Master Calculus: The Tangent Line And Its Applications

tangent line (2.1)

a line that touches a point on a curve, but does not cross it

In calculus, the tangent line (2.1) of a curve at a specific point is a straight line that just touches the curve at that point. It can be thought of as the instantaneous rate of change of the curve at that particular point. Mathematically, the tangent line at point a can be found by taking the derivative of the function at that point, f'(a), and using it to form the equation of the line in point-slope form:

y – f(a) = f'(a)(x – a)

where (a, f(a)) is the point on the curve and f'(a) is the slope of the tangent line.

Alternatively, another way to find the tangent line is to use the limit definition of the derivative:

f'(a) = lim h->0 (f(a+h) – f(a)) / h

This formula gives the slope of the tangent line at point a. Once you have the slope and the point on the curve, you can easily find the equation of the line.

The tangent line is useful in many applications of calculus, especially in optimization problems where you need to find the maximum or minimum of a function. In physics, the tangent line represents the instantaneous velocity of a moving object at a specific time.

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