Problem 19 - Calculus with Concepts in Calculus

Question

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. Then sketch the graph of the function. $$ f(x)=\sec x $$

Step-by-Step Solution

Answer

Concave upward on \((k\pi, k\pi + \frac{\pi}{2})\), downward on \((k\pi - \frac{\pi}{2}, k\pi)\).

Step 1: Find the Second Derivative

To determine concavity, we begin by finding the second derivative of the function. First, we calculate the first derivative of \(f(x) = \sec x\). Using the derivative rule for \(\sec x\), we have:\[\frac{d}{dx}(\sec x) = \sec x \tan x.\]Now, find the derivative of the first derivative to get the second derivative. Applying the product rule to \(\sec x \tan x\), we have:\[f''(x) = \frac{d}{dx}(\sec x \tan x) = \sec x \tan^2 x + \sec^3 x.\]

Step 2: Analyze the Second Derivative

Determine where \(f''(x)\) is positive or negative. We solve the inequality \(\sec x (\tan^2 x + \sec^2 x) > 0\) to find where the function is concave upward, and \(\sec x (\tan^2 x + \sec^2 x) < 0\) to find where it is concave downward. Since both \(\sec x\) and \(\tan x\) change signs at odd multiples of \(\frac{\pi}{2}\), consider the intervals between these points.

Step 3: Determine Concave Intervals

Next, evaluate the sign of \(f''(x)\) in each interval:- For \(x \in (0, \frac{\pi}{2})\), \(f''(x) > 0\), so the graph is concave upward.- For \(x \in (-\frac{\pi}{2}, 0)\), \(f''(x) < 0\), so the graph is concave downward.- The pattern repeats every \(\pi\) phase shift due to the periodicity of \(\sec x\).Thus, the graph is concave upward on intervals of the form \((k\pi, k\pi + \frac{\pi}{2})\), where \(k\) is an integer, and concave downward on \((k\pi - \frac{\pi}{2}, k\pi)\).

Step 4: Sketch the Graph

Use the intervals of concavity to sketch the graph of \(f(x) = \sec x\):- Note that \(\sec x\) has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer.- Between the asymptotes, sketch upward or downward parabola-like segments according to the intervals of concavity.

Key Concepts

Second derivative test

The second derivative test is a crucial tool in calculus for understanding how the shape of a graph behaves. It helps us determine the concavity of a function: whether a curve is concave upward or downward. Here's how you can use this test:

• First, check the first derivative: This tells us the slope or the rate of change of the function at different points.
• Then, find the second derivative: This derivative indicates how this rate of change itself is changing.

When the second derivative at a point is positive, the graph is concave upward, resembling a cup. Conversely, when it's negative, the graph is concave downward, similar to a cap.

For the function \(f(x) = \sec x\), its second derivative \(f''(x) = \sec x \tan^2 x + \sec^3 x\) reveals its concavity. Positive values of \(f''(x)\) indicate intervals where the graph curves upwards; these intervals are \( (k\pi, k\pi + \frac{\pi}{2}) \).

Negative values indicate downward concavity; these are \( (k\pi - \frac{\pi}{2}, k\pi) \).Remember that the graph shifts due to the periodic nature of the secant function.

Graph sketching

Sketching the graph of a function is like building a visual story from mathematical data. Here's a handy guide to doing this effectively:

• Identify key points: Always start with critical points like intercepts and vertical asymptotes.
• Use concavity: As provided by the second derivative test, intervals of concavity help shape the graph.

For \(f(x) = \sec x\), note that vertical asymptotes appear at points \(x = \frac{\pi}{2} + k\pi\). These are places where the function is undefined and the graph cannot touch.

In between these asymptotes, draw sections of the graph that align with the concavity intervals you’ve determined:

• Draw upward curves in intervals where the graph is concave upward, such as \( (k\pi, k\pi + \frac{\pi}{2}) \).
• Draw downward curves in concave downward sections like \( (k\pi - \frac{\pi}{2}, k\pi) \).

This creates parabola-like segments that fit within the asymptote boundaries, ensuring that the sketch is both accurate and informative.

Trigonometric functions

Trigonometric functions, like \(\sec x\), play an essential role in calculus and graph sketching. Some properties of these functions are worth keeping in mind:

• Periodicity: Functions like sine and cosine, and thus secant as well, repeat their patterns at regular intervals. For secant, the period is \(\pi\).
• Relation to cosine: The secant function is the reciprocal of cosine. Thus, wherever cosine is zero, secant will have a vertical asymptote because division by zero is undefined.
• Symmetry: Secant, like cosine, is an even function, meaning \(\sec(-x) = \sec(x)\). This symmetry can help anticipate the graph's appearance.

Understanding these elements aids in predicting the overall shape and behavior of the graph of \(\sec x\). Vertical asymptotes appear where cosine crosses zero, aligning with points like \(x = \frac{\pi}{2} + k\pi\).

By mastering these concepts, you not only graph with confidence but also gain a deeper understanding of trigonometric behavior and its calculus applications.