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# 2_7 - Other rates of change Given y = f x let • 4 x = x...

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Section 2.7: Tangents, Velocities and Other Rates of Change Instructor: Ms. Hoa Nguyen Tangents Given a point P = ( a, f ( a )) on a curve C : y = f ( x ), the equation of the tangent line at the point P is y - f ( a ) = m ( x - a ) with the slope : m = lim x a f ( x ) - f ( a ) x - a Notice : 1) slope = rise over run slope of a tangent can be thought as the steepness of the hill. It is the change in height for a horizontal step forward. 2) If we zoom in far enough toward the point P , the curve C becomes indistinguishable from the tangent line at the point P . Example 2 of Section 2.7 (textbook): Velocities Given a position function s = f ( t ) of an object, the instantaneous velocity v ( a ) at time t = a is: v ( a ) = lim h 0 f ( a + h ) - f ( a ) h This means that the velocity at time t = a is equal to the slope of the tangent line at P = ( a, f ( a )) on a curve of the position function s = f ( t ). Example 3 of Section 2.7
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Unformatted text preview: Other rates of change Given y = f ( x ), let: • 4 x = x 2-x 1 : the change in x from x 1 to x 2 . • 4 y = f ( x 2 )-f ( x 1 ): the corresponding change in y . Then, the average rate of change of y with respect to x over the interval [ x 1 , x 2 ] is: 4 y 4 x = f ( x 2 )-f ( x 1 ) x 2-x 1 Let 4 x → 0, the limit of the average rates of change = the (instantaneous) rate of change of y with respect to x at x = x 1 : lim 4 x → 4 y 4 x = lim x 2 → x 1 f ( x 2 )-f ( x 1 ) x 2-x 1 Notice : average rate = change in y divided by the change in x ⇒ the instantaneous rate of change (at x 1 ) can be thought as how much the function y = f ( x ) is changing at the point x 1 . Example 6 of Section 2.7 (textbook): 1...
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