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lecture11

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Lecture 11 (Section 2.7): Tangents, Velocities and Rates of Change 6 - ? Def. The tangent line to y = f ( x ) at the point P ( a; f ( a )) is the line through P with slope m = provided that the limit exists. NOTE: Slope of the curve y = f ( x ) at the point ( a; f ( a )):
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ex. Find the equation of the tangent line to y = x 3 at the point (2 ; 8). 6 - ?
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Alternate De nition of the Slope of a Tangent Line Let h = x a . Then and m =
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ex. Find the equation of the tangent line to y = 1 x 2 at x = 4. 6 - ?
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ex. Find the slope of the tangent line to y = p 10 3 x at x = a . 1) at x = 2 m = 2) at x = 1 3 m = 3) at x = 5 m =
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Velocity Suppose an object moves along a straight line according to an equation of motion s = f ( t ) where s is the displacement (directed distance) from the starting point at time t . We call f ( t ) the position function of the object. Average Velocity on interval t = a to t = a + h NOTE: This is the same as the slope of the secant line through ( a; f ( a )) and ( a + h; f ( a + h )). 6 - ? t s
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Instantaneous Velocity at time t = a A useful formula: The position of an object subject to the force of gravity only is given by the function s ( t ) = where s is in feet and t in seconds. where v 0 = h 0 = The formula can also be expressed in meters/second:
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