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The Concept of the Derivative

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Tangents to a Curve

We begin with the familiar notion of the tangent to a circle, depicted below:

Figure %: Tangent to a circle

Calculus, to a certain extent, concerns itself with the study of tangents to a curve. Depicted below is the graph of a polynomial function with tangents drawn at various points:

Upon observation, two important properties of the tangents to a curve may become apparent:

Figure %: Tangents to a graph

1) At the point at which it is tangent to the curve, the tangent line touches the curve, but does not "cross" it. This is to say that tangent lines are distinct from lines such as the one below, which also touches the graph at only one point, but which clearly "crosses" it:

Figure %: A line that is not tangent to a graph

2) The second important property of a tangent line is that it has the same slope as the point of the graph that it touches. Although a formal definition for the slope of a curve at a point has not yet been presented, it should be visually clear that the slope of the tangent line matches the slope of the curve at the point of tangency.

The Slope of a Curve at a Point

"Slope" is a concept that can easily be applied to linear functions. It is the change in y divided by the change in x . To calculate the slope of a line, we pick any two points on that line and divide the difference in their y -values by the difference in their x - values.

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