2.6.notes - f and f ′ are given on the right. Which is...

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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 On spotting f and f graphically The graphs of f , f and f ′′ are given in the diagram on the right. Which is which? Answer . After a little pondering, we observe that the red graph is about a quadratic, the brown graph is about a line, and the blue one is about a constant. Now, we know that di±erentiating a quadratic (say, f ( x ) = x 2 - 2 x ) will give a line ( f ( x ) = 2 x - 2), di±erentiating a line will give a constant ( f ′′ ( x ) = 2). Thus, we see that the graphs of f, f , f ′′ are given respec- tively by the red, brown and blue graphs. How about the following problem? The graphs of
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Unformatted text preview: f and f ′ are given on the right. Which is which? Answer . There are a few tell-tale signs from which we can draw our conclusion: (a) Note that the red graph has a horizon-tal tangent when x = 0, while the blue graph is 0 when x = 0. Since the deriva-tive f ′ gives the slopes of the tangents of f , we conclude that the red graph is for f and the blue graph is for f ′ . (b) The red graph is increasing on (-2 , 0) and is decreasing on (0 , 4), while the blue graph is positive on (-2 , 0) and is nega-tive on (0 , 4). Since the derivative f ′ of f is about the rate of change of f , we see that the red graph is for f and the blue graph is for f ′ ....
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