In Exercises 21-36, find the absolute maximum and minimum...

In Exercises $21-36,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$ f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4} $$

In Exercises $21-36,$ find the absolute maximum and minimum values
of each function on the given interval. Then graph the function. Identify the points
on the graph where the absolute extrema occur, and
include their coordinates.
$$
f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}
$$

Key Concepts

Behavior of the Tangent Function

The tangent function has unique properties, including periodicity and vertical asymptotes, which must be taken into account when analyzing its behavior on a specific interval. Understanding these characteristics helps in determining where the function is continuous and differentiable, as well as anticipating its growth or decay near the endpoints of the interval.

Extreme Value Theorem

The Extreme Value Theorem is a fundamental concept in calculus that guarantees a continuous function defined on a closed interval will attain both an absolute maximum and an absolute minimum within that interval. This theorem provides the foundation for finding global extrema by ensuring that evaluation at critical points and endpoints covers all possibilities for extreme values.

Critical Points

Critical points are points in the interior of the domain of a function where the derivative is zero or undefined. These points are important in the analysis of extrema because they are potential candidates for local maxima or minima, which might also turn out to be absolute extrema on a closed interval.

Endpoint Evaluation

When finding absolute maximum and minimum values on a closed interval, it is necessary to evaluate the function at the endpoints of the interval as well as at its critical points. The absolute extrema are determined by comparing the function values at these points, which is crucial for a complete analysis on a closed interval.

Graphical Analysis

Graphical analysis involves plotting the function to visually assess its behavior over the interval. This can provide insight into where the graph achieves its highest and lowest values, aiding in identifying and verifying the locations of absolute extrema. It also helps in understanding the overall shape and trends of the function.

Transcript

00:01 Okay, here we have f of theta.

00:04 It's just tan theta defined on the interval negative pi over 3 to pi over 4.

00:18 And recall that tan will be continuous because this interval lies between one of its periods, negative pi over 2 to pi over 2.

00:28 Okay, so we have an absolute max in men.

00:31 And so if we find the critical points, 2 squared theta, the second square theta is never equal to zero.

00:45 It's also never, well, it's going to be undefined when cosine is zero, but that's exactly when tan is undefined.

00:54 So in this interval, the derivative is actually going to be not only just not zero, it's going to be strictly positive and finite.

01:05 So it's just going to, this is saying because the derivative is positive here, tangent is increasing.

01:12 But the point is we have no critical.

01:13 Points and again i mean secan squared is 1 over cosine squared the only way a a fraction can be 0 is if the numerator is 0 okay so we just need to test the endpoints so tangent of negative pi over 3 that is going to be negative 1 over root 3 believe and then tangent of pi over 4 is 1 so this is the absolute men at negative pi over 3 is the opposite max.

02:02 Okay, looks good.

02:16 Actually, no, this isn't correct.

02:20 Double check my trigly.

02:23 So yeah, tangent of negative pi over 3 is cosine of negative pi over 3 over sorry, sign of negative pi over 3 over cosine negative pi over 3.

02:35 And so that's negative root 3 over 2 over 1 half, which is root 3...

Please give Ace some feedback