page 40 the graph of a function the graph of a function can have many features: holes, isolated...
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Page 1The 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:
a b a b a b
In figure 14-a, we say a function is increasing on the open interval of (a, b) if for any x1 and x2 in (a, b), with x1 < x2, we have f(x1) < f(x2). In layman’s 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 x1 and x2 in (a, b), with x1 < x2, we have ________. In layman’s 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 x1 and x2 in (a, b), we have _________. In layman’s 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
x1 x2
f(x1)
f(x2)
a bFigure 15
x
y
x1 x2
f(x1)
f(x2)
a bFigure 16
x
y
x1 x2
f(x2)f(x1)
a bFigure 17
Page 2Concave 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 x3 – x2 – 3x + 1 f ´(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) = x2 – 4x + 2 f ´(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 _____________
Page 3Formal Definitions of Inc./Dec. and C.U./C.D
Ex 3. f(x) = 2x – 3 f ´(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).
Page 4
f(x) f (x) Comments
Increasing Positive
Decreasing Negative
Local Extrema Zero
f(x) f (x) Comments
Concave Up Positive
Concave Down Negative
Inflection Points Zero
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) = x3 f(x) = x4
Recall the relationship between f(x) and f (x)
Recall the relationship between f(x) and f (x)
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f(x) = 1/6 x3 + 1/4 x2 – 3x – 2 f (x) =
f(x) = 1/6 x3 + 1/4 x2 – 3x – 2 f (x) =
Increasing/Decreasing and Concave Up/Concave Down