7T Find the average rate of change of f from 0 to 96. f(x)%3Dtan x

Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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**Title: Calculating the Average Rate of Change for the Function \( f(x) = \tan x \)**

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**Problem Statement:**

Find the average rate of change of \( f \) from 0 to \( \frac{7\pi}{6} \).

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**Given Function:**
\[ f(x) = \tan x \]

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**Solution:**

The average rate of change is given by the formula:

\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]

For the given problem:
- \( a = 0 \)
- \( b = \frac{7\pi}{6} \)

So, we need to find \( \frac{\tan \left(\frac{7\pi}{6}\right) - \tan(0)}{\frac{7\pi}{6} - 0} \).

1. Calculate \( f\left(\frac{7\pi}{6}\right) = \tan\left(\frac{7\pi}{6}\right) \).
2. Since \( \tan(0) = 0 \).

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By substituting the values, we find:

\[ \text{Average Rate of Change} = \frac{\tan\left(\frac{7\pi}{6}\right) - 0}{\frac{7\pi}{6}} \]

Given the result:

\[ \text{The average rate of change is} \quad \frac{2\sqrt{3}}{7\pi} \]

**Note:** Simplify your answer, including any radicals. Type an exact answer, using \( \pi \) as needed.

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This calculation highlights the process of determining the average rate of change for a trigonometric function over a specified interval. Understanding and simplifying expressions involving trigonometric values and fractions is crucial in this exercise.
Transcribed Image Text:--- **Title: Calculating the Average Rate of Change for the Function \( f(x) = \tan x \)** --- **Problem Statement:** Find the average rate of change of \( f \) from 0 to \( \frac{7\pi}{6} \). --- **Given Function:** \[ f(x) = \tan x \] --- **Solution:** The average rate of change is given by the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] For the given problem: - \( a = 0 \) - \( b = \frac{7\pi}{6} \) So, we need to find \( \frac{\tan \left(\frac{7\pi}{6}\right) - \tan(0)}{\frac{7\pi}{6} - 0} \). 1. Calculate \( f\left(\frac{7\pi}{6}\right) = \tan\left(\frac{7\pi}{6}\right) \). 2. Since \( \tan(0) = 0 \). --- By substituting the values, we find: \[ \text{Average Rate of Change} = \frac{\tan\left(\frac{7\pi}{6}\right) - 0}{\frac{7\pi}{6}} \] Given the result: \[ \text{The average rate of change is} \quad \frac{2\sqrt{3}}{7\pi} \] **Note:** Simplify your answer, including any radicals. Type an exact answer, using \( \pi \) as needed. --- This calculation highlights the process of determining the average rate of change for a trigonometric function over a specified interval. Understanding and simplifying expressions involving trigonometric values and fractions is crucial in this exercise.
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