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lecture16 - y = f x is called a secant line If a f a and a...

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Math 006 (Lecture 16) Rate of Change Example 1. The revenue from the sale of x plastic boxes is given by R ( x ) = 20 x - 0 . 02 x 2 , 0 x 1 , 000 . (a) What is the change in revenue if production is changed from 100 boxes to 400 boxes? (b) What is the average change in revenue for this change? Definition 1. For y = f ( x ), the average rate of change from x = a to x = a + h is f ( a + h ) - f ( a ) ( a + h ) - a = f ( a + h ) - f ( a ) h , h = 0 . Example 2. A small ball dropped from a tower will fall a distance of y feet in x seconds, as given by the formula y = f ( x ) = 16 x 2 . (a) Find the average velocity from x = 2 seconds to x = 3 seconds. (b) Find the average velocity from x = 2 seconds to x = 2 + h seconds, h = 0. (c) Find the expression from part (2) as h 0, if it exists. 1
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Definition 2. For y = f ( x ), the instantaneous rate of change at x = a is lim h 0 f ( a + h ) - f ( a ) h if the limit exist. Example 3. Find the instantaneous rate of change of the revenue function in example 1 when x = 100. Slope of Tangent line
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Unformatted text preview: y = f ( x ) is called a secant line . If ( a, f ( a ) and ( a + h, f ( a + h )) are two points on the graph of y = f ( x ), we have (slope of secant line from x = a to x = a + h ) = f ( a + h )-f ( a ) ( a + h )-a = f ( a + h )-f ( a ) h . Thus, the slope of secant line can be intercepted as the average rate of change. Example 4. Given y = f ( x ) = x 2 , (a) Find the slope of secant line for a = 1, and h = 2 and 1, respectively. (b) Find the slope of secant line for a = 1 and h for any nonzero number. (c) Find the limit of expression in part (2) as h → 0. Definition 3. Given y = f ( x ), the slope of the tangent line of f ( x ) at the point x = a is given by lim h → f ( a + h )-f ( a ) h if the limit exists. 2...
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