Question

In Exercises \(13-20\) , find all points of inflection of the function. $$y=\tan ^{-1} x$$

Short answer

The inflection point of the function \( y = \tan^{-1} x \) is (\(0\), \(0\))

Step-by-step solution

Find the first derivative

Differentiate \( y = \tan^{-1}(x) \) with respect to \(x\). The derivative of \( \tan^{-1}(x) \) is \( \frac{1}{1+x^{2}} \) . So, \( \frac{dy}{dx} = \frac{1}{1+x^{2}} \)

Find the second derivative

Differentiate \( \frac{dy}{dx} = \frac{1}{1+x^{2}} \) with respect to \(x\) to find the second derivative. The derivative of \( \frac{1}{1+x^{2}} \) is \( \frac{-2x}{(1+x^{2})^{2}} \). So, \( \frac{d^{2}y}{dx^{2}} = \frac{-2x}{(1+x^{2})^{2}} \)

Find the potential points of inflection

Find all \(x\) values that make the second derivative equal to zero \(x=0\).

Determine if the potential points are points of inflection

If the sign of \( \frac{d^{2}y}{dx^{2}} \) changes at \(x\), then \(x\) is a point of inflection. Since the sign of \( \frac{d^{2}y}{dx^{2}} \) changes at \(x=0\), \(x=0\) is a point of inflection of \( y = \tan^{-1} x \)

Key concepts

Second Derivative Test

The second derivative test is a useful tool in calculus for determining the concavity of a function at a particular point and whether that point is a maximum, a minimum, or a point of inflection. A point of inflection is where a function changes concavity, from convex to concave or vice versa.

In mathematical terms, to apply the second derivative test, you take the second derivative of the function and analyze its value at the point of interest. If the second derivative is positive at that point, the function is concave up, resembling a U-shape. If it is negative, the function is concave down, resembling an upside-down U. If the second derivative is zero and changes signs at that point, you have found a potential point of inflection. However, it is essential to check the sign change because if there's no sign change, the test is inconclusive at that point.

For our exercise with the arctangent function, we derived the second derivative and found it to be \( \frac{-2x}{(1+x^{2})^{2}} \). We then solved for when this second derivative equals zero, which occurs when \(x=0\). Checking the sign change across \(x=0\), we confirmed a true inflection point.

Arctangent Function

The arctangent function, denoted as \(\tan^{-1}(x)\) or \(\text{arctan}(x)\), is the inverse of the tangent function. It is used to find the angle whose tangent is a given number. The function has a range of \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) and can be used in various applications, such as trigonometry and calculus.

The arctangent function has a unique characteristic; it is an increasing function that approaches \(\frac{\pi}{2}\) as \(x\) approaches positive infinity and \( -\frac{\pi}{2}\) as \(x\) approaches negative infinity. This property makes it a function with no local maxima or minima but potentially with points of inflection, where the concavity of the function changes. In our problem, we investigated the concavity of the arctangent function by finding the second derivative, which is necessary to determine potential inflection points.

Concavity of a Function

The concavity of a function refers to the direction in which a function curves. There are two types of concavity: concave up, where the function curves like a cup (\(\bigcup\)), and concave down, where the function curves like a frown (\(\bigcap\)). This concept is closely tied to the curvature of the graph and affects the overall shape and behavior of the function.

To determine concavity, we observe the sign of the second derivative. If the second derivative is positive for a range of values, the function is concave up in that interval. Conversely, if the second derivative is negative, the function is concave down. Changes in concavity can signify important features in the graph like points of inflection. In the context of our arctangent function, we examined the second derivative to detect changes in concavity and successfully identified a point of inflection at \(x=0\).

Calculus

Calculus is a branch of mathematics that deals with rates of change (differential calculus) and accumulation of quantities (integral calculus). It is a tool that enables us to model and solve problems involving dynamic systems, such as motion, growth, and decay.

In differential calculus, the concept of derivatives is central. Derivatives represent the rate of change of a function concerning its variable. They are fundamental in finding the slope of a tangent line to a curve, maximizing or minimizing functions, and identifying points of inflection, like in our arctangent function example. Integral calculus, on the other hand, revolves around the concept of integrals, which are used to calculate areas under curves, volumes, and other quantities that accumulate over an interval.

In summary, calculus provides powerful methods for analyzing and understanding changes and has applications that span across sciences, economics, engineering, and beyond. Understanding its concepts, including those applied in our discussed problem, is essential for solving complex problems in these fields.