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Magazine Chapter 5: Problem 28
Determine all inflection points.
\(f(x)=\tan x,-\frac{\pi}{2}
Short Answer
Expert verified
The inflection point of \( f(x) = \tan x \) on \( -\frac{\pi}{2} < x < \frac{\pi}{2} \) is at \( x = 0 \).
Step by step solution
01
Understand the Function and Its Domain
The function given is \( f(x) = \tan x \) and the domain is restricted to \( -\frac{\pi}{2} < x < \frac{\pi}{2} \). This means we need to consider values of \( x \) between these limits. The tangent function has vertical asymptotes at \( x = -\frac{\pi}{2} \) and \( x = \frac{\pi}{2} \), which makes them points of discontinuity.
02
Find the First Derivative
To find inflection points, we need the second derivative. Start by finding the first derivative: \[ f'(x) = \sec^2 x. \]
03
Find the Second Derivative
Differentiate \( f'(x) \) with respect to \( x \) to get the second derivative: \[ f''(x) = 2\sec^2 x \tan x. \]
04
Determine Where the Second Derivative is Zero or Undefined
An inflection point occurs where the second derivative is zero or changes sign. However, \( f''(x) = 2\sec^2 x \tan x = 0 \) implies \( \tan x = 0 \), since \( \sec x eq 0 \) for all defined \( x \) in this interval. \( \tan x = 0 \) when \( x = 0 \).
05
Confirm Inflection Point by Checking Sign Change
To confirm \( x = 0 \) is an inflection point, check the sign of \( f''(x) \) around \( x = 0 \). Just to the left of 0, say at \( x = -0.1 \), \( f''(x) = 2\sec^2(-0.1) \tan(-0.1) < 0 \). Just to the right of 0, say at \( x = 0.1 \), \( f''(x) = 2\sec^2(0.1) \tan(0.1) > 0 \). Since \( f''(x) \) changes from negative to positive at \( x = 0 \), it confirms an inflection point.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Tangent Function Insights
The tangent function, denoted as \( \tan x \), is a fundamental trigonometric function used commonly in calculus and beyond. It is defined as the ratio of the sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \). This function is periodic with a period of \( \pi \) and has vertical asymptotes where the cosine function equals zero, resulting in points of discontinuity.
- In the interval \( -\frac{\pi}{2} < x < \frac{\pi}{2} \), the tangent function takes on all real values, making it continuous within these bounds except at the boundaries themselves.
- The graph of the tangent function shows a characteristic pattern where it rises rapidly, passes through zero, and continues its rise. This pattern repeats every \( \pi \) units.
- Understanding the behavior of \( \tan x \) is essential when analyzing its derivatives, which help in identifying changes in curvature or slope.
Exploring Calculus Concepts
Calculus is a branch of mathematics that studies how things change. Among its many tools are derivatives, which provide information about the rate at which something is changing. Here are a few important calculus concepts that relate to this exercise:
- Derivative: The derivative of a function, like \( f'(x) \), indicates the rate of change or the slope of the function at any given point. For the tangent function \( \tan x \), its first derivative is \( \sec^2 x \), which tells us how rapidly \( \tan x \) is increasing or decreasing.
- Inflection Points: Inflection points occur where there's a change in the concavity of the function, from concave up to concave down, or vice versa. Identifying these points often involves examining where the second derivative is zero or undefined.
- Critical Thinking: Understanding derivatives and their behavior is critical for solving real-world problems where rates of change are involved. This includes physics, engineering, and economics.
Understanding the Second Derivative
The second derivative is a derivative of the first derivative, often denoted as \( f''(x) \). It provides deeper insight into the behavior of a function. Specifically, it helps to determine the concavity of the graph and identify inflection points. When analyzing \( f(x) = \tan x \), finding the second derivative is crucial:
- What Does It Do? The second derivative reveals whether a function is concave up or concave down in a particular section. If \( f''(x) > 0 \), the function is concave up (like a cup facing upwards), and if \( f''(x) < 0 \), it is concave down (like a frown).
- Calculating \( f''(x) \): For the tangent function, we have \( f'(x) = \sec^2 x \). Differentiating again yields \( f''(x) = 2\sec^2 x \tan x \), which gives us the information needed to analyze concavity.
- Using the Second Derivative: In the specific example of \( f''(x) = 2\sec^2 x \tan x \), finding where it equals zero helps locate inflection points. Here, \( \tan x = 0 \) at \( x = 0 \), makes it an inflection point when \( f''(x) \) changes sign around that point.
