Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Verify that \(\tan x\) has points of inflection at \(x=\pi k, k\) an integer, by showing that the sign of its second derivative changes at these points.

Step-by-Step Solution

Answer

The function \(\tan x\) has points of inflection at \(x = \pi k\), where \(k\) is an integer, as the sign of its second derivative changes at these points due to the periodicity of the tangent and secant functions.

Step 1: Find the First Derivation

Start with the function \(\tan x\). Its derivative can be found using standard trigonometric derivation rules: \\ The derivative of \(\tan x\) is \(\frac{d}{dx}\tan x = \sec^2 x\).

Step 2: Find the Second Derivation

Next, differentiate the first derivative to find the second derivative: \\ The derivative of \(\sec^2 x\) is \(\frac{d}{dx} \sec^2 x = 2 \sec x \sec x \tan x = 2 \tan x \sec^2 x\).

Step 3: Evaluate the Second Derivative at the Given Points

Substitute \(x = \pi k\), where \(k\) is an integer, into the second derivative to find its values at the given points. \\ Substituting into the second derivative gives, \(2 \tan (\pi k) \sec^2 (\pi k) = 2 \cdot 0 \cdot \infty = 0\).

Step 4: Verify the Change in Sign of the Second Derivative

Inspect the values just before and after \(x = \pi k\) in the second derivative for sign change. \\ For \(x = \pi k + \varepsilon\) and \(x = \pi k - \varepsilon\), where \(\varepsilon\) is a small positive number, the signs need to be different for inflection points. \\ Due to the periodicity of the tangent and secant functions, the values will oscillate, causing the sign of the second derivative to change.

Key Concepts

Second Derivative

The second derivative of a function gives us valuable information about the concavity of the function's graph. It indicates how the slope of the tangent line to the curve is changing. This is crucial in identifying points of inflection, which are points where the graph changes concavity.

For the function \[ an x \], its second derivative is calculated by differentiating the first derivative, which is \[ rac{d}{dx} an x = ext{sec}^2 x \].
When we compute the second derivative, \[ rac{d}{dx} ext{sec}^2 x = 2 an x ext{sec}^2 x \], it helps us understand how the concavity of the tangent function changes.

By evaluating this at specific points, such as \[ x = \pi k \],where \[ k \] is an integer, we can determine if the sign changes, indicating a point of inflection. The sign change of the second derivative is the mathematical clue we need to verify the presence of an inflection point.

Trigonometric Functions

Trigonometric functions, like \[ an x \], perform periodic oscillations which are highly relevant in the study of points of inflection. Understanding their derivatives is crucial for analyzing their behavior and graph characteristics.

The tangent function is defined as \[ an x = rac{ ext{sin} x}{ ext{cos} x} \]. Its derivative, \[ ext{sec}^2 x \], showcases how rapidly the tangent line's slope changes as the input value \( x \) increases or decreases.
Using trigonometric identities and derivatives, we explore deeper properties of these functions, such as how they lead to inflection points naturally through their oscillating nature. The derivative \[ ext{sec}^2 x \] looks simple, but when plugged into another derivative, it aids in solving problems involving curvature and slope changes in \[ an x \].

Periodicity of Functions

Periodicity is a core feature of trigonometric functions, such as \( \tan x \),and it describes the repetition of the function's value in regular intervals.

For \( \tan x \),the periodicity is \( \pi \),meaning every \( \pi \) units, the function starts repeating its output. This is crucial when analyzing the second derivative for changes in sign, as it indicates that certain behaviors will recur consistently.
When examining inflection points, the periodicity implies that similar changes in concavity will also repeat periodically. Thus, finding points of inflection at \( x = \pi k \), where \( k \) is an integer, aligns perfectly with the periodic nature of the tangent function.
Using the concept of periodicity helps us simplify complex analyses by focusing on one period and concluding for all repetitions. This property assists mathematicians and students in visualizing and solving trigonometric problems efficiently.