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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics Box 2-3 SYLLABUS OUTLINE MATHEMATICS – HL The mathematical methods standard level programme consists of the study of eight core topics and one option. PART I: Core 105 hours All topics in the core are compulsory. Candidates are required to study all the sub-topics in each of the six topics in this part of the syllabus as listed in the Syllabus Details. 1 Number and Algebra 20 hours 2 Functions and Equations 25 hours 3 Circular Functions and Trigonometry 25 hours 4 Vector Geometry 25 hours 5 Matrices and Transformations 20 hours 6 Statistics 10 hours 7 Probability 20 hours 8 Calculus 50 hours PART II: Options 35 hours Candidates are required to study all the sub-topics in one of the following options as listed in the Syllabus Details. 9 Statistics 35 hours 10 Sets, Relations and Groups 35 hours 11 Discrete Mathematics 35 hours 12 Analysis and Approximation 35 hours 13 Euclidean Geometry and Conic Sections 35 hours PORTFOLIO 10 hours Five assignments, based on different areas of the syllabus, representing the following three activities: mathematical investigation extended closed-problem solving mathematical modeling mathematical research SOURCE: IB Mathematics - HL, September 1997
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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics Box 2-4 SYLLABUS OUTLINE FURTHER MATHEMATICS – SL The mathematical methods standard level programme consists of the study of six core topics and one option. PART I: Core 105 hours All topics are compulsory. Candidates are required to study all the sub-topics in each of the five topics in this part of the syllabus as listed in the Syllabus Details. As these five topics are identical to the options in part II of the mathematics HL syllabus, it will be presumed that candidates will have studied one of these topics as part of that programme. Consequently, this portion of the further mathematics SL programme is regarded as having a total teaching time of 140 hours. 1 Statistics 35 hours 2 Sets, Relations and Groups 35 hours 3 Discrete Mathematics 35 hours 4 Analysis and Approximation 35 hours 5 Euclidean Geometry and Conic Sections 35 hours PORTFOLIO 10 hours Three assignments, based on different areas of the syllabus, representing at least two of the following four activities: mathematical investigation extended closed-problem solving mathematical modeling mathematical research SOURCE: IB Mathematical Methods SL, September 1997
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554 g CONTENT PANEL REPORT + Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors. function. · Optimization, both absolute (global) and relative (local) extreme. · Modeling rates of change, including related rates problems. · Use of implicit differentiation to find the derivative of an inverse · Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. + Geometric interpretation of differential equations via slope fields and the relationship between slope fields and derivatives of implicitly defined functions. + Numerical solution of differential equations using Euler's method. + L'Hopital's Rule and its use in determining convergence of improper integrals and series. Computation of derivatives. · Knowledge of derivatives of basic functions, including power, expo- nential, logarithmic, trigonometric, and inverse trigonometric functions. · Basic rules for the derivative of sums, products, and quotients of functions. · Chain rule and implicit differentiation. + Derivatives of parametric, polar, and vector functions. m. Integrals Interpretations and properties of definite integrals. · Computation of Riemann sums using left, right, and midpoint evalu- ation points. · Definite integral as a limit of Riemann sums over equal subdivisions. · Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: Ib f'(x)dx = f (b) - f (a) a · Basic properties of definite integrals. (Examples include additivity and linearity.) *Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the empha- sis is on using the integral of a rate of change to give accumulated change or
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MATHEMATICS using the method of setting up an approximating Riemann sum and repre- senting its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve (including a curve given in parametric form). Fundamental theorem of calculus. · Use of the Fundamental Theorem to evaluate definite integrals. · Use of the Fundamental Theorem to represent a particular antideriva- tive, and the analytical and graphical analysis of functions so defined. Techniques of antidifferentiation. · Antiderivatives following directly from derivatives of basic functions. + Antiderivatives by substitution of variables (including change of lim- its for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only). + Improper integrals (as limits of definite integrals). Applications of antidifferentiation. · Finding specific antiderivatives using initial conditions, including ap- plications to motion along a line. · Solving separable differential equations and using them in modeling. In particular, studying the equation y' = by and exponential growth. + Solving logistic differential equations and using them in modeling. Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values. *IV. Polynomial Approximations and Series *Concept of series. A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence. *Series of constants. + Motivating examples, including decimal expansion. Geometric series with applications. + The harmonic series. ~ Alternating series with error bound. 555
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556 .~ CONTENT PANEL REPORT + Terms of series as areas of rectangles and their relationship to im- proper integrals, including the integral test and its use in testing the conver- gence of ~series. + The ratio test for convergence and divergence. + Comparing series to test for convergence or divergence. *Taylor series. + Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.) + Maclaurin series and the general Taylor series centered at x= a. + Maclaurin series for the functions en, sin x, cos x, and 1 / (1 - x). + Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the for- mation of new series from known series. + Functions defined by power series. + Radius and interval of convergence of power series. + Lagrange error bound for Taylor polynomials. SOURCE: College Entrance Examination Board (1999a, 1999c, pp. 9-141.
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