Finding the horizontal tangent of the arctangent function requires understanding its derivative and the concept of horizontal tangents. Let's break down the process step-by-step.
Understanding Horizontal Tangents
A horizontal tangent occurs where the slope of a function is zero. The slope of a function at a given point is represented by its derivative. Therefore, to find horizontal tangents, we need to find where the derivative of the function equals zero.
The Derivative of Arctan(x)
The derivative of the arctangent function, arctan(x), is:
d/dx [arctan(x)] = 1 / (1 + x²)
This is a crucial piece of information for finding our horizontal tangent.
Finding the Horizontal Tangent
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Set the derivative equal to zero: To find where the slope is zero (i.e., the horizontal tangent), we set the derivative equal to zero:
1 / (1 + x²) = 0
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Solve for x: Notice that there's no solution for x in this equation. The numerator is always 1, and the denominator is always positive (since x² is always non-negative). Therefore, the fraction can never equal zero.
Conclusion: Arctan(x) Has No Horizontal Tangents
This analysis demonstrates that the arctangent function, arctan(x), does not have any horizontal tangents. Its derivative is always positive, meaning the function is always increasing. This is a key characteristic of the arctangent function.
Further Exploration and Related Concepts
While arctangent itself lacks horizontal tangents, understanding this process helps solidify concepts related to:
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Derivatives: The ability to find and interpret derivatives is fundamental in calculus and many areas of science and engineering.
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Horizontal and Vertical Tangents: Knowing how to identify these special points on a function's graph is essential for sketching and analyzing its behavior.
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Increasing and Decreasing Functions: The sign of the derivative indicates whether a function is increasing or decreasing. Arctan(x)'s always-positive derivative confirms it's a strictly increasing function.
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Application of Calculus: This problem highlights a practical application of calculus—analyzing the behavior of a function using its derivative.
By understanding these concepts, you can effectively tackle more complex problems involving tangents and derivatives of other functions. Remember, practice is key to mastering these techniques.
