Question

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

Short answer

\( \frac{6}{\pi} \)

Step-by-step solution

Define the formula for average rate of change

The average rate of change of a function between two points is given by the formula:\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]In this case, we need to find it from 0 to \( \frac{\pi}{6} \). Therefore, \(a = 0\) and \(b = \frac{\pi}{6}\).

Evaluate the function at the endpoints

Substitute \(a\) and \(b\) into the function \(f(x) = \sec(2x)\):\[f(0) = \sec(2 \cdot 0) = \sec(0) = 1 \]\[f\left( \frac{\pi}{6} \right) = \sec \left( 2 \cdot \frac{\pi}{6} \right) = \sec \left( \frac{\pi}{3} \right) = 2 \]

Substitute the values back into the average rate of change formula

Using the values from Step 2: \[ \text{Average Rate of Change} = \frac{f\left( \frac{\pi}{6} \right ) - f(0)}{ \frac{ \pi }{ 6 } - 0 } \]We get: \[ \text{Average Rate of Change} = \frac{2 - 1}{ \frac{\pi}{6} } = \frac{1}{ \frac{\pi}{6} } = \frac{6}{\pi} \]

Key concepts

Secant Function

In trigonometry, the secant function, denoted as \(\text{sec}\), is the reciprocal of the cosine function. The secant function is defined by the formula \[ \text{sec}(x) = \frac{1}{\text{cos}(x)} \]. This means that for any angle \(x\), the secant of \(x\) is the ratio of 1 to the cosine of \(x\). It’s essential to understand this relationship because it frequently appears in various trigonometric calculations, especially when dealing with angles and their transformations.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles in triangles. The primary trigonometric functions are sine, cosine, and tangent, which are divided into ratios of the sides of a right-angled triangle. These functions extend to the unit circle, enabling the analysis of angles beyond just those within triangles. In the context of this exercise, knowing how to navigate secant and other related functions (cosine, sine) will help simplify and solve the problem.

Rate of Change Formula

The average rate of change formula is essentially a measure of how a function responds to changes over specified intervals. It is given by \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]. Here, \(a\) and \(b\) represent two different points along the function \(f(x)\). To illustrate, if we evaluate this formula over an interval (0, \(\frac{\text{π}}{6}\)) using the function \(f(x) = \text{sec}(2x)\), we observe how the function changes from point \(a\) to point \(b\).