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Tangent lines

Learn how to take a snapshot of curves at various points by watching this detailed breakdown of tangent lines.

Now let’s turn our focus to tangent lines. Here we will define the point of tangency and explain how derivatives are related to tangents.

  • A tangent line for a function (f(x)) at a given point (x=a) is a straight line that touches a function at only one point (x=a) and has the same slope as the curve does at that point. It represents the instantaneous rate of change of the function at that one point. The slope of the tangent line at a point on the function is equal to the derivative of the function at the same point.

A graph featuring an s-curve with the secant line and a tangent line at x0, along with the formula f'(x0)=slope of tangent line

  • Reminder: A secant line is a straight line joining two points on a function. It is also equivalent to the average rate of change, or simply the slope between two points.
  • The derivative is not the same thing as a tangent line. Instead, the derivative is a tool for measuring the slope of the tangent line at any particular point.
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Applications of Calculus

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