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Chapter 4: Problem 36

25-40 Find the critical numbers of the function. $$g(\theta)=4 \theta-\tan \theta$$

Short Answer

Expert verified
Critical numbers are the angles where \(\cos\theta = \pm \frac{1}{2}\): \(\theta = \pm\frac{\pi}{3} + 2k\pi\) and \(\theta = \pm\frac{2\pi}{3} + 2k\pi\) for integers \(k\).

Step by step solution

01

Find the Derivative

To find the critical numbers, we first need the derivative of the function. Given the function \(g(\theta) = 4\theta - \tan\theta\), the derivative is found by differentiating each term. The derivative of \(4\theta\) is \(4\), and the derivative of \(-\tan\theta\) is \(-\sec^2\theta\). Thus, the derivative is \(g'(\theta) = 4 - \sec^2\theta\).
02

Set the Derivative to Zero

Critical numbers occur where the derivative is either zero or undefined. First, set \(g'(\theta) = 4 - \sec^2\theta = 0\). This simplifies to \(4 = \sec^2\theta\) or \(\sec^2\theta = 4\).
03

Solve for \(\theta\)

From \(\sec^2\theta = 4\), note that \(\sec\theta = \pm 2\). Since \(\sec\theta = \frac{1}{\cos\theta}\), it follows that \(\cos\theta = \pm \frac{1}{2}\). The angles satisfying \(\cos\theta = \frac{1}{2}\) are \(\theta = \frac{\pi}{3} + 2k\pi\) and \(\theta = -\frac{\pi}{3} + 2k\pi\), where \(k\) is any integer. For \(\cos\theta = -\frac{1}{2}\), the angles are \(\theta = \frac{2\pi}{3} + 2k\pi\) and \(\theta = -\frac{2\pi}{3} + 2k\pi\).
04

Check where \(g'(\theta)\) is undefined

\(g'(\theta)\) is undefined wherever \(\sec^2\theta = \infty\), which happens when \(\cos\theta = 0\). This occurs at odd multiples of \(\frac{\pi}{2}\), i.e., \(\theta = (2k + 1)\frac{\pi}{2}\) for integer \(k\). However, these values do not lead to finite critical numbers, as the function is not defined there.

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Key Concepts

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

Derivative
A derivative tells us how a function changes as its input changes. It's like the slope of a curve at a specific point, showing us the steepness or angle. To find the critical numbers of a function like \(g(\theta) = 4\theta - \tan\theta\), we need its derivative.
  • The derivative of \(4\theta\) is \(4\).
  • The derivative of \(-\tan\theta\) is \(-\sec^2\theta\).
Thus, the derivative of \(g(\theta)\) is\[g'(\theta) = 4 - \sec^2\theta\] Finding the derivative helps us locate where the function has critical points by analyzing where it equals zero or where it doesn't exist.
Trigonometric Functions
Trigonometric functions like tangent and secant are periodic and have interesting properties. In our exercise, we encounter both \(\tan\theta\) and \(\sec\theta\).
  • \(\tan\theta\) is the ratio of \(\sin\theta\) to \(\cos\theta\).
  • \(\sec^2\theta\) is the square of \(\frac{1}{\cos\theta}\).
These functions repeat their values at regular intervals, known as periods. For example, the period of \(\tan\theta\) is \(\pi\), meaning it repeats every \(\pi\) units.Trigonometric functions are key to solving trigonometric equations, like \(4 = \sec^2\theta\), to find critical points.
Undefined Derivative
A derivative can become undefined if the slope of the curve becomes vertical or if the function itself isn't defined for a certain value. In our case, the derivative \[g'(\theta) = 4 - \sec^2\theta\] becomes undefined when \(\sec^2\theta\) approaches infinity. This occurs if
  • \(\cos\theta = 0\)
  • This happens at odd multiples of \(\frac{\pi}{2}\)
where \(\theta = (2k + 1)\frac{\pi}{2}\) for integer \(k\). Nonetheless, these points don't give us usable critical numbers since the original function features undefined values at these points.
Critical Points
Critical points are where a function's derivative is either zero or doesn't exist. Finding these points helps us understand the behavior of the function, like where it is increasing or decreasing.For \(g(\theta)\), we need
  • Solutions to \(g'(\theta) = 0\)
  • Where the derivative is undefined
Setting \(g'(\theta) = 4 - \sec^2\theta = 0\), we find\[\sec^2\theta = 4 \]Solving gives us multiple angles at which this equation holds, explained in terms of cosine: \(\cos\theta = \pm \frac{1}{2}\). These critical points suggest where attention is required when sketching or analyzing the function's graph.

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