Problem 233 - Calculus Volume 1

Question

For the following exercises, determine $$ \begin{array}{l}{\text { a. intervals where } f \text { is increasing or decreasing, }} \\ {\text { b. local minima and maxima of } f} \\ {\text { c. intervals where } f \text { is concave up and concave }} \\ {\text { down, and }} \\ {\text { d. the inflection points of } f \text { . Sketch the curve, then }} \\ {\text { use a calculator to compare your answer. If you }} \\\ {\text { cannot determine the exact answer analytically, use }} \\ {\text { a calculator. }}\end{array} $$ $$ f(x)=\sin x+\tan x \text { over }\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

Increasing and decreasing intervals are defined by critical points, and concavity is analyzed with the second derivative. Inflection points occur where concavity changes.

Step 1: Find the First Derivative

First, find the derivative of the function \( f(x) = \sin x + \tan x \) to determine where the function is increasing or decreasing. We have: \[ f'(x) = \cos x + \sec^2 x \] This derivative will help us find critical points and analyze the behavior of the function.

Step 2: Critical Points for Increasing/Decreasing Intervals

Find the critical points by setting the first derivative equal to zero. \[ \cos x + \sec^2 x = 0 \] Solve the equation. However, note this may not have simple solutions, so consider using a calculator or graphing tool to find approximate solutions if necessary.

Step 3: Determine Intervals of Increase and Decrease

Evaluate test points in the intervals defined by the critical points within the domain \((-\pi/2, \pi/2)\) to determine the signs of \( f'(x) \). Positive values indicate increasing intervals, and negative values indicate decreasing intervals.

Step 4: Find the Second Derivative

Take the second derivative of \( f(x) = \sin x + \tan x \) to analyze concavity. \[ f''(x) = -\sin x + 2\sec^2 x \tan x \] This will help in determining concavity and inflection points.

Step 5: Determine Concavity

Examine the signs of \( f''(x) \) over the domain to ascertain intervals of concave up and concave down. If \( f''(x) > 0 \), the function is concave up. If \( f''(x) < 0 \), the function is concave down.

Step 6: Identify Inflection Points

Inflection points occur where \( f''(x) \) changes sign. Evaluate points where \( f''(x) = 0 \) and confirm sign changes to find potential inflection points.

Step 7: Characterize Local Extrema

Using the critical points found earlier, determine if they correspond to local minima or maxima by analyzing the behavior of the function around these points, or use the second derivative test for confirmation.

Step 8: Sketch and Compare with Calculator

Sketch the curve based on the findings in previous steps and use a graphing calculator as necessary to verify critical points, check inflection points, and observe concavity of the function within \((-\pi/2, \pi/2)\).

Key Concepts

Increasing and Decreasing Functions

In calculus, understanding whether a function is increasing or decreasing over certain intervals is crucial to grasp its overall behavior. This is often done by analyzing the first derivative of the function. For the function \( f(x) = \sin x + \tan x \), the first derivative is \( f'(x) = \cos x + \sec^2 x \). This derivative allows us to determine where the function increases or decreases.
To find out where this function is increasing or decreasing, we set the first derivative equal to zero, \( \cos x + \sec^2 x = 0 \), to find critical points. These are the points where the function changes its behavior. Once the critical points are pinpointed, we test values in the intervals around these points to determine the sign of \( f'(x) \):

• A positive derivative indicates the function is increasing in that interval.
• A negative derivative means the function is decreasing.

This step is essential in creating a more complete understanding of how the function behaves across its domain.

Local Minima and Maxima

Local minima and maxima refer to points on the graph of a function where it reaches a local low or high, respectively. To find these points for \( f(x) = \sin x + \tan x \), we use critical points found from the first derivative where \( f'(x) = 0 \). After locating these critical points, we need to determine the behavior around them.
To do this, we can:

• Use the first derivative test: Examine the sign changes of \( f'(x) \) before and after a critical point.
• Apply the second derivative test: If \( f''(x) < 0 \), the point is a local maximum; if \( f''(x) > 0 \), it’s a local minimum.

For this function, the first or second derivative tests may reveal where the curves take a turn for a local peak or trough. These insights are indispensable for sketching the shape of the graph.

Concavity and Inflection Points

Concavity describes the direction in which a graph curves. Calculus uses the second derivative of a function, \( f''(x) \), to determine this aspect. For \( f(x) = \sin x + \tan x \), we found \( f''(x) = -\sin x + 2\sec^2 x \tan x \).
By examining the second derivative, we determine:

• Concave up intervals (\( f''(x) > 0 \)): The graph is shaped like a "cup".
• Concave down intervals (\( f''(x) < 0 \)): The graph resembles an "umbrella".

Inflection points further detail where the concavity of the function changes. These points occur where \( f''(x) = 0 \) and there is a change in sign, indicating a shift in the curvature of the graph. By identifying these areas, we gain deeper insights into the function's dynamic nature.

Sin and Tan Functions

Both \( \sin x \) and \( \tan x \) are fundamental trigonometric functions with unique properties that influence the behavior of combined functions like \( f(x) = \sin x + \tan x \).
The \( \sin x \) function oscillates between -1 and 1, exhibiting periodic behavior that results in regular peaks and troughs. Meanwhile, \( \tan x \) has asymptotes, specifically where \( x = \frac{\pi}{2} \) and \( x = -\frac{\pi}{2} \), due to division by zero.
When combined in this function, \( \sin x \) adds oscillation while \( \tan x \) contributes to steep inclines and declines near its asymptotes. Understanding how these basic functions interact helps interpret the overall behavior of the function, especially within the domain \((-\frac{\pi}{2}, \frac{\pi}{2})\). This grasp of behavior guides analysis for derivatives and concavity, crucial for accurately sketching or modeling the curve.