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2.1.notes - MAC 2233/001-006 Business Calculus Fall 2011...

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MAC 2233/001-006 Business Calculus, Fall 2011 CRN: 81700–81702, 81705, 81716, 81715 Assistants : Matthew Fleeman, Arbin Rai, Tadesse Zerihun, Vindya Kumari, Junyi Tu, Jing- han Meng Brief notes on derivatives I shall briefly recapture how I approach derivatives in class. We shall use the graph (see below) of a function y = f ( x ) as an example. e assume that the point P is ( x,f ( x )) and that Q is obtained from P by adding Δ x to x . That is, the point Q is ( x + Δ x,f ( x + Δ x )). The diagram above actually shows the graph of y = f ( x ) = 1 5 ( x 2 + 4 x + 5). The point P is at x = 1, and Q is at x = 3. Hence Δ x = 3 ( 1) = 4 in the example. We start by considering the secant line through P and Q . (1) The slope of the secant line is the difference quotient : f ( x + Δ x ) f ( x ) Δ x . (2) The slope of the secant line represents the average rate of change of f between P and Q . Next, we consider the situation when Δ x approaches 0. We see that Q will move closer to P through Q 1 , Q 2 ,... . We also have the following:
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(a) The secant line now looks like the tangent line at P .
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