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IB MATHEMATICS SL Course Description: IB Mathematics SL is an advanced study of mathematics, designed to prepare the student for the IB Math SL Exam and additional Calculus, either AP Calculus AB or BC. It is a rigorous course of study specifically designed for that student who expects to go on to study subjects which have significant mathematical content. The class will include a portfolio, consisting of two assignments, one on mathematical investigation and the other on mathematical modelling. The following are the 7 Areas of study with approximate recommended hours of instruction: Algebra 8 hrs Functions & Equations 24 hrs Circular functions & trigonometry 16 hrs Matrices 10 hrs Vectors 16 hrs Statistics & probability 30 hrs Calculus 36 hrs Total = 140 hrs AIMS & SUBTOPICS 1. ALGEBRAIC CONCEPTS 1.1.a Arithmetic sequences & series a n a1 (n 1)d Sn n (a1 a n ) 2 1.1.b Geometric sequences & series an a1r n 1 a (1 r n ) Sn 1 r a Sn for n , r 1 1 r 1.1.c Sigma Notation 1 1.2.a Exponents & Logarithms 1.2.b Laws of Exponents & Logarithms 1.2.c Change of Base Formula and application log b a log c a log c b 1.3.a Binomial Theorem: expansion of ( a b) , n N n 2. FUNCTIONS & EQUATIONS 2.1.a Concept of function f : x f ( x) with Domain & Range 2.1.b Composite functions ( f g )( x) f ( g ( x)) 1 2.1.c Inverse function f 2.2.a The graph of a function = f(x) 2.2.b Window settings on GDC to view both global and local behavior 2.2.c Solutions of equations graphically with “root” concept with f(x) = 0 or “intersect” concept with f(x) = g(x) 2.3.a Transformations of graphs 2.3.a.1 Vertical translations y = f(x) + b 2.3.a.2 Horizontal translations y = f(x –a) 2.3.a.3 Vertical dilations (stretches) y = pf(x) 2.3.a.4 Horizontal dilations y = f(x/q) 2.3.a.5 X-axis Reflections y = - f(x) 2.3.a.6 Y-axis Reflections y = f( -x ) 1 2.3.b Graphs of y f ( x) as a reflection of y = f(x) in the line y = x 2 2.4.a The reciprocal function y = 1/x and its selfinverse nature 2.5.a The quadratic function f(x) = ax bx c with its graph, the y-intercept (0, c) and 2 b 2a the axis of symmetry x = 2.5.b The quadratic function in the form f(x) = a ( x h) k with vertex at (h,k) 2 2.5.c The quadratic function in the form f(x) = a(x – p)(x – q) with x-intercepts at ( p, 0) and ( q, 0) 2.6.a Solutions to ax bx c 0 with use of the 2 b 4ac and the discriminant quadratic formula. 2 2.7.a The exponential function y a , a 0 and its graph 2.7.b The logarithmic function y log a x, x 0 and its graph 2.7.c Application of x log a a x x and a log a x x, x 0 2.7.d Solution of a b using logarithms x 2.8.a The exponential function y e 2.8.b The natural log function y ln x, x 0 x 2.8.c Manipulation/Application of a e with compound interest and growth/decay models x x ln a 3. CIRCULAR FUNCTIONS & TRIGONOMETRY 3.1.a Radian measure versus degree measure of an angle 3.1.b Length of an arc of a circle s r 3.1.c Area of a sector of a circle A 1 2 r 2 3 3.2.a Definitions of sin and cos in terms of the unit circle 3.2.b Definition of tan and use of y = x ( tan ) 3.2.c The Pythagorean Identity sin 2 cos 2 1and its transformations 3.3.a Double angle formulas for sin 2 and cos 2 3.4.a Domain, Range , and graphs of the basic three circular functions sin x, cos x, and tan x 3.4.b Composite functions of the form f(x) = a sin (b(x + c)) + d 3.5.a Solutions of trig equations in a restricted interval using both graphical and analytic means 3.5.b Equations of the type a sin(b(x + c)) = k with graphical interpretation 3.5.c Quadratic equations involving trig functions 3.6.a Solving Right triangle: finding all parts 3.6.b Law of Cosines 3.6.c Law of Sines and the Ambiguous Case 3.6.d The area of a triangle using A 1 ab sin C 2 with application 4. MATRICES 4.1.a Definition of a matrix with understanding of the terms “element”, “row”, “column”, and “order” 4.2.a Algebra of Matrices: Addition, Subtraction, Multiplication by a scalar 4.2.b Multiplication of matrices 4.2.c Identity and Zero matrices 4.3.a Determinant of a square matrix 4.3.b Calculation of 2 X 2 and 3 X 3 determinants 4.3.c Inverse of a 2 X 2 matrix 4.3.d Conditions for the existence of the inverse of a matrix 4 4.4.a Solution of systems of linear equations using inverse matrices with a maximum of three equations in three unknowns 5. Vectors 5.1.a Vectors in two dimensions and 3-D 5.1.b Components of a vector with column representation v1 v v2 v1i v2 j v3k with i,j, & k v3 unit vectors 5.1.c Algebraic and Geometric approaches to: 5.1.c.1 Sum and Difference of two vectors 5.1.c.2 Multiplication by a scalar, kv 5.1.c.3 Magnitude of a vector 5.1.c.4 Unit vectors, base vectors i, j , and k 5.1.c.5 Position vectors OA a 5.2.a Scalar product or dot product of two vevtors: v w v w cos ; v w v1w1 v2 w2 v3w3 5.2.b 5.2.c 5.3.a 5.3.b 5.4.a Parallel and perpendicular vectors Angle between two vectors Representation of a line r = a + tb The angle between two lines Distinguishing between intersecting and parallel lines 5.4.b Finding points where lines intersect 6. STATISTICS & PROBABILITY 6.1.a Concepts of population, sample, random sample and frequency distribution of discrete and continuous data 5 6.2.a Presentation of data including: 6.2.a.1 Frequency tables & diagrams 6.2.a.2 Box & whisker plots 6.2.a.3 Grouped data 6.2.a.4 mid-interval values 6.2.a.5 interval width 6.2.a.6 upper and lower interval boundaries 6.2.a.7 frequency histograms 6.3.a Mean, Median, Mode, Quartiles, & Percentiles 6.3.b Range, inter-quartile range, variance, and standard deviation 6.4.a Cumulative frequency and cumulative frequency graphs used to find median, quartiles, and percentiles 6.5.a Concepts of trial, outcomes, equally likely outcomes, sample space, and event 6.5.b Probability of an event P( A) n( A) n(U ) 6.5.c Complementary events A and A’ (not A); P(A) + P(A’) = 1 6.6.a Combined events, the formula: P( A B) P( A) P( B) P( A B) where P( A B) 0 for mutually exclusive events 6.7.a Conditional Probability, the formula: P( A B) P( A B) P( B) 6.7.b Independent Events, the definition: P( A B) P( A) P( A B') 6.8.a Use of Venn diagrams to solve problems 6.9.a Concept of discrete random variables and their probability distributions 6.9.b Expected value (mean), E(X) for discrete data 6.10.a Binomial distribution 6.10.b Mean of the binomial distribution 6 6.11.a Normal distribution 6.11.b Properties of the normal distribution 6.11.c Standardization of normal variables 7. CALCULUS - an informal introductory approach is used in developing the history of and utility for modern day calculus A. The derivative definitions: f(x) - f(a) ’ 1. f (a) = lim -------------x --->a x - a f(x + h) - f(x) 2. f (x ) = lim ----------------h ---> 0 h ’ B. derivatives of elementary functions C. derivatives of sums, products, and quotients D. the chain rule E. derivatives of implicitly defined functions F. related rates applications G. use of graphing calculator to explore the tangent question H. instantaneous change vs. average rate of change => the tangent vs. the secant I. Indefinite integration with application to acceleration and velocityfunctions J. Integration by substitution K. Numerical approximation techniques including the trapezoidal rule 7