Vertical Tangents In Calculus: Rare Points Of Non-Differentiability

What does a Vertical Tangent look like?

vertical tangent image

A vertical tangent is a mathematical term used in calculus to describe a specific point on a curve. It occurs when the slope (derivative) of the curve becomes infinitely large or small, which means that the line tangent to that point is a vertical line.

Visually, a vertical tangent looks like a point on a curve where the tangent line is a perfectly vertical line. This means that the curve approaches that point from two different directions, but does not cross the line of the vertical tangent. In other words, the curve changes direction abruptly at this point without touching the tangent line.

For example, the function f(x) = √x has a vertical tangent at x = 0, because the slope (derivative) of the curve becomes infinitely large as x approaches 0 from the right side. At that point, the tangent line is a vertical line, and the curve changes direction abruptly without crossing the line.

It is important to note that vertical tangents are rare and only occur when a curve has a cusp or a sharp bend. They are significant in calculus because they indicate a point where a function is not differentiable, which means that the derivative cannot be calculated at that point.

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