Engineering Calculus Notes 188

Engineering Calculus Notes 188 - p ( x ) is one-to-one; its...

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176 CHAPTER 2. CURVES 4. The tangent line at P is the graph of the degree one Taylor polynomial of f at x 0 T x 0 f ( x ) := f ( x 0 ) + f ( x 0 )( x x 0 ) which has frst-order contact with f ( x ) at x = x 0 : f ( x ) T x 0 f ( x ) = o ( x x 0 ) , i.e. , lim x x 0 f ( x ) T x 0 f ( x ) x x 0 = 0 . To compare diFerent parametrizations of the same curve, we consider the example of the upper semicircle of radius 1 centered at the origin in the plane x 2 + y 2 = 1 , y > 0 which is the graph of the function y = f ( x ) = r 1 x 2 , 1 < x < 1 and hence can be parametrized by −→ p ( x ) = ( x, r 1 x 2 ) = ( x ) −→ ı + ( r 1 x 2 ) −→  , x ( 1 , 1) . Clearly, the vector-valued function −→
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Unformatted text preview: p ( x ) is one-to-one; its velocity vector at each parameter value x = x ( 1 &lt; x &lt; 1) v ( x ) = p x r 1 x 2 P is nonvanishing and parallel to the tangent line, which in turn is the graph of y = T x f ( x ) = R 1 x 2 x r 1 x 2 ( x x ) = 1 x x r 1 x 2 . An equivalent equation for the tangent line is y R 1 x 2 + xx = 1 ....
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