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chapter5Notes - Chapter 5 What the Derivative tells us...

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Chapter 5 What the Derivative tells us about a function The derivative of a function contains a lot of important information about the behaviour of a function. In this chapter we will focus on how properties of the first and second derivative can be used to help us refine curve-sketching techniques. Most of the material in this chapter places an emphasis on analytic approaches to understanding the behaviour of functions. By analytic, we mean “pen and paper” computations of derivatives and solving for special points on the curves. Other, complementary approaches to understanding functions are the geometric and/or the computational approaches. Most students will have learned how to use graphics calculators to plot functions, and would thus be familiar with one type of “computational approach”. Alongside the lectures in this course, the student is also exposed to a second computational approach, based on numerical differentiation. This appears in some of the lab assignments 1 . 5.1 The shape of a function: from f prime ( x ) and f primeprime ( x ) Consider a function given by y = f ( x ). We first make the following observations: 1. If f prime ( x ) > 0 then f ( x ) is increasing . 2. If f prime ( x ) < 0 then f ( x ) is decreasing . Naturally, we read graphs from left to right, i.e. in the direction of the positive x axis, so when we say “increasing” we mean that as we move from left to right, the value of the function gets larger. We can use the same ideas to relate the second derivative to the first derivative. 1. If f primeprime ( x ) > 0 then f prime ( x ) is increasing . This means that the slope of the original function is getting steeper (from left to right). The function curves upwards: we say that it is concave up . See Figure 5.1(a). 1 In particular, the completed assignments for Lab 3 can also be used to plot functions discussed in this chapter, and to display their derivatives on the same graph. This could be used as a check or an additional tool for understanding the connection between the functions and their first (and higher) derivatives. v.2005.1 - September 4, 2009 1
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Math 102 Notes Chapter 5 x x f(x) f(x) (a) (b) x x f ’ (x) f ’ (x) Figure 5.1: In (a) the function is concave up, and its derivative thus increases (in the positive direction). In (b), for a concave down function, we see that the derivative decreases. 2. If f primeprime ( x ) < 0 then f prime ( x ) is decreasing . This means that the slope of the original function is getting shallower (from left to right). The function curves downwards: we say that it is concave down . See Figure 5.1(b). We see examples of the above two types in Figure 5.1. In Figure 5.1(a), f ( x ) is concave up, and its second derivative (not shown) would be positive. In Figure 5.1(b), f ( x ) is concave down, and second derivative would be negative. To summarize, the second derivative of a function provides information about the curvature of the graph of the function, also called the concavity of the function.
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