Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the average rate of change of \(f(x)=\cos ^{-1} x\) from \(x=-\frac{1}{2}\) to \(x=\frac{1}{2}\)

Short answer

The average rate of change is \(-\frac{\pi}{3}\).

Step-by-step solution

Identify the formula for the average rate of change

The average rate of change of a function between two points can be determined using the formula: \[\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}\]where \(a\) and \(b\) are the two points.

Determine values of \(f(a)\) and \(f(b)\)

For our function \(f(x) = \cos^{-1} x\), we need to find the values at \(x = -\frac{1}{2}\) and \(x = \frac{1}{2}\):\[f\left(-\frac{1}{2}\right) = \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3}\]\[f\left(\frac{1}{2}\right) = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}\]

Plug the values into the formula

Now we apply the values we found for \(f(a)\) and \(f(b)\) into the average rate of change formula:\[\frac{f\left(\frac{1}{2}\right) - f\left(-\frac{1}{2}\right)}{\frac{1}{2} - (-\frac{1}{2})} = \frac{\frac{\pi}{3} - \frac{2\pi}{3}}{1} = \frac{-\frac{\pi}{3}}{1} = -\frac{\pi}{3}\]

Key concepts

inverse trigonometric functions

Inverse trigonometric functions are functions that 'undo' the trigonometric functions. For example, if \(\tan(\theta) = x\), then \(\tan^{-1}(x) = \theta\). These functions help us find angles when given a trigonometric ratio. There are six inverse trigonometric functions corresponding to the six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant).

However, they have certain limitations in their ranges to ensure they are functions (pass the vertical line test). These restrictions are important because they allow us to find unique outputs. For instance, the range of \[\sin^{-1}(x)\] \is \(-\frac{\pi}{2}\leq y \leq \frac{\pi}{2}\)\.

cosine inverse

The cosine inverse function, denoted as \(\cos^{-1}(x)\), specifically deals with finding the angle whose cosine is a given number x. It is also known as arccosine. The standard range of \[\cos^{-1}(x)\] \is \[0 \,\text{to}\, \pi\]\. This means \(\cos^{-1}(x)\) will return an angle within this interval.

In our specific exercise, we needed to calculate \(\cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3}\) and \(\cos^{-1}(\frac{1}{2}) = \frac{\pi}{3}\). Recognizing these values often comes from familiarity with the unit circle and knowing the specific angles where cosine takes on these values.
When faced with calculating these values, remember that \(\cos^{-1}(\cdot)\) essentially means finding the angle whose cosine equals the input value.

step-by-step solution

Let’s break down the problem step-by-step:

1. **Identify the average rate of change formula**: We use the formula for the average rate of change, \[\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}\]\. In this formula, \(a\) and \(b\) are the interval boundaries.

2. **Determine \(f(a)\) and \(f(b)\)**:

• First, for \(x = -\frac{1}{2}\), the value is \(\cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3}\).
• Second, for \(x = \frac{1}{2}\), the value is \(\cos^{-1}(\frac{1}{2}) = \frac{\pi}{3}\).

3. **Apply these values into the formula**:
\[\text{Average rate of change} = \frac{f\left(\frac{1}{2}\right) - f\left(-\frac{1}{2}\right)}{\frac{1}{2} - (-\frac{1}{2})} = \frac{\frac{\pi}{3} - \frac{2\pi}{3}}{1} = \frac{-\frac{\pi}{3}}{1} = -\frac{\pi}{3}\]\.

This step-by-step process, closely following the problem, ensures we capture all the details needed for full comprehension. Once you understand each step, you can apply the same logic to other similar problems.