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HARVARD MATH Xb - Final Exam Review Guide

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Math Xb Spring 2004Final Exam Review Guide1 TopicsThe final exam will cover chapters 17 through 25 (omitting §18.3 and §20.7), as well as §27.2, §15.2, §31.1,and Appendices F and G in our textbook, Calculus: An Integrated Approach to Functions and Their Ratesof Change. In particular, you will be responsible for the following topics.Chapter 17: Implicit Differentiation and Its Applications• To use logarithmic differentiation to find the derivatives of functions of the form f (x)g (x)• To use implicit differentiation to finddydxgiven an equation involving x and y• To find the slope of a tangent line to a given curve (described by an equation involving x and y) at agiven point• To find the p oints on a curve (described by an equation involving x and y) at which the tangent linehas a given slope• To understand how to use a geometric relationship between two or more variables that depend on timeto find a relationship between the rates of change of those variablesChapter 18: Geometric Sums and Series• To recognize a finite geometric sum and identify its common ratio.• To express a geometric sum in closed form.• To compute a numeric geometric sum.• To determine if a given geometric series converges or diverges.• To find the sum of a given convergent geometric series.• To express and analyze a geometric sum or se ries using summation notation.• To be able to use geometric sums and series to solve problems in a variety of contexts.Chapters 19 & 20: Trigonometry• To understand sine and cosine as functions of arc length on the unit circle.• To approximate sine, cosine, and tangent values given a calibrated unit circle.• To be familiar with the graphs of the sine, cosine, and tangent functions.• To understand the periodicity of the sine and cosine functions.• To identify the balance value, amplitude, and period of a sinusoidal function given its formula or graph.• To use trig functions to model other functions.• To understand the interpretation of tan x as the slope of a certain line.• To understand the relationship between angles and arc length.• To take advantage of circle symmetry when finding trig function values.• To understand the relationship between sine, cosine, and tangent and right triangles.• To know the sine, cosine, and tangent values ofπ6,π4, andπ3.• To “solve” triangles, that is, to determine all angles and sides of a triangle from some given information.• To understand the inverse trig functions sin−1, cos−1, and tan−1and their domains and ranges.• To simplify expressions involving inverse trig functions by using triangles.• To solve equations involving trig functions on both restricted and unrestricted domains.• To be able to apply the Law of Cosines and the Law of Sines.• To know the identities listed on the Trig Identities handout from March 10th.Chapter 21: Differentiation of Trig Functions• To know the derivatives of the six trigonometric functions.• To find derivatives of more complex functions that involve trigonometric functions.• To solve related rates problems involving trig functions.• To solve optimization problems involving trig functions.• To solve curve-sketching problems involving trig functions.• To know how one can find the derivatives of the inverse trig functions using implicit differentiation.• To know the derivatives of sin−1, tan−1, and cos−1.Appendices F and G: L’Hˆopital’s Rule and Newton’s Method• To recognize and evaluate the indeterminate forms00,∞∞, 0·∞, 1∞, ∞0, and 00using L’Hˆopital’s Rule.• To understand how Newton’s Metho d works and how it can go wrong.• To approximate roots of functions iteratively using Newton’s Method.Chapter 22: Net Change, Area, and the Definite Integral• To understand the relationship between the definite integral and the questions1. Given a rate function, how do we calculate the net change in amount?2. How do we calculate the signed area between the graph of a function and the horizontal axis?and to use these relationships to solve problems (s uch as §22.1 #1, §22.2 #5, and §22.3 #3).• To approximate a definite integral (or net change or signed area) using left- and right-hand sums.• To approximate the net change in amount given sample points on a rate function using left- andright-hand sums.• To evaluate a definite integral (or net change or signed area) using geometric area formulas.• To understand the definition of the definite integral.• To use properties of definite integrals to assist in simplifying definite integrals.Chapters 23 & 24: The Area Function and the FTC• To find the area functioncAffor a function f .• To interpret the area function as a net change function.• To know why two area functionscAfanddAfdiffer by a constant.• To determine where an area function Afis increasing, decreasing, concave up, and concave down byexamining f .• To know the Fundamental Theorem of Calculus (Versions 1 and 2).• To apply the Fundamental Theorem of Calculus to evaluate definite integrals.• To use properties of definite integrals to assist in evaluating definite integrals.• To compute the average value of a function on a given interval.• To interpret the average value of a function in a particular context (e.g., velocity or speed).Chapter 25 & §27.2: Applications and Computation of the Integral• To evaluate simple definite and indefinite integrals. (See the list on page 784.)• To evaluate more c omplex definite and indefinite integrals using algebra and/or the substitution rule.• To understand how the area between two curves can be thought of as a Reimann sum.• To find the area between two curves by integrating with resp ect to x or y.• To find the area between two curves when f (x) ≥ g(x) for some values of x and g(x) ≥ f(x) for othervalues of x.§15.2 & §31.1: Differential Equations• To use the differential equationdydt= ky and its solution y(t) = Cektto model and analyze exponentialgrowth and decay problems.• To solve the differential equationdydt= ky given an initial condition.• To interpret a given differential equation in the context of a particular application.• To write a differential equation which models a given situation.2 Suggested ExercisesChapter 17• §17.1 #4• §17.2 #1, 5• §17.3 #2, 4, 5, 7, 10• §17.4 #1, 6, 8, 10, 13Chapter 18• §18.1 #1, 4, 11, 12, 18, 24, 27, 32• §18.2 #3, 4, 6, 8, 11• §18.4 #7, 8, 11,


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