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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Question
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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.
---
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd018cc6-2cd8-4cb3-91a6-c6c76d6754a6%2Fc972aba9-b302-44e9-af42-24175ae32503%2Fphl1c9e.jpeg&w=3840&q=75)
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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