defined and continuous.

Real Numbers in Calculus

In calculus, the domain of a function is often comprised of real numbers, denoted by \( \mathbb{R} \). Real numbers include all rational and irrational numbers, meaning they cover every point on the continuous number line without gaps.When evaluating functions such as \( f(x) = \tan(2\pi x) \), it’s crucial to identify which real numbers form the domain — the set of values where the function is both defined and continuous. In this exercise, the function encompasses all real numbers except those specifically where it becomes undefined.Because of the periodicity and characteristics of the tangent function, certain real numbers will lead to undefined values (asymptotes) at points \( x = \frac{1}{4} + \frac{n}{2} \), leaving those out of the domain. Otherwise, \( f(x) \) encompasses the vast space of the real number continuum outside these discontinuities, maintaining its continuity wherever it is defined.Recognizing the interplay between real numbers, continuity, and the calculus of functions helps avoid pitfalls and ensures a sound understanding of when functions behave predictably across their domain.