Problem 34 - Calculus Single Variable

Question

Sketch the graph of each of the functions in Exercise \(25-40,\) exhibiting and labeling: a) all local and globa extrema; b) inflection points; c) intervals on which the func tion is increasing or decreasing; d) intervals on which the function is concave up or concave down; e) all horizontal an vertical asymptotes. $$ f(x)=\sin (x)+\tan (x),-\pi

Step-by-Step Solution

Answer

The function has critical points, inflection points, intervals of increasing/decreasing and concavity, and vertical asymptotes at \(x = \pm \frac{\pi}{2}\).

Step 1: Find Critical Points

To find the critical points, we need the derivative of the function. The function is given as \(f(x) = \sin(x) + \tan(x)\). First, differentiate:\[f'(x) = \cos(x) + \sec^2(x)\]Set \(f'(x) = 0\) to find critical points. This equation doesn't have an elementary solution, so approximate using numerical methods or graph the derivative to locate critical points within the interval \(-\pi < x < \pi\). Note potential critical points where \(f'(x)\) crosses zero.

Step 2: Determine Extrema

Evaluate \(f(x)\) at the critical points found in Step 1 as well as at the endpoints \(x = -\pi\) and \(x = \pi\) if within the domain. Since trigonometric functions are periodic, ensure that extremes at or near the boundary are accounted. Determine local minima and maxima based on the values of \(f(x)\) calculated.

Step 3: Find Inflection Points

The second derivative, \(f''(x)\), is needed. Differentiate \(f'(x)\):\[f''(x) = -\sin(x) + 2\sec^2(x)\tan(x)\]Set \(f''(x) = 0\) and solve to find points of inflection. Like before, approximate solutions numerically or visually inspect the graph of \(f''(x)\) for approximate locations of zero crossings.

Step 4: Determine Intervals of Increase/Decrease

Analyze the sign of \(f'(x)\) to determine where the function is increasing or decreasing. Where \(f'(x) > 0\), the function is increasing, and where \(f'(x) < 0\), it is decreasing. Map these intervals based on intersections with the x-axis from Step 1.

Step 5: Determine Concavity

Examine the second derivative \(f''(x)\) from Step 3. When \(f''(x) > 0\), the function is concave up, whereas when \(f''(x) < 0\), it is concave down. Identify these intervals.

Step 6: Identify Asymptotes

To identify vertical asymptotes, seek for where \(\tan(x)\) is undefined, specifically where \(x = \pm \frac{\pi}{2}\). For horizontal asymptotes, evaluate the behavior of \(f(x)\) as \(x\) approaches infinity, but here focus on the interval since it's restricted. Observe near endpoints for suspect asymptotic behavior.

Key Concepts

Critical Points

Critical points are essential in analyzing the behavior of functions like the given trigonometric function: \(f(x) = \sin(x) + \tan(x)\). These points occur where the derivative of the function equals zero or is undefined.
Finding the critical points begins with taking the derivative:

• Differentiate the function to get \(f'(x) = \cos(x) + \sec^2(x)\).
• Set \(f'(x) = 0\) to find potential critical points.
• In some cases, this equation does not lend itself to easy solutions.

That's why numerical methods or graphical inspections can help locate these points. Critical points can indicate where functions reach their local maxima or minima. Additionally, these points are also where the behavior of the function may change such as switching from increasing to decreasing or vice versa.

When dealing with periodic functions like trigonometric ones, it becomes crucial to also examine the endpoints of the interval because these points give more insights into the function's behavior over a specified range.

Derivatives

The concept of derivatives is fundamental in calculus and trigonometry. It helps explain how functions change. When looking at the function \(f(x) = \sin(x) + \tan(x)\), its derivative tells us about the slope of the curve at any point.

• The derivative can also inform us about the function's rate of increase or decrease.
• For instance, if \(f'(x) > 0\), the function is increasing in that interval.
• If \(f'(x) < 0\), the function is decreasing.

Using the derivative \(f'(x) = \cos(x) + \sec^2(x)\), we can examine where the function increases and decreases between \(-\pi\) and \(\pi\).

By plugging in different values or examining the graph, we can identify intervals where \(f(x)\) ascends or descends. This is especially useful in sketching accurate graphs and understanding the function's overall behavior.

Inflection Points

Inflection points occur where the function changes its concavity, i.e., from concave up to concave down, or vice versa. For the function \(f(x) = \sin(x) + \tan(x)\), finding inflection points involves the second derivative.

• Compute \(f''(x) = -\sin(x) + 2\sec^2(x)\tan(x)\).
• Inflection points occur where \(f''(x) = 0\) or is undefined.
• Typically, inflection points are more challenging to solve algebraically and may require numerical approximation or visual methods.

Identifying these points gives insights into the curvature of the function. It's where the graph of a function changes from being curved upward to downward or vice versa. This understanding is vital in comprehensively analyzing the graph's shape and understanding how the function behaves.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. There are vertical and horizontal asymptotes, particularly relevant for functions involving trigonometric terms like \(f(x) = \sin(x) + \tan(x)\).

Vertical Asymptotes:

• The function \(\tan(x)\) becomes undefined where \(x = \pm \frac{\pi}{2}\).
• In these locations, the function tends toward infinity, hence the vertical asymptotes.

Horizontal Asymptotes:

• Often found by examining the behavior as \(x\) approaches infinity.
• However, in bounded intervals like \(-\pi < x < \pi\), observing endpoints gives similar insights.

Recognizing these lines is crucial because they indicate boundaries where the function grows or decreases without limits. Asymptotes guide us to predict how the graph behaves in extreme regions, ensuring more accurate and insightful graph interpretations.