/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Find the average rate of change ... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 44

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

Short Answer

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

Step by step solution

01

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 \(b = \frac{\pi}{6}\).
02

Evaluate the function at the endpoints

Substitute \(a\) and \(b\) into the function \(f(x) = \sec(2x)\):\[f(0) = \sec(2 \cdot 0) = \sec(0) = 1 \]\[f\left( \frac{\pi}{6} \right) = \sec \left( 2 \cdot \frac{\pi}{6} \right) = \sec \left( \frac{\pi}{3} \right) = 2 \]
03

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: \[ \text{Average Rate of Change} = \frac{2 - 1}{ \frac{\pi}{6} } = \frac{1}{ \frac{\pi}{6} } = \frac{6}{\pi} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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, \(\frac{\text{蟺}}{6}\)) using the function \(f(x) = \text{sec}(2x)\), we observe how the function changes from point \(a\) to point \(b\).

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Most popular questions from this chapter

True or False The graph of the sine function has infinitely many \(x\) -intercepts.

A point on the terminal side of an angle \(\theta\) in standard position is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ \left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right) $$

Determine the amplitude and period of each function without graphing. $$ y=-\frac{1}{7} \cos \left(\frac{7}{2} x\right) $$

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \tan 390^{\circ} $$

Given \(\sin 30^{\circ}=\frac{1}{2},\) use trigonometric identities to find th exact value of (a) \(\cos 60^{\circ}\) (b) \(\cos ^{2} 30^{\circ}\) (c) \(\csc \frac{\pi}{6}\) (d) \(\sec \frac{\pi}{3}\)
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