How to Find the Horizontal Tangent of arctan(x)
Finding the horizontal tangent of a function involves identifying points where the derivative (the slope of the tangent line) is equal to zero. This signifies a flat, horizontal line. Let's explore how to do this for the arctangent function, arctan(x), also written as tan⁻¹(x).
Understanding the Arctangent Function
The arctangent function, arctan(x), gives the angle whose tangent is x. It's the inverse function of the tangent function, restricted to the interval (-π/2, π/2) to ensure a single-valued output. Understanding this domain is crucial.
Finding the Derivative
The first step to finding horizontal tangents is to determine the derivative of arctan(x). The derivative of arctan(x) is:
d(arctan(x))/dx = 1 / (1 + x²)
Setting the Derivative to Zero
To find horizontal tangents, we need to solve for x when the derivative equals zero:
1 / (1 + x²) = 0
Notice that there's no value of x that can make this equation true. The numerator is always 1, and the denominator (1 + x²) is always positive (since x² is always non-negative). Therefore, the fraction 1/(1+x²) will never equal zero.
Conclusion: No Horizontal Tangents
This means that the arctangent function, arctan(x), does not have any horizontal tangents. Its derivative is always positive, indicating a continually increasing function with a slope that approaches zero as x approaches positive or negative infinity, but never actually reaching zero.
Visualizing the Graph
Graphing arctan(x) will visually confirm this. The function steadily increases from -π/2 (as x approaches negative infinity) to π/2 (as x approaches positive infinity) without ever having a completely flat section (horizontal tangent).
Key Takeaways
- The derivative of arctan(x) is 1/(1 + x²).
- There are no values of x that make this derivative equal to zero.
- Therefore, the function arctan(x) has no horizontal tangents.
This understanding helps solidify your knowledge of derivatives, inverse trigonometric functions, and how to find horizontal tangents in general. Remember to always consider the domain and range of the function when analyzing its behavior.
