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OCR for page 60
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
OCR for page 61
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
OCR for page 62
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
OCR for page 63
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
OCR for page 64
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.
Representative terms from entire chapter:
differential equations
