Understanding Horizontal Tangents in Mathematics: A Guide to Flat Slopes and Zero Derivatives

Horizontal tangent

In mathematics, a horizontal tangent refers to a tangent line that is parallel to the x-axis on a graph

In mathematics, a horizontal tangent refers to a tangent line that is parallel to the x-axis on a graph. This means that the derivative of the function at that point is equal to zero.

To understand this concept better, let’s consider a function f(x) and its graph. The tangent line to the graph at a particular point, let’s say (a, f(a)), represents the instantaneous rate of change of the function at that point.

To determine if a tangent line is horizontal, we need to find the derivative of the function and set it equal to zero. If the derivative is zero at a specific x-coordinate, then the tangent line at that point will be horizontal.

To illustrate this with an example, let’s consider the function f(x) = x^3 – 2x. We want to find the x-coordinate(s) where the tangent line is horizontal.

Step 1: Find the derivative of the function f'(x):

f'(x) = 3x^2 – 2

Step 2: Set the derivative equal to zero and solve for x:

3x^2 – 2 = 0

By solving this equation, we find:

3x^2 = 2
x^2 = 2/3
x = ±√(2/3)

Therefore, the x-coordinates where the tangent line is horizontal are approximately x = √(2/3) and x = -√(2/3).

To confirm that these points have horizontal tangents, we can calculate the derivative at these x-values:

f'(√(2/3)) = 3(√(2/3))^2 – 2 ≈ 0
f'(-√(2/3)) = 3(-√(2/3))^2 – 2 ≈ 0

Since both derivatives are approximately equal to zero, we can conclude that the tangent lines at x = √(2/3) and x = -√(2/3) are horizontal.

In summary, a horizontal tangent occurs when the derivative of a function is equal to zero at a specific x-coordinate. It represents a point where the slope of the function is flat or has no change with respect to x.

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