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What Are Tangent Lines?

Choose a point on an object. The tangent line—or simply tangent—to the object at this point is the straight line that goes through the point and only touches the object at that single point. This object can for example be a circle or the graph of a function.

Tangent line to a circle

Tangent line to a function

The point which the tangent line intersects the object is called the point of tangency.

You have probably tried to balance a ruler on your finger. You can think of the ruler as being a tangent line and the point it balances on your finger as the point of tangency.

To find tangent lines for graphs, you’ll need the point-slope equation and derivation.

Think About This

How many tangent lines pass through a point on a sphere?

There are infinitely many tangent lines passing through a given point on a sphere. All these tangent lines form a whole plane. This makes sense! A sphere is to a circle as a plane is to a line. When the object increases in dimension by 1, the tangent increases in dimension as well.

Tangent plane to a sphere

Think About This

For most points on a graph, there is precisely one tangent line. If the graph has a “cusp”, it may have more tangents through that point.

Two tangents to a function at the same point

Two tangents to a function at the same point

When there is more than one tangent through the same point, the derivative of that function is not well-defined there. We say that the function is not differentiable at that point.

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