MA242 012 Spring 2009 Test 2 review sheet Disclaimer This review sheet is only intended to give you some guidance and thus should not be taken as an 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 of the 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 on the test in terms of content and difficulty In general you must show all steps and or give sufficient justification for any answer in order to receive credit for a problem No credit will be given for guesses 10 2 I will not directly test you on the material from this section You still need to know how to differentiate and integrate v v f s but because these concepts will show up in other problems there won t be problems dedicated exclusively to them 10 3 You need to be able to compute the arc length of a three dimensional vector valued function which is 2 term essentially the same as the arc length formula from calc 2 for parametric equations with a dz dt added under the square root Unless I explicitly tell you not to you will be expected to completely evaluate the integral which typically means manipulating whatever is under the radical to look like some quantity squared then simplify and evaluate You need to be able to compute the curvature of both a v v f using the formula on pg 711 and an equation of the form y f x using the formula from page 712 You need to be able to find the principal unit normal vector N t which means you also need to remember how to find the unit tangent vector T t You do not need to know how to find B t 10 4 You need to know how v v f s for position velocity and acceleration relate to each other just differentiate or integrate and given one of them be able to find the others You need to be able to compute the tangential and normal components of acceleration aT aN formulas 9 and 10 at the bottom of pg 721 are probably 0 0 t r 00 t r 00 t aN r t Because of the amount of going to be the easiest way to do this aT r r 0 t r 0 t computation that can be involved in these problems I will probably combine finding aT aN with a problem from 10 3 so that you can reuse previous work 11 1 The only thing you need to be able to do from this section is match a graph with its level curves contour map such as 11 1 31 36 parts b You will not have to match a function with its graph 1 11 2 As 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 show that it exists either by evaluating along one path to find out what the limit is and then using the squeeze theorem as in example 11 2 4 and problems 11 2 11 12 or by doing some sort of algebraic manipulation to get to the point where you can just plug the point in as in problems 11 2 15 16 I will not ask any questions about continuity 11 3 You need to be able to find any order of partial derivatives for functions of tow or three variables and evaluate 2 f at a given point You should know what both forms of the notation mean e g fx f x fxy y x etc You need to take partial derivatives in the correct order even if certain mixed partials are equal so I know that you know what the notation means Since partial derivatives appear in almost everything in the rest of chapter 11 I m not going to ask you to just take partial derivatives of multivariate polynomials or something easy 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 4 You need to know how to find an equation of the plane tangent to a surface at a given point then how to use the linear approximation to estimate the function value at some point nearby keep in mind that the linearization L x y can always be simplified to something like ax by c so the arithmetic needed to find an approximation should be fairly easy You need to know what the total differential is for functions of two or three variables and how to find it 11 5 You need to be able to use the chain rule to differentiate a function given in terms of its intermediate and independent variables You will be asked to draw at least one tree diagram but you may also want to use tree diagrams on other problems if you ve opted not to memorize the book s special cases You should be able to do implicit differentiation the problems will be much easier if you use the formulas in this section rather than try to do it the way we did in section 11 3 but you may use whichever method you like 11 6 You need to be able to compute directional derivatives and gradients and use them to find the rate of increase in a given direction and or find the direction and rate of maximum increase I could potentially ask you a word problem such as 11 6 29 30 11 7 Given a function f x y you need to be able to find any relative maxima or minima and any saddle points using the function D You will have to work out one problem completely from scratch but I might also give you something like problem 11 7 1 where you are given all of the partial derivative values and just have to put them into D and see which case you fall into 2