Question

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

Short answer

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

Step-by-step solution

Understand the Formula for Average Rate of Change

The average rate of change of a function between two points is given by the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] where \(a\) and \(b\) are the endpoints of the interval.

Identify the Interval and Function Values

In this problem, the interval is from 0 to \( \frac{\theta}{6} \) and the function is \( f(x) = \text{sec} \theta \). So, \(a = 0\) and \(b = \frac{\theta}{6}\). Next, we need to find the values of the function at these points: \( f(0) \) and \( f\big(\frac{\theta}{6}\big) \).

Evaluate the Function at the Endpoints

Evaluate the function \( f(x) = \text{sec} \theta \) at the endpoints: \[ f(0) = \text{sec}(0) = 1 \ \text{sec}( \frac{\theta}{6}) = \frac{1}{\text{cos} ( \frac{\theta}{6})} \] Now compute: \[ \text{cos}\big( \frac{\theta}{6}\big) = \text{cos}\big( \frac{30^{\textdegree}}{6}\big) = \frac{\text{sqrt}(3)}{2} \]\[ f\big(\frac{\theta}{6}\big) = \text{sec}\big( \frac{\theta}{6}\big) = \frac{1}{ \frac{\text{sqrt}(3)}{2}} = \frac{2}{\text{sqrt}(3)} \]

Substitute the Values into the Formula

Now substitute these values into the average rate of change formula:\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} = \frac{\frac{2}{\text{sqrt}(3)} - 1}{\frac{\theta}{6} - 0} \]

Simplify the Expression

Simplify the numerator and denominator:\[ \frac{\frac{2}{\text{sqrt}(3)} - 1}{\frac{\theta}{6}} = \frac{2\text{sqrt}(3) - \text{sqrt}(3)}{\frac{\theta}{6}\text{sqrt}(3)} = \frac{\text{sqrt}(3)}{\frac{\theta}{6}} \]

Final Simplification

Now, perform the final simplification: \[ \frac{\text{sqrt}(3)}{\frac{\theta}{6}} = 6\text{sqrt}(3) \ \text{Therefore, the average rate of change is } 6\text{sqrt}(3) \]

Key concepts

Secant Function

Let's dive into the secant function, also written as \(\text{sec}(x)\). This function is closely related to the cosine function, as it is defined as the reciprocal of the cosine function: \[\text{sec}(x) = \frac{1}{\text{cos}(x)} \]In simpler terms, wherever cosine is zero, the secant function will not be defined because dividing by zero is undefined. For example, \(\text{sec}(0) = 1\) because \(\text{cos}(0) = 1\). Understanding the secant function is crucial, especially in trigonometry, as it helps in evaluating angles and solving various mathematical problems involving triangles.
It's important to remember that the secant function can have large values when cosine is close to zero, leading to steep changes in the function's behavior.

Trigonometric Evaluation

Evaluating trigonometric functions at specific points is a fundamental skill. In this exercise, we need to find the secant of \(\frac{\pi}{6}\). First, we simplify the angle: \(\text{cos}\left( \frac{\pi}{6} \right) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2}\). Using this, we compute the secant as: \[\text{sec}\left( \frac{\pi}{6} \right) = \frac{1}{\text{cos}\left( \frac{\pi}{6} \right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \] This simplification is necessary for finding the values needed in the average rate of change calculation. Remember that simplifying expressions and knowing trigonometric values for standard angles allow for solving more complex problems efficiently.

Interval Notation

Interval notation is a method used to write the set of values that lie between two endpoints. In our problem, the interval is from \(0\) to \(\frac{\pi}{6}\), represented as \[ [0, \frac{\pi}{6}] \] By using interval notation, it's easy to identify our start (\(0\)) and end (\(\frac{\pi}{6}\)) points. This clarity is essential for applying the average rate of change formula because we need to substitute these values for \(a\) and \(b\) respectively. Using intervals helps eliminate any confusion about which points to use in evaluations and ensures accuracy in mathematical problems.

Simplification of Expressions

The final, but crucial, part of this exercise involves simplifying expressions. Upon substituting the function values and interval endpoints into the average rate of change formula, we get: \[ \frac{\frac{2}{\sqrt{3}} - 1}{\frac{\pi}{6} - 0} \] Simplifying the numerator: \[ \frac{2}{\sqrt{3}} - 1 = \frac{2 - \sqrt{3}}{\sqrt{3}} \] And further simplification gives us the result: \[ \frac{ \frac{\sqrt{3}}{ \frac{\pi}{6} } } = 6 \sqrt{3} \] It is important to manipulate the expressions correctly to avoid errors. Breaking down each step ensures that the calculations remain clear and that the answers are accurate. Practice in simplification builds confidence and precision in solving mathematics problems.