121616949-math.52 - 38 Chapter 2 Instantaneous Rate Of...

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38 Chapter 2 Instantaneous Rate Of Change: The Derivative Sometimes one encounters a point in the domain of a function y = f ( x ) where there is no derivative , because there is no tangent line. In order for the notion of the tangent line at a point to make sense, the curve must be “smooth” at that point. This means that if you imagine a particle traveling at some steady speed along the curve, then the particle does not experience an abrupt change of direction. There are two types of situations you should be aware of—corners and cusps—where there’s a sudden change of direction and hence no derivative. EXAMPLE 2.15 Discuss the derivative of the absolute value function y = f ( x ) = | x | . If x is positive, then this is the function y = x , whose derivative is the constant 1. (Recall that when y = f ( x ) = mx + b , the derivative is the slope m .) If x is negative, then we’re dealing with the function y = - x , whose derivative is the constant - 1. If x = 0, then the function has a corner, i.e., there is no tangent line. A tangent line would have
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