MATH 212 QUIZ # 1 REVIEW

Below is a short summary of each section that will be on your upcoming quiz. Enjoy! (1) Section 1.1: Approximating Areas. This section introduces the “area problem” and its solution: to find the area underneath a curve, first approximate that area with some boxes, and then take the limit of that estimation as the number of boxes goes to infinity. These approximations are called Riemann Sums, and are denoted as Ln or Rn depending on how many boxes you use (that’s n) and whether or not that estimation is a left-handed (Ln) or right-handed (Rn) approximation. So, R4 is a right-handed approximation using 4 boxes. See class notes and the book for more details on how to compute these quantities, this process is important. We didn’t spend a lot of time actually doing the limit of these approximations, so I wouldn’t worry about that. We also learned about summation notation, and useful summation formulas. I would be fluent in this notation. (2) Section 1.2: The Definite Integral. This section points out that there is a special name we use for the limit of these Riemann sums: the definite integral. Putting together a mathematical definition of the integral requires us, however, to interpret the number we arrive at as a “signed area”, in the sense that whenever the curve dips below the x−axis the resulting area there is “negative”. We will discuss how to compute the actual area later, but for now I would know what this definite integral notation means, and its basic properties. We also discussed computing the average value of a function over an interval, which is something I would want you to know how to do. We also discussed the mean value theorem for integrals, which will be of help in the next section. (3) Section 1.3: The Fundamental Theorems of Calculus. There are two fundamental theorems of calculus, and you need to know how to use both. The first tells you that d R x a rather oddly defined function has a very familiar derivative: dx [ a f(t)dt] = f(x). We will twist this around a bit by way of asking related questions. The second fundamental theorem of calculus tells you, basically, that all you need to compute definite integrals is an antiderivative (and then to compute something with it). Thus, finding antiderivatives will become a central feature of this course and much of it will be related to that for this reason. So I would know how to apply this theorem in a variety of straightforward ways. (4) Section 1.4: Integration formulas. This section reveals some integration formulas that will help you compute antiderivatives. I would expect to see some questions asking you to compute some relatively straightforward antiderivatives.

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