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Harvard - SOC - 183
SOC 183 Race and Ethnic Relations Session 9: A Comparative Perspective: Race in Latin America Dr. Seth D. HannahAgendaKey Questions Where is the Racial Divide In the U.S. today? Telles's "Race in Another America"Alternative Ways of Constructing Racial
Harvard - SOC - 183
Social Policy and EthnoEthnoRacial Inequality Ineq alitAlicia D. Simmons simmonsa@wjh.harvard.edu April 19, 2010Game Plan Group representation in American politics ReparationsWhen Affirmative Action Was WhiteInequality & Democratic Responsiveness (
Harvard - SOC - 183
SOCIOLOGY 183 Race and Ethnic Relations Session 11 The Future of the Color Line Dr. Seth D. HannahAssimilation Patterns Straightline Assimilation: "acculturation and assimilation are viewed as secular trends that culminate in the eventual absorption of
Harvard - MATH - 20
Object 1skip to main | skip to sidebarBlog ArchiveOlder Post 2006 (5) July (1) Saturday, July 01, 2006 The Dot Product and Cosine The Dot Product and Cosine June (4) Interesting The Dot Next, we'll show thatProduct product of two vectors is the produc
Harvard - MATH - 20
Answers to even-numbered problems in Mathematics for Economic AnalysisKnut Sydster Peter HammondPrefaceMathematics for Economic Analysis, Prentice Hall, 1995 has been out for a long time, and over the years we have had many request for supplying soluti
Harvard - MATH - 20
Harvard University, Math 20 Fall 2011, Instructor: Rachel Epstein1Review for Final ExamYou should review all the material for the midterms, as well as the following new material below. 1. Optimization without constraints (Chapter 17.4-9): (a) Know what
Harvard - MATH - 20
Math 20 Final Exam ReviewCarolyn Stein Exam Date: December 13, 2011Unconstrained OptimizationFinding Stationary PointsTo find the stationary points or critical points, set fx = 0, fy = 0 and find all x, y that satisfy the system of equations. A statio
Harvard - MATH - 20
Key Terms: KNOW THESE INSIDE AND OUT! Leontief Definition (p.378), Examples (Leontief worksheet, 12.1 #1,4)Linear Combination Definition (p.381), Example (12.2 #8)RREF/REF Bretscher supplement online look at examplesDot Product Definition (p.389), Exam
Harvard - MATH - 20
Name:Math 20 Fall 2010 Final Exam1(1) (10 points) For a system of linear equations in 2 variables x and y, how many solutions could it have? For each case, give an example of a system of linear equations in two variables with that many solutions.2(2)
Harvard - MATH - 20
Harvard University, Math 20 Fall 2011, Instructor: Rachel Epstein1Notes on Vector Spaces, Bases, and DimensionDefinition 0.1. A vector space is a set of vectors V Rn that satisfies the following properties: 1. If v, w are in V , then v + w is in V . 2.
Harvard - MATH - 20
Harvard University, Math 20 Fall 2011, Instructor: Rachel Epstein1Practice & Review of Single Variable CalculusFall, 2011We are about to begin our study of multi-variable calculus. Here are some single-variable calculus topics and exercises for you to
Harvard - MATH - 20
Harvard University, Math 20 Fall 2011, Instructor: Rachel Epstein1Lines and PlanesSeptember 16, 2011Find the following line and planes, using either parametric form or an equation. Let P = (1, 2, 3, 4), Q = (2, 3, 4, 4), and R = (0, 0, 2, 0). 1. Find
Harvard - MATH - 20
Harvard University, Math 20 Fall 2010, Instructor: Rachel Epstein1Review sheet 21. Vector spaces, bases, and dimension (from supplement) (a) Know the definitions of vector space, basis, and dimension. Be able to identify when something is or is not a v
Harvard - MATH - 20
Math 20 Midterm 2 ReviewCarolyn Stein Exam Date: November 9, 2011Eigenvectors and EigenvaluesWhat are Eigenvectors and Eigenvalues?"Eigen" is German for "own" or "characteristic" If A~ = ~ , we say ~ is an eigenvector of matrix A with an associated ei
Harvard - MATH - 20
Math 20 Midterm 2 Review (Solutions)Carolyn Stein Exam Date: November 9, 2011Eigenvectors and EigenvaluesWhat are Eigenvectors and Eigenvalues?"Eigen" is German for "own" or "characteristic" If A~ = ~ , we say ~ is an eigenvector of matrix A with an a
Harvard - MATH - 20
Math 20 Midterm ReviewCarolyn Stein October 4, 2011Systems of Equations and the Leontief ModelExample: Table 1: A Farming Economy Input for 1 unit Input for 1 unit Input for 1 unit of tomatoes of tomato seeds of labor 0 0.33 0.2 0.5 0 0 0.5 0.2 0Exter
Harvard - MATH - 20
Name: Linear Algebra and Multivariable Calculus Math 20 Fall 2011 Midterm 2 Please write neatly and show all your work, using proper notation. Don't hesitate to ask me questions if anything isn't clear. There are 100 points total. The point values of each
Harvard - MATH - 20
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Harvard - MATH - 20
Here is a list of topics that may be covered on the first midterm, and problems to help you prepare. The intention is not that you will do all the problems, but that you will use this to identify the areas that you need to study and try to do some problem
Harvard - MATH - 20
1. True. The constraint is a circle, which is a closed, bounded set. The EVT says the functions on closed bounded sets have a max and min. 2. False. This constraint is a parabola, which is unbounded. The EVT only holds for closed, bounded sets. 3. False.
Harvard - MATH - 1a
Math 1a 1Velocities, Secants & TangentsFall, 2009My father lives a two-and-a-half hour (150 minute) drive away. On a recent trip to visit him I recorded the trip odometer at regular intervals: Time (minutes) 0 30 60 90 120 150 Distance (km) 0 30 80 135
Harvard - MATH - 1a
Math 1a 1Limits of FunctionsFall, 2009The graph below shows the plot of a function y = f (x). Determine whether the limits shown below exist. If the given limit does exist, find this limit.. . . . . 2 . . . . . . . . . . . -2 2 4 6 8(a) lim f (x)x0
Harvard - MATH - 1a
Math 1a'The Chain Rule Another Differentiation RuleFall, 2009$The Chain Rule: Suppose F (x) = f (g(x) (or F = f g). Further suppose that g is differentiable at x and f is differentiable at g(x). Then F is differentiable at x and F (x) = f (g(x)g (x)
Harvard - MATH - 1a
Math 1a'Implicit DifferentiationFall, 2009$Often functions are defined explicitly, like y = x3 ex or y = cos(x + 1). But sometimes functions are defined implicitly, like the circle x2 +y 2 = 1. This is really two functions y = 1 - x2 and y = - 1 - x2
Harvard - MATH - 1a
Math 1a 1Derivatives & LogarithmsFall, 2009In this problem, we'll figure out the derivative of ln(x) and loga (x) (where a is a positive constant other than 1). We'll do this in the same way we found the derivatives of arcsin(x), arccos(x), and arctan(
Harvard - MATH - 1a
Math 1a'Linear Approximations Linear ApproximationFall, 2009$A linear approximation is f (a) dy y . If y = f (x), then this can be re-written as dx x or y f (a) + f (a)(x - a)y y - f (a) = x x-anear the point (x, y) = (a, f (a). This is the tangen
Harvard - MATH - 1a
Math 1a 1Related RatesFall, 2009Two cars are approaching an intersection. A red car, approaching from the north, is traveling 30 feet per second and is currently 60 feet from the intersection. A blue car, approaching from the east, is traveling 20 feet
Harvard - MATH - 1a
Math 1aMaxima and MinimaFall, 2009For each of the following functions, find the absolute maximum or minimum on the given closed interval [a, b] by following these steps: 1. Find the critical numbers of f (that is, the numbers c so that f (c) is zero or
Harvard - MATH - 1a
Math 1a 1 Show that1 2 xMVT and Two Derivative TestsFall, 2009= 5- 3 2for some x between 3 and 5.2A differentiable function f (x) satisfies f (x) 2 for all x. If f (0) = 1, what can we say about f (3)?3Show that 1 x + 1 1 + 2 x for x 0.4Suppo
Harvard - MATH - 1a
Math 1aGraphingFall, 2009Some questions to answer while graphing: (1) What is the domain of y = f (x)? (2) Where does the graph of y = f (x) cross the axes? (3) Where is the tangent to y = f (x) vertical or horizontal? (4) Where is the graph of y = f (
Harvard - MATH - 1a
Math 1a 1OptimizationFall, 2009(a) Suppose a rectangular region has fixed perimeter of 40 cm. What is the largest area the region can have?(b) Suppose now that the region was in the shape of a right triangle, not a rectangle. If the perimeter is still
Harvard - MATH - 1a
Math 1aOptimization Day TwoFall, 2009. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Harvard - MATH - 1a
Math 1a'L'H^pital's Rule o Indeterminate FormsFall, 2009$We're considering lim Typexaf (x) . We begin with several indeterminate forms: g(x)0 : lim f (x) = 0 and lim g(x) = 0 xa 0 xa Type : lim f (x) = or - and lim g(x) = or - xa xa L'H^pital's Ru
Harvard - MATH - 1a
Math 1a'L'H^pital's Rule Day Two o More Indeterminate FormsFall, 2009$Now let's consider several new indeterminate forms: Type 00 : lim f (x)g(x) with lim f (x) = 0 and lim g(x) = 0xa xa xa xaType : lim f (x)xa0g(x)with lim f (x) = and lim g(x)
Harvard - MATH - 1a
Math 1aThe Definite IntegralFall, 2009We define the definite integral of y = f (x) from x = a to x = b asb nf (x) dx = limanf (x )x ii=1where xi-1 x xi . Note that x could be xi (in which case we have the limit of Rn ) or xi-1 (in i i which case
Harvard - MATH - 1a
Math 1a'Calculating Definite Integrals The Definite IntegralIf F (x) is any anti-derivative of f (x) (that is, if F (x) = f (x), then the definite integral of f (x) from a to b isb bFall, 2009$f (x) dx = F (x)a a= F (b) - F (a).%&Compute the f
Harvard - MATH - 1a
Math 1a'It's Fundamental The Fundamental Theorem:Fall, 2009$Suppose f (x) is continuous on an interval I = [a, b].x1. If g(x) =a bf (t) dt, then g (x) = f (x).2.af (x) dx = F (b) - F (a), where F (x) is any anti-derivative of f (x) (that is, F
Harvard - MATH - 1a
Math 1a 1One Last Fundamental Theorem ProblemxFall, 2009(a) Let f (x) =0tn dt for some fixed n > 0. Find f (x).xn(b) Let g(x) =0t1/n dt for the same fixed n > 0. Find g (x).xxn(c) Let F (x) = f (x) + g(x) =0t dt +0nt1/n dt for this value
Harvard - MATH - 1a
Math 1a'SubstitutionFall, 2009$The Substitution Rule:Suppose u = g(x) is a differentiable function with domain an interval I and f (x) is continuous on I. Then f (g(x)g (x) dx = f (u) du. Thus we can make substitutions and treat du and dx like diffe
Harvard - MATH - 1a
Math 1a 31 x2 dx x3 + 1More Substitution32 xex dx2Fall, 200933dx 2 x (ln x) + 4x ln x + 4x34e x dx x1/935-1(x + 1)3 dx360sin(3x) dx3370dx (2x + 1)2/238/4cot(x) dx/61390sin3 (2x) cos(2x) dx4001 dx 1 + 4x24410x dx 1 + 4x2
Harvard - MATH - 1a
ANTIDERIVATIVESRecall that an antiderivative of f is a function whose derivative is f . For example, 1 if F (x) = x3 , then F (x) = x2 ), thus F (x) is an antiderivative of x2 . We should 3 1 notice, however, that the function G(x) = x3 + 1 also satisfie
Harvard - MATH - 1a
AREAS AND DISTANCESSuppose that we want to find the area under a curve. First of all, we need to define what the area is. If we have a rectangle, it is relatively easy, because we can simply define the area as the product of the length and the width. We
Harvard - MATH - 1a
CHAIN RULERecall the bottle calibration problem. If we increase the amount of water dripped into a bottle twice as much, then, no matter what the shape of the bottle is, the height of the water will raise twice as fast. This suggests that, if we have a c
Harvard - MATH - 1a
CONTINUITYWe have seen that the limit of a function as x approaches a can sometimes be found by calculating the value of the function at x = a. Functions with this property are called continuous at a. Mathematical definition is as follows. Continuity A f
Harvard - MATH - 1a
THE DEFINITE INTEGRALWe saw a limit of the formn nlim [f (x )x + f (x )x + + f (x )x] = lim 1 2 nnf (x )x ii=1Because this form arises frequently in a wide variety of situations, we give this type of limit a special name and notation. Definition 1.
Harvard - MATH - 1a
DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONSFor basic functions, we have differentiation rules as follows. d 1. (c) = 0. dx 2. 3. d (x) = 1. dx d n (x ) = nxn-1 . dxIf we know derivatives of certain functions, we can calculate derivatives of new
Harvard - MATH - 1a
DERIVATIVE AS A FUNCTIONIf we replace a by x in the definition of the derivative of a function f at a number a, we can get f (x + h) - f (x) f (x) = lim . h0 h So we can define a function that gives us the slope of the tangent line at each point, and we
Harvard - MATH - 1a
WHAT DOES f SAY ABOUT fSince f (x) represents the slope of the curve y = f (x), we can find the direction in which the curve proceed at each point. Thus, we can find some information about f (x) from information about f (x). In particular, we will see ho
Harvard - MATH - 1a
THE DERIVATIVEWhenever we calculate the slope of a tangent line, the velocity of an object, or any rates of change such as a rate of reaction in chemistry, a marginal cost in economics, or a population growth in biology, we encounter limits of a typexa
Harvard - MATH - 1a
EVALUATING DEFINITE INTEGRALSAlthough we could calculate some definite integrals, it was quite tedious and time-consuming. Sir Issac Newton, the creator of Calculus, found a much simpler way to evaluate definite integrals by using antiderivatives. It con
Harvard - MATH - 1a
THE FUNDAMENTAL THEOREM OF CALCULUSThe first part of the Fundamental Theorem of Calculus describes functions defined by an equation of the formxg(x) =af (t)dtwhere f is a continuous function on [a, b] and x varies between a and b.xExample 1. Let g
Harvard - MATH - 1a
Math 1a: Lecture 2September 11, 20091. The following are some odometer readings during my bike ride to Harvard: Time (hh:mm) Distance (meters) 10:00 0 10:01 610 10:02 980 10:04 1240 10:05 1250 10:06 1260 10:08 1520 10:09 1890 10:10 2500(a) What was my
Harvard - MATH - 1a
Math 1a: Limit LawsSeptember 16, 20091. Let f be a function such that f (2.01) = 0, f (2.001) = 0, f (2.0001) = 0, and so on. Can we conclude that limx2 f (x) = 0?2. Suppose we know that limx5 f (x) = 3. Which of the following must be true? (a) f (5) =
Harvard - MATH - 1a
Math 1a: ContinuitySeptember 18, 20091. At which values of x are the following functions continuous? (a) a(x) = x , the greatest integer less than or equal to x.(b) b(x) = the taxi fare (in dollars) for distance x (in miles). Assume that the meter tick
Harvard - MATH - 1a
Math 1a: Intermediate Value TheoremSeptember 21, 2009Sample problem: Prove that there is a number c such that c2 = 2. Sample good answer: Let f (x) = x2 . We have f (0) = 0 and f (2) = 4; hence f (0) < 2 < f (2). Since f (x) is continuous on [0, 2], the
Harvard - MATH - 1a
The DerivativeSeptember 28, 20091. Find the equation of the tangent line to the following functions at the given point. 1. f (x) = x3 + 4x at x = 1.2. f (x) =1 at x = 0. x+43. f (x) =x at x = 2.2. Let f (x) = x2 sin(1/x) for x = 0 and f (0) = 0. Wh
Harvard - MATH - 1a
The derivative functionSeptember 30, 20091. Let f (x) = x(x - 1)(x - 2) = x3 - 3x2 + 2x. Determine f (x) using the limit definition. Sketch f (x) and f (x), and compare the graphs.2. Let h(x) = |x - 1| + |x + 1|. On what intervals is h(x) continuous? W
Harvard - MATH - 1a
Math 1a: What does f say about f ?October 02, 20091. The following is the graph of the velocity s(t) (in m/s) of a particle as a function of time t (in s).st 42246ta. Sketch a graph of the acceleration a(t).b. Sketch the graph of the position p(
Harvard - MATH - 1a
Math 1a: Product/quotient rules and applicationsOctober 9, 20091. Determine the following derivatives. d 1. (tet ). dt2.d 3 s (s 2 ). ds3.d dpp2 . p2 + 12. Differentiate the following functions of x in two different ways. Check that your answers a
Harvard - MATH - 1a
Math 1a: Derivatives of trigonometric functionsOctober 14, 2009d d sin x = cos x and cos x = - sin x to compute dx dx c. d sec x dx1. Use a.d tan x dxb.d cot x dxd.d csc x dx2. Find the following limits. a. limh0tan h hb. limh0sin 2h sin h3.
Harvard - MATH - 1a
Math 1a: The chain ruleOctober 16, 20091. Differentiate following functions. 1. (x - 1)(x + 3)112. ex23. f (x) = 2x tan x4.x+x+x2. Some liquid is poured in a (conical) glass. Denote the volume of the liquid in the glass at time t by V (t). At wh