Calculus > Vertical tangent in calculus

A tangent of a curve is a line that touches the curve at one point. It has the same gradient as the curve at that point. A **vertical tangent** touches the curve at a point where the gradient (slope) of the curve is infinite and undefined.

To find the vertical tangent in calculus and the gradient of a curve, you need to differentiate the function that describes the curve. For example, if the curve is y = √(x – 2), then differentiating this equation to find dy/dx will give you the gradient of the curve. Next, find a value of x that makes dy/dx infinite. The vertical tangent of the curve is a vertical line at this value of x.

## Vertical Tangent in Calculus Example

Sample Problem: Find the vertical tangent of the curve y = √(x – 2).

Step 1: Differentiate y = sqrt(x – 2). Note that the square root can can be written as a power of one half, i.e., y = sqrt(x – 2) = (x – 2)^{(1/2)}. Therefore, the result of differentiation is dy/dx = 1/2 (x – 2)^{(-1/2)}, which is the same as dy/dx = 1/(2 √(x – 2)).

Step 2: **Look for values of x** that would make dy/dx infinite. In other words (for this sample problem), look for values of x that make the bottom part of dy/dx equal to zero. If x = 2, then sqrt(x – 2) = 0. This means that the gradient of the curve is infinite (i.e., vertical) when x = 2. The vertical tangent of the curve is x = 2.

*That’s it!*

Tip: If you aren’t able to immediately see where your function might return zero, you’ve got two options: **graph the function** — so you can see where the graph might have a vertical tangent, or make a **table of values **and test for several values of x. The second option can be very time consuming — strong algebra skills (like knowing when an equation might result in division by zero) will help you to avoid having to make a table.

Warning: Some curves will have** more than one vertical tangent**. Always make sure you have found all the values of x that make the gradient infinite. You can use graph-plotting software to check by eye for places where the gradient becomes vertical.

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