Main topics for the First Midterm Exam

Transcription

1 Main topics for the First Midterm Exam The final will cover Sections.-.0, , and 4.. This is roughly the material from first three homeworks and three quizzes, in addition to the lecture on Monday, October 7. In particular, I will expect you to know how to solve each problem from the homeworks and the quizzes. In addition, you will certainly need to know the material from 50/30s. We have covered a lot of different concepts and formulas so far. Especially if you haven t seen vectors before, you may have found this material confusing. In situations like this, it is easy to fall into the trap of simply trying to memorize steps required to solve a specific type of problem, without really understanding why you are taking these stepts. There two main issues with this approach. First, when you try memorizing a bunch of facts, you will occasionally misremember things. If you don t understand the underlying concepts, then you will have no way to realize that you have done something wrong, and so you will get the problem completely wrong, and never realize you ve made a mistake. Second, if you have only learnt to solve specific types of problems, then if I were to give you a problem that uses the same concepts as problem you ve seen before, but in a slightly different way (which I may very well do on the midterm!) you will likely be at a loss for what to do. If you frequently find that you can easily do all of the computations in a problem, but you are essentially guessing at how to get the final answer, it is likely that you have fallen into this trap. Many of the concepts we have learned so far are fairly geometric. If you want to be able to use them effectively, you must understand the geometry behind them. Of the main challenges of a problem may be to translate a geometrical situation into the language of vectors. For instance you likely know by now how to find the angle between vectors but that knowledge won t helo you unless you can look at a problem and figure out which two vectors you want to find the angle between. If you ve been largely ignoring the geometrical side of things, because you feel you have troubles visualizing things and the algebraic side of things seems easy to understand, then it is doubly important that you lear to think geometrically. If you are having trouble with something (say convergence tests), focus on doing problems, not just learning formulas. The only way to really feel this stuff is by doing problems. Your textbook has plenty of examples and exercises - try doing some of them to help you learn the material. Remember, just learning a bunch of theorems doesn t help, unless you also know how to use them (and how to notice when you should use them!!!)

2 Be sure to watch out for the following common mistakes: When computing limits, avoid imprecise arguments like n + n 3 n 3, so we can ignore n. Remember, you can usually make these arguments work by simply dividing by (what you suspect is) the largest term. Don t forget that you can divide by the largest term inside radicals as well! Remember when calculating limits, to take the whole limit at once. In particular, the folloiwng ( ) k is wrong: lim 2 k k = lim k 2 k =. Remember that if a k converges, then a k 0, but not the other way around. If it were that easy to tell if a series converges, most of what we did in the second half of the course would have been pointless. We have several different convergence tests. Make sure you can keep them all straight. When can each of them be used? What do you need to do for each one? What do they tell you? Do they tell that the series converges, or diverges? Do they just tell you that you need to consider a different series (or integral)? Also make sure you know whether you are considering a sequence or a series at each step. For instance, in the root test, are you considering the series a /k k of the limit lim? Why does it matter? k a/k k Remember that the formula you do if the sum starts at k =, k = 2, or k = 000? x k = x only works if the sum starts at k = 0. What would Remember the chain rule when taking derivatives of single variable functions (e.g. what is the derivative of f(x) n?) Remember what is a number (e.g. dot product a a, comp a b,coefficient c n = f (n) (0) n! in f(x) = c n x n ), what is a vector (e.g. cross product a b, proj a b), what is a function (e.g. f (n) (x) n! ) Remember to check the endpoints to determine the interval of convergence of power series 2

3 Make sure you are comfortable with the following: Sequences: Know what it means for a sequence to be increasing/decreasing or to be bounded (one sided or both sided) Know how to show that a sequence is increasing by either checking the ratio a n+ a n or calculating the derivative f (x) for a n = f(n). (In particular, know in which case which method to use). Know the order of increasing of the following expressions log n, n c, c n, n!. Limits of Sequences: Know what is meant by the expression lim n a n = L (both intuitively, and the ε, N- definition. Know the basic properties of limits: If a n L and b n M, then what is the limit of a n + b n, ca n, a n b n, a n /b n? Know how you can use the pinching theorem to sometimes compute more complicated limits than this. 4n Know how to find the limit of a rational function (e.g. lim 3 5n n 2n 3 7 ). Avoid making imprecise statements like 4n 3 5n 4n 3. Try dividing my the largest power of n, and using the fact that /n a 0 for a > 0. Also remember that you can often do this for functions which aren t quite rational (such as ones involving square roots of certain expressions.) You can even do this if your limit involve things like ln n or a n. Remember that ln n is smaller than n a, which is smaller than a n, which is smaller than n!. Explicitely, ln n n a 0 and na a n When trying to compute the limit of f(n) 0 and an n! 0. (for any a > 0.) g(n), identify which term is the largest, and divide the numerator and denominator by that term. That should make both the numerator and denominator have finite limits. If f is continuous, then understand why f( lim ) = lim f(a n) n n Make sure you know how to compute limits (as n ) of the following standard sequences: ( ) n x /n, x n, n, ln n a n, a na a, n xn n!, n/n and + x n Series: All of these show up constantly when talking about series. If a n is a sequence, understand what the notation isn t exactly a sum, it s a limit. diverge. a k means. Remember that this k= Know what it means for this series to converge or Know how to compute the sum of some simple series. In particular: 3

4 A geometric series x k. When does this converge? What are the partial sums? What does the series converge to? Don t just focus on memorizing the formulas for these. Make sure you can recognize these when you see them. How can you recognize a geometric series? (Hint: if a n is geometric, what can you say about a n+ a n?) What can you do if the first term of the series is not? What about if it is something of the form 2 k+3 /7 k 2? Convergence Tests: Know how to relate the series k= f(k) to the integral f(x)dx. What conditions does f need to satisfy in order to use the integral test? Remember that this isn t giving us a way to compute the series. The integral converges iff the series converges, but they aren t equal. Know how to use the integral test to figure out when remember when the p-series converges. Undestand why it doesn t matter where we start the sum i.e. k= converges. In particular, kp a k will converge iff k= a k converges. Undestand why it is okay to just write a k without including the k=j bounds. Know that if a k and b k both converge so does (a k + b k ) and ca k Know how to use the comparison test to figure out if a series converges or diverges (this is almost exactly the same as for improper integrals) Know how to use the limit comparison text. This is useful if you know what sequence you want to compare to, but it s difficult to figure out which sequence is larger. Also remember that you need to have a k /b k L where L > 0 in order to use this. Know how to use the root and ratio tests to check whether the given series converges without using another assistant series. Remember that in either of these tests, if the limit is or doesn t exist, then you can t use that test, and you need to try something esle. Also remember that n /n, this is very useful when using the root test. Don t just focus on memorizing these tests, think about how you can tell which test to apply to a given series. Does it look easy to integrate the terms? Does it look like it can be compared to a simpler series (like k )? If the terms involve only powers, it may p be simpler to work with a /k l. If the terms involve factoriels or factoriels and powers, it may be easier to work with a k+ a k. Don t forget the Divergence test. It is an easy check whether a series a k converges. Rememer that the converse is not true. What series a k diverges even though lim a k = 0? 4

5 Absolute and Conditional Convergence: Remember that most of out convergence tests only work for sequence with nonnegative terms. For general series, we need to try something else. Know what it means for a series to absolutely converge or to conditionally converge. Understand why an absolutely convergent series must converge, but a series can converge without absolutely converging. Know how to tell if a series absolutely conveges (Hint: You are now talking about a sequence with nonnegative terms.) Know how to tell if an alternating series converges. Remember that you really do need the sequence to be decreasing. Know how to use partial sums to estimate the sum of alternating series. How close is N ( ) k a k to ( ) k a k? (Alternating Series Estimation Theorem) k= k= Know when to use the Absolute Root or Ratio test and what they tell you about the absolute convergence or divergence of given series Power Series: Know how to find the set of values of x for which a k x k converges (i.e. the interval of convergence.) Understand why this set must always be an interval. Understand why knowing that a k x k converges (or diverges) at some specific x = c tells us that the series converges (or diverges) at other values of x. Understand what the radius of convergence of a power series a k x k is. Why must any power series have one? Remember that if the radius of convergence is r, the interval of convergence could be any of ( r, r), [ r, r), ( r, r], or [ r, r]. How do you determine which is the actual interval of convergence? Know how to use the Absolute root or Absolute ratio test to find the radius of convergence of a power series. If f(x) = a k x k, know how to write f (x) and f(x)dx as power series. These all have the same radius of convergence. Do they (necessarily) have the same interval of convergence? Know how to get power series of new functions using substitutions from the power series x of familiar functions: e.g. know how to get power series for from the power series 5+3x 2 y = y k (y = 3/5x 2 ). How do you need to change the interval of convergence in this case? Taylor Series: 5

6 Understand why we would expect the polynomial T n (x) = approximation to the function f(x) for x 0. f (k) (0) x k to be a good Understand why just knowing that f(x) T n (x) isn t good enough, we need to know how good an approximation this is. That is, we need to know how big the error term R n (x) (given by f(x) = T n (x) + R n (x)) is. Know the Taylor s inequality R n (x) M (n+)! x a n+ for all x a d if f (n+) (x) M for all x a d. As a tip for remembering this, this looks almost like the next term in the Taylor series. Know how to use this formula to estimate R n (x). If f is a nice function (like e x or sin x) you should be able to approximate f(x) using Taylor polynomials T n (x) by showing that R n (x) is smaller than the desired accuracy. Know how to recognize when you need to use power series to estimate a number e.g. / 0 e or sin(x 4 )dx Don t forget to show that the error term R n (X) is less than the 0 desired accuracy! n f (k) (0) Understand why the fact that f(x) x k f (k) (0) does not imply that f(x) x k. This only happens when R n (x) 0. It is possible for the series not to converge at some point where f(x) is defined. It is even possible for f(x) to be defied and for the series f (k) (0) x k to converge, but these numbers need not be equal. Know the basic Taylor series: e x x k = + x + x2 2! + x3 3! + x4 4! + = cos x = x2 2! + x4 4! x6 6! + = sin x = x x3 3! + x5 5! x7 7! + = x = + x + x2 + x 3 + x 4 + = ( ) k (2k)! x2k ( ) k (2k + )! x2k+ x k ln( + x) = x x2 2 + x3 3 x4 4 + = k= ( ) k arctan x = x x3 3 + x5 5 x7 7 + = ( ) k 2k + x2k+ and know for which values of x they converge. Know how to compute Taylor series in (x a) instead of in x (this should be almost the same as for ordinary Taylor series.) Know how to use Taylor series to compute limits, as an alternative to L Hopital s rule and how to use them to easily compute f (k) (0) for large k. 6 k x k

7 If f(x) = a k x k know why this implies that a k x k is the Taylor series of f(x) This makes it much easier to compute the Taylor series of more complicated functions, like e x2. Usually if you are asked to compute the Taylor series of some function, you should try doing something like this. If you know the basic Taylor series listed above, you should be able to compute the Taylor series of most other functions you encounter, without going through the tedious process of computing f (k) (0) for all k and then estimating the remainder. Three dimensional Coordinates (2.): Know how to use three coordinates (x, y, z) to describe a point in three dimensional space. Know what is meant by: the origin, the x, y, x axes, the xy, yx, xz planes. How can you tell if a point (x, y, z) lies on one of these things? What does it mean in terms of the x, y, z coordinates? In two dimensions, giving a single equations will (usually) describe a curve. What would a single equation describe in three dimensions? Know how to use the distance formula to find the distance between two points in three dimensions. Know how to write the equation of a sphere with a given center and radius. If you were given some equation, how would you figure out if it was the equation of a sphere? How would you find the center and radius? If f(x, y, z) = C describes a surface, what type of object would an equality like f(x, y, z) < C or f(x, y, z) > C describe? If you are given some sort of solid object in three dimensions, know how to write doesn inequalities describing it. Vectors (2.2): Understand what a vector is - it is an object with a length and a direction. How is this different from a scalar? Make sure you understand exactly what information is contained in a vector. Does it matter where the starting point of the vector is? Is tere a difference between parallel vectors of same length? What is the difference between a point and a vector? Understan geometrically what it means to add two vectors, or to take the scalar multiple of a vector. If c < 0, then what does ca mean? Know how to express a vector in terms of components. If A = (x, y, z ) and B = (x 2, y 2, z 2 ), how can you express the vector AB? Now what if v = a, b, c and v starts at A = (x, y, z )? Where is the endpoint of v? How do yoy use the components of a vector to calculate a + b, ca? What about the length of a? 7

8 Know what a unit vector is. How do you find a unit vector with the same direction as a? Know what the vectors i, j, k are. How can you express any three dimensional vector in terms of these? Dot Product (2.3): Know what the dot product is, and know how to use it to find the angle between two vectors. Know the basic properties of dot products, such as (a+b) c = a c+b c or a b = b a. Know how to show these by looking at components. Remember that a b is a scalar (i.e. a number) not another vector. Remember what the quantity a a means geometrically. What is a b when a, b are perpendicular? What about when they are parallel? Know how to use dot product to find the angle between objects other than vectors (such as lines, or graphs). Your first step should be to describe the angle between them in terms of vectors (how do you do this?) Understand the scalar projection comp a b and vector projection proj a b of b onto a. How do you compute these? What do they mean geometrically? WHat can they be used for? What is the scalar projection of a onto i, j, k? How does this relate to the components of a? Cros Product (2.4): Know what the cross product of two vectors is, and understand why it is perpendicular to a and b. We call the cross product a product a product, what common properties of a product does it satisfy? Does (a + b) c = a c + b c? Does a b = b a? Does (a b) c = a (b c)? Notice that the cross product is a different sort of object from the dot product - a b is a scalar, whereas a b is a vector. Also remember that, unline any of the other vector concepts we ve talked about, the cross product makes sense only in three dimensions. You can t thake the cross product of n-dimensional vectors. Know what the vector a a is. If you know about determinants, try to learn the 3 3 determinant formula for the cross product. If you don t know about determinants, don t worry about this. Just remember the formula for the cross product paying attention to index patterns. Know how to take the cross products of i, j, and j. Remember that i j = k, j k = i, but i k = j, not j. Instead k i = j If you have problems memorizing the formula for the cross product, know how to find the cross product of any two vectors expressing them in terms of i, j, k, and distribute. This may take longer than using the formula, but will at least give you the right answer. 8

9 We know that a b is orthogonal to a, b, but there are still two possible directions. Know how to use the right hand rule to find the direction of a b. Know how to epress a b in terms of a, b and the angle θ between a, b. What does this mean geometrically? Know how to use cross product to find the area of a parallelogram. What about a triangle? Know how to use cross product to determine if two vectors are parallel. Know how to use the triple product to find the volume of a parallelepiped, and to figure out if three vectors are coplanar. Equations of Lines and Plances (2.5): Know how to find the parametric, vector, and symmetric equations for a line through a point A, parallel to a vector v. What about the line through two points A and B? Know what it means for two lines to be parallel. What about to be skew? Know how to use cross product to find a normal vector to a plane. Know how to use the normal vector to a plane to find an equation for a plane. You should be able to use this to find the equation of a plane through three points. (What if the cross product you try to compute gives you the zero vector, what does that mean about the points you picked?) Know how to find the intersection point of a line and a plane. Know how to find the equation for intersection of two planes. Know how to use normal vectors to find the angle between two planes. When are the planes perpendicular? Parallel? Know how to find the distance from a point to a plance. What about the distance between two parallel planes? The distance between two parallel lines? Two skew lines? Functions of Mulitple Variables (4.): Know what it means for f to be a function of two variables. What about 3 or more? What does it mean to say z = f(x, y)? What does it mean to say the domain of such a function is D? What is the range? Know how to find the domain of a function like f(x, y) = x ln(y 2 x) or f(x, y, z) = x + y + z. Pay attention to what happens at the boundaries! Know what it means to graph a multivariable functions. How many dimensions do you kneed to draw the graph of f(x, y)? What about f(x, y, z)? Know what a level curve (or level surface) of a function is. Know how to use these to draw the contour map of a function. Also know how to interpret a contour map. 9