IB Math SL Topics for Review

IB Mathematics SL Topics    1   a 1.1   r    b   e 1.2   g    l    A 1.3    2   s   n   o    i    t   a   u   q   e    d   n   a   s   n   o    i    t   c   n   u    F

2.1 2.2

2.3 2.4 2.5

2.6 2.7 2.8

   3   i   g   r    t    &   s   n   o    i    t   c   n   u    f   r   a    l   u   c   r    i    C

3.1 3.2 3.3 3.4

3.5

3.6

   4   s 4.1   e   c    i   r 4.2    t   a    M 4.3

4.4

  s 5.1    5   r   o    t   c   e    V

arithmetic sequences geometric sequences exponents logarithms the binomial theorem

sums of finite arithmetic sequences sums of infinite geometric series laws of exponents laws of logarithms expansion of ( a + b)n

sigma notation

the concept of function  f : x # f ( x  x) the graph of a function its equation  y = f ( x)

domain, range composite functions function graphing skills use of graphing calculator

transformations transformations of graphs: translations

stretches reflections its graph

identity function inverse function  f –1 investigation of key features of  graphs (incl. asymptotes) solving of equations graphically –1  f  as reflection in  y = x

The reciprocal function  x #

1

change of base Pascal’s triangle

its self-inverse nature

 x

the quadratic function 2  x # ax + bx + c its graph

its y-intercept axis of symmetry  x = −

b

2a

the solution of ax2 + bx + c = 0 the exponential function  x # a x with a > 0 the exponential function  x # e x

the quadratic formula its inverse function  x # loga x for  x > 0 the logarithmic function  x # ln x for x > 0

the circle radian measure of angles definition of cos θ  and sin θ  in terms of the unit circle

length of an arc

double angle formulae: the circular functions sin  x, cos x, and tan x their domains and ranges solution of trigonometric equations in a finite interval

tan θ  as

sin θ  

cosθ   sin 2θ  = 2 sin θ  cos θ   their periodic nature their graphs includes those of the type a sin( b( x + c)) = k 

solution of triangles the law of cosines

the law of sines

definition of a matrix

the terms element , row, column, and order  multiplication multiplication by a scalar multiplication multiplication of matrices

algebra of matrices: equality addition, subtraction determinant of a square matrix calculation of 2 × 2 and 3 × 3 determinants solution of systems of linear equations using inverse matrices vectors as displacements in the plane vectors as displacements in three dimensions components of a vector column representation

the form x # a( x – h)2 + k with vertex (h, k )  x – q) with the form x # a( x – p)( x  x-intercepts ( p, 0) and (q, 0) use of the discriminant ∆ = b2 – 4ac their graphs solution of  a x = b using logarithms

area of a sector the identity cos 2θ  + sin2θ  = 1 cos 2θ  = cos2θ  – sin2θ   composite functions of the form  f ( x) = a sin(b( x + c)) + d  quadratic equations in terms of  trigonometric functions graphical interpretation 1 the area of a triangle as ab sin C  2

identity matrices zero matrices

inverse of a 2 × 2 matrix conditions for the inverse of a matrix (maximum of three equations in three unknowns)

finding the inverse of a 3 × 3 matrix with a graphing calculator

 v1    v =  v2  = v1i + v2 j + v3k v   3

the vector – v multiplication multiplication by a scalar k v the magnitude of a vector v

algebraic and geometric approaches to the sum and difference of  vectors the zero vector

unit vectors base vectors i, j,  j, k      

position vectors OA = a   

  

  

 AB = OB − OA = b − a

5.2    5   s   r   o    t   c   e 5.3    V

parallel vectors the scalar product of two vectors

v · w = v w cos θ  

representation of a line as r =  a + t  b distinguishing between coincident and parallel lines

the angle between two lines

concepts of population, sample, random sample presentation of data as frequency tables presentation of data as diagrams mean, median, mode quartiles, percentiles cumulative frequency

frequency distribution of discrete and continuous data presentation of data as box and whisker plots grouped data mid-interval values range interquartile range cumulative frequency graphs

6.5

concepts of trial, outcome, equally likely outcomes sample space ( U ) and event

the probability of an event  A as n( A) P( A) = n(U )

6.6

Combined events

P( A ∪ B) = P( A) + P( B) – P( A ∩ B)

6.7

conditional probability

5.4

  y    6   t    i    l    i    b   a    b   o   r   p    d   n   a   s   c    i    t   s    i    t   a    t    S

6.1 6.2

6.3 6.4

perpendicular vectors the angle between two vectors

= v1w1 + v2w2 + v3w3

finding points where lines intersect

definition P ( A | B ) =

P ( A ∩ B) P ( B)

6.8

use of Venn diagrams

use of tree diagrams

6.9

concept of discrete random variables binomial distribution normal distribution

their probability distributions

6.10 6.11

   7   s 7.1   u    l   u   c    l   a    C 7.2 7.3 7.4

7.5

7.6 7.7

informal ideas of limit and convergence n derivatives of  x , sin x, cos x,  x tan x, e , and ln x differentiation of sums and real multiples of simpler functions local maximum and minimum points indefinite integration as antidifferentiation antidifferentiation with a boundary condition to determine the constant term kinematics problems graphical behavior of functions: tangents and normals behavior for large  x horizontal and vertical asymptotes

mean of the binomial distribution properties of the normal distribution definition of derivative as   f ( x + h) − f ( x)   f ′( x) = lim   h →0  h  the chain rule for composite functions use of first and second derivative in optimization problems n indefinite integrals of  x , sin x, 1  x cos x, ,e  x definite integrals

displacement ( s), velocity (v), and acceleration ( a) the significance of the second derivative distinction between maximum and minimum points

interval width upper and lower interval boundaries frequency histograms variance standard deviation use to find median, quartiles, percentiles the complementary events  A and A′ P( A) + P( A′) = 1 P( A ∩ B) = 0 for mutually exclusive events independent events definition P( A |B) = P( A) = P( A |B′)

use of tables of outcomes to solve problems expected value (mean)  E ( X ) for discrete data standardization of normal variables derivative interpreted as gradient (slope) function derivative interpreted as rate of  change the product and quotient rules the second derivative

the composites of these with the linear function ax + b areas under and between curves volumes of revolution

points of inflexion (inflection) with zero and non-zero gradients (slopes)