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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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- Fall '08