Question

Find the average rate of change of from 0 to \(\frac{\pi}{6}\). $$ f(x)=\tan (2 x) $$

Short answer

The average rate of change is \( \frac{6\sqrt{3}}{\pi} \).

Step-by-step solution

Understand the function

The given function is \(f(x) = \tan(2x)\). The goal is to find the average rate of change from \(x = 0\) to \(x = \frac{\pi}{6}\).

Recall the formula for average rate of change

The average rate of change of a function \(f(x)\) over the interval \([a, b]\) is given by \[\frac{f(b) - f(a)}{b - a}\]

Determine \(a\) and \(b\)

In this problem, \(a = 0\) and \(b = \frac{\pi}{6}\).

Evaluate \(f(a)\)

Calculate \(f(0)\): \[f(0) = \tan(2 \times 0) = \tan(0) = 0\]

Evaluate \(f(b)\)

Calculate \(f\left(\frac{\pi}{6}\right)\): \[f\left( \frac{\pi}{6} \right) = \tan\left( 2 \times \frac{\pi}{6} \right) = \tan\left( \frac{\pi}{3} \right) = \sqrt{3}\]

Apply the average rate of change formula

Substitute \(a, b, f(a)\) and \(f(b)\) into the average rate of change formula: \[\frac{f\left( \frac{\pi}{6} \right) - f(0)}{\frac{\pi}{6} - 0} = \frac{\sqrt{3} - 0}{\frac{\pi}{6}} = \frac{\sqrt{3}}{\frac{\pi}{6}} = \frac{6\sqrt{3}}{\pi}\]

Key concepts

Tangent Function

The tangent function, denoted as \(\tan(x)\), is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine functions: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). The tangent function is periodic with a period of \( \pi \), which means it repeats its values every \( \pi \) units along the x-axis. This function has vertical asymptotes where \( \cos(x) = 0\), causing the function to approach infinity.

For transformations, such as \( \tan(2x) \) in this exercise, the multiplication inside the function (by 2) affects the period of the tangent function. Instead of repeating every \( \pi \) units, \( \tan(2x) \) repeats every \( \frac{\pi}{2} \).

Understanding the behavior of \( \tan(2x) \) is crucial when evaluating it over specific intervals, as the function can have wide variations within short spans.

Interval Evaluation

Interval evaluation involves determining the value of a function at specific points within a given range. In this exercise, the interval is from \( 0 \) to \( \frac{\pi}{6} \).

First, identify your interval endpoints: \( a = 0 \) and \( b = \frac{\pi}{6} \). Evaluating the function \( f(x) = \tan(2x) \) at these endpoints provides the necessary values for calculating the average rate of change.

For \( x = 0 \):\( f(0) = \tan(2 \times 0) = \tan(0) = 0 \).
For \( x = \frac{\pi}{6} \):\( f(\frac{\pi}{6}) = \tan(2 \times \frac{\pi}{6}) = \tan(\frac{\pi}{3}) = \sqrt{3} \). Using these values, we apply the formula for the average rate of change to find our result.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in the context of periodic phenomena. These functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), among others.

The function \( \tan(x) \) can be expressed in terms of sine and cosine: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Each trigonometric function has specific properties:

• Periodicity: Sine and cosine have a period of \( 2\pi \), while tangent has a period of \( \pi \).
• Asymptotes: Tangent has vertical asymptotes wherever cosine equals zero.
• Amplitudes and Ranges: Sine and cosine range from -1 to 1, while tangent ranges from negative infinity to positive infinity.

In solving problems with these functions, it is important to understand their behavior, particularly their periodicity and undefined points. For instance, within the interval from 0 to \( \frac{\pi}{6} \), knowing where \( \tan(2x) \) is defined and its values at those points is crucial for accurate calculations.