Engineering Calculus Notes 187

Engineering Calculus Notes 187 - and this vector varies...

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2.4. REGULAR CURVES 175 Graphs of Functions We begin with the simplest example of a curve in the plane: the graph of a function f ( x ) deFned on an interval I gr ( f ) := { ( x,y ) | y = f ( x ) , x I } . In order to apply calculus to this curve, we assume that the function f ( x ) is continuously di±erentiable, 13 or C 1 . Such a curve has a natural parametrization, using the input x as the parameter: −→ p ( x ) = ( x ) −→ ı + ( f ( x )) −→ = ( x,f ( x )) . We note several properties of this parametrization: 1. Di±erent values of the input x correspond to distinct points on the graph. This means we can use the value of the parameter x to unambiguously specify a point on the curve gr ( f ). In other words, the vector-valued function −→ p ( x ) is one-to-one : 14 x n = x −→ p ( x ) n = −→ p ( x ) . 2. The vector-valued function −→ p ( x ) is C 1 : it is di±erentiable, with derivative the velocity vector −→ v ( x ) = V p ( x ) = −→ ı + ( f ( x )) −→
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Unformatted text preview: and this vector varies continuously with x . Also, since its Frst component is always 1, it is always a nonzero vector: we express this by saying that p ( x ) has nonvanishing velocity . 3. At each point P ( x ,f ( x )) on the curve, gr ( f ) has a tangent line : this is the line through P with slope dy dx = f ( x ) . The velocity vector v ( x ) points along this line, so it can be used as a direction vector for the tangent line. 13 Although in principle only diferentiability is needed, we also assume the derivative is continuous; dealing with examples oF Functions with possibly discontinuous derivatives could distract us From our main goal here. 14 A synonym For one-to-one, derived From the rench literature, is injective ....
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