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5.1 concavity.pdf

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Dan Dolderman

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Calc 1 Fri 10/17/08 5.1 - Analysis of Functions Okay, let’s analyze some functions! A few of these things we’ve gone over before, but we’re going to go over them again, because they are very important in this chapter. Let f be a continuous function on [a,b], and diff on (a,b). Then if f is Increasing on [a,b]: f (x) > 0 on (a,b). ′ Decreasing on [a,b]: f (x) < 0 on (a,b). Constant on [a,b]: f(x) = 0 on (a,b). 1 3 2 Example: Find the intervals on which f(x) = x − 2x 3 3x − 4 is increasing and decreasing. Solution: f (x) = x − 4x + 3 = (x − 3)(x − 1). The idea with this type of problem is to see when ′ ′ f (x) = 0. The only times f (x) will change signs (because that’s what we’re lookinig for) are when f (x) = 0 or undefined. Well, this is a polynomial, which means there are no vertical ′ asymptotes or any other problems, so we should be fine! So, f (x) = 0 when x = 1,3. Now all we need to do is check the signs! +++++++ 1 −−−−−−− 3 +++++++ 0 2 4 All I did here was plug in 0, 2, and 4 into f (x) and looked to see if they were positive or negative. ′ Remember, f (x) positive means f is increasing. So, f is increasing on the intervals (−∞,1] and on [3,∞). Meanwhile, f is decreasing on the interval [1,3]. Example: Find the intervals on which f(x) = x +x+7 is increasing and decreasing. x−1 Solution: Well, let’s take that derivative again! (x − 1)(2x + 1) − (x + x + 7) x − 2x − 8 f (x) = = (x − 1)2 (x − 1)2 And, well, the top factors into (x + 2)(x − 4), so that’s when f (x) = 0, but we also can’t forget about when the derivative is not defined (VA’s), because sign changes may in fact occur there! This would be when x = 1. So, let’s do that number line again! +++++++ −2 −−−−−−− 1 −−−−−−− 4 +++++++ −3 0 2 5 So, f is increasing on the intervals (−∞,−2] ∪ [4,∞) and decreasing on (−2,1) ∪ (1,4). Notice how we still have to throw out x = 1 from that interval of decreasing because it’s not in our original domain of f(x)! 1 Concavity: We shall now take a closer look at the second derivative. The concavity of a function is the study of its curvature. There are two basic types of concavity: concave up and concave down. To determine if a function f is concave up or down, we just take a look at the sign of the second derivative at that point. If f (x) > 0, then f is concave up (usually abbreviated CCU) If f (x) < 0, then f is concave down (usually abbreviated CCD) The only times that f can change concavity is when f (x) = 0 or f (x) is not defined (ie, VA). ′′ If f changes concavity at x an0 f (x ) = 0, then x is cal0ed an inflection point. What does concave up look like? It looks like a giant smiley face. It’s happy to be concave up. Or, if you want to think in terms of functions, it basically looks like y = x . Notice that a function can be both increasing and decre
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