Chapter 3 Section 04

Chapter 3 Section 04 - 3.4 Derivative as a rate of change...

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1 3.4 Derivative as a rate of change page 150
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Objectives Use the derivative to describe the rate of change in different application problems and answer questions about that rate of change. 2
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3 The derivative as a rate of change If x changes from x 1 to x 2 , then we say x = x 2 – x 1 . The corresponding change in y is y = f(x 2 ) – f(x 1 ). The difference quotient can then be written 1 2 1 2 ) ( ) ( x x x f x f x y - - =
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4 1 2 1 2 ) ( ) ( x x x f x f x y - - = The difference quotient is the “ average rate of change ” of y with respect to x over the interval [x 1 , x 2 ]. This is also the slope of the secant line from (x 1 , f(x 1 )) to (x 2 , f(x 2 )). As x → 0, we get the “ instantaneous rate of change (ROC) .” This is also the derivative, f’(x). And it is the slope of the tangent line at (x 1 , f(x 1 )).
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5 x." " for units the by divided y" " for units the are dx dy for units The lim following get the we So, 0 x y dx dy x =
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object along a straight line. For linear motion, the velocity function v(t) is the ROC of the position function s(t) with respect to time, that is, v(t) = s’(t). Speed is defined as the absolute of velocity
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This note was uploaded on 09/28/2009 for the course MATH 1550 taught by Professor Wei during the Spring '08 term at LSU.

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Chapter 3 Section 04 - 3.4 Derivative as a rate of change...

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