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Page 40 The Graph of a Function The graph of a function can have many features: holes, isolated points, gaps, vertical asymptotes and/or horizontal asymptotes. Nevertheless, most functions only consist only linear pieces and/or curves. If a function consists of linear pieces only, then it can only consist one (or more) of the following: ababab In figure 14-a, we say a function is increasing on the open interval of (a, b) if for any x 1 and x 2 in (a, b), with x 1 < x 2, we have f(x 1 ) < f(x 2 ). In laymans terms, if we look from left to right, the function is going up on (a, b). This is shown in figure 15 also. In figure 14-b, we say a function is decreasing on the open interval of (a, b) if for any x 1 and x 2 in (a, b), with x 1 < x 2, we have ________. In laymans terms, if we look from left to right, the function is going down on (a, b). This is shown in figure 16 also. In figure 14-c, we say a function is constant on the open interval of (a, b) if for any x 1 and x 2 in (a, b), we have _________. In laymans terms, if look from left to right, the function is a horizontal linear piece. This is shown on figure 17 also. As we can see, the difference in figures 14-a and 15 (14-b and 16) is: not only we can talk about increasing/decreasing when the function consists of linear pieces, but also when it is a curve (non-linear). Therefore, when we talk about the graph of a function (especially when it consists of curves), besides increasing and decreasing, we also talk about something else. Figure 14 (a)(b)(c) y x x y x1x1 x2x2 f(x1)f(x1) f(x2)f(x2) ab Figure 15 x y x1x1 x2x2 f(x1)f(x1) f(x2)f(x2) ab Figure 16 x y x1x1 x2x2 f(x2)f(x2)f(x1)f(x1) ab Figure 17

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Page 41 Concave Up and Concave Down In the figure on the left, we say the function is concave up since it opens up, and in the one on the right, the function is ___________ since it opens down. Before we present you the formal definition of concave up/concave down and increasing/decreasing, we will present you the graph of some functions along with the graph of their first and second derivatives: Ex 1. f(x) = 1 / 3 x 3 – x 2 – 3x + 1f ´ (x) =f ´´ (x) = f(x) is increasing on _____________ f(x) is decreasing on _____________ f(x) is concave up on _____________ f(x) is concave down on _____________ Ex 2. f(x) = x 2 – 4x + 2f ´ (x) =f ´´ (x) = f(x) is increasing on _____________ f(x) is decreasing on _____________ f(x) is concave up on _____________ f(x) is concave down on _____________

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Page 42 Formal Definitions of Inc./Dec. and C.U./C.D Ex 3. f(x) = 2x – 3f ´ (x) =f ´´ (x) = f(x) is increasing on _____________ f(x) is decreasing on _____________ f(x) is concave up on _____________ f(x) is concave down on _____________ Definition: 1.We say a function is increasing on an open interval (a, b) if f (x) > 0 (i.e., positive) for all x (a, b). 2. We say a function is decreasing on an open interval (a, b) if ______ (i.e., negative) for all x (a, b). 3. We say a function is concave up on an open interval (a, b) if f (x) > 0 (i.e., positive) for all x (a, b). 4. We say a function is concave down on an open interval (a, b) if ______ (i.e., negative) for all x (a, b).

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Page 43 f(x)f(x) f (x) Comments IncreasingPositive DecreasingNegative Local ExtremaZero f(x)f(x) f (x) Comments Concave UpPositive Concave DownNegative Inflection PointsZero Q: If for some real constant a, both f (a) = 0 and f (a) = 0, will it be a local extreme or an inflection point? A: Depends Examples: f(x) = x 3 f(x) = x 4 Recall the relationship between f(x) and f (x) f(x) = 1 / 6 x 3 + 1 / 4 x 2 – 3x – 2 f (x) = f(x) = 1 / 6 x 3 + 1 / 4 x 2 – 3x – 2 f (x) = Increasing/Decreasing and Concave Up/Concave Down

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In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs,

In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs,

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