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NCSU MA 242 - MA242_review_sheet_2

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MA242-012, Spring 2009Test 2 review sheetDisclaimer: “This review sheet is only intended to give you some guidance, and thus should not be taken asan exhaustive listing of possible test question topics. Do not expect this every time.”Calculators will not be allowed on this test (I’ve made some additional comments about this under some ofthe sections). No notes or other aids will be allowed on this test.The test will cover the second part of §10.2, §§10.3, 10.4, and 11.1-7.General comments: The homework should hopefully give you a good indication of that to expect onthe test in terms of content and difficulty. In general, you must show all steps and/or give sufficient justifi-cation for any answer in order to receive credit for a problem. No credit will be given for guesses.§10.2I will not directly test you on the material from this section. You still need to know how to differentiateand integrate v.v.f.s, but because these concepts will show up in other problems, there won’t be problemsdedicated exclusively to them.§10.3You need to be able to compute the arc-length of a three dimensional vector-valued function, w hich isessentially the same as the arc-length formula from calc. 2 for parametric equations with a “dzdt2” termadded under the square root. Unless I explicitly tell you not to, you will be expected to completely evaluatethe integral (which typically means manipulating w hatever is under the radical to look like some quantitysquared, then simplify and evaluate). You need to be able to compute the curvature of both a v.v.f. (usingthe formula on pg. 711) and an equation of the form y = f(x) (using the formula from page 712). You needto be able to find the principal unit normal vector, N(t) (which means you also need to remember how tofind the unit tangent vector, T(t)). You do not need to know how to find B(t).§10.4You need to know how v.v.f.s for position, velocity, and acceleration relate to each other (just differentiate orintegrate) and given one of them, be able to find the others. You need to be able to compute the tangentialand normal components of acceleration: aT, aN(formulas 9 and 10 at the bottom of pg 721 are probablygoing to be the easiest way to do this :aT=~r0(t)·~r00(t)|~r0(t)|, aN=|~r0(t)×~r00(t)||~r0(t)|). Because of the amount ofcomputation that can be involved in these problems, I will probably combine finding aT, aNwith a problemfrom §10.3 so that you can reuse previous work.§11.1The only thing you need to be able to do from this section is match a graph with its level curves/contourmap (such as 11.1.31-36, parts (b)). You will not have to match a function with its graph.1§11.2As I mentioned in class, for the multivariate limit problems, I will tell you whether the limit exists or not.If it doesn’t, then you need to find two paths that give different limits. If it does, then you need to showthat it exists, either by evaluating along one path to find out what the limit is and then using the squeezetheorem (as in example 11.2.4 and problems 11.2.11, 12), or by doing some sort of algebraic manipulationto get to the point where you can just plug the point in (as in problems 11.2.15, 16). I will not ask anyquestions about continuity.§11.3You need to be able to find any order of partial derivative s for functions of tow or three variables, and evaluateat a given point. You should know what both forms of the notation mean (e.g., fx=δfδ x, fxy=δ2fδ yδx,etc).You need to take partial derivatives in the correct order (even if certain mixed partials are equal) so I knowthat you know what the notation m eans. Since partial derivatives appear in almost everything in the rest ofchapter 11, I’m not going to ask you to just take partial derivatives of multivariate polynomials or somethingeasy like that, so be sure you’re comfortable with the more complicated and higher order homework problems(the easier functions will show up in problems from later sections).§11.4You need to know how to find an equation of the plane tangent to a surface at a given point, then how touse the linear approximation to estimate the function value at some point nearby (keep in mind that thelinearization L(x, y) can always be simplified to some thing like ax + by + c, so the arithmetic needed to findan approximation should be fairly easy). You need to know what the total differential is (for functions oftwo or three variables) and how to find it.§11.5You need to be able to use the chain rule to differentiate a function given in terms of its intermediate andindependent variables. You will be asked to draw at least one tree diagram, but you may also want to usetree diagrams on other problems if you’ve opted not to memorize the book’s “special cases”. You should beable to do implicit differentiation; the problems will be much easier if you use the formulas in this sectionrather than try to do it the way we did in section 11.3, but you may use whichever method you like.§11.6You need to be able to compute directional derivatives and gradients, and use them to find the rate ofincrease in a given direction, and/or find the direction and rate of maximum increase. I could potentiallyask you a word problem such as 11.6.29-30.§11.7Given a function f (x, y), you need to be able to find any relative maxima or minima, and any saddle pointsusing the function D. You will have to work out one problem completely from scratch, but I might also giveyou something like problem 11.7.1, where you are given all of the partial derivative values and just have toput them into D and see which case you fall


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