Math+SL+Objectives

Syllabus content Topic 1—Algebra 8 hrs Aims The aim of this section is to introduce students to some basic algebraic concepts and applications. Number systems are now in the presumed knowledge section. Details © International Baccalaureate Organization 2006 12  Content Amplifications/inclusions Exclusions 1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Sigma notation. Examples of applications, compound interest and population growth. Exponents and logarithms. Elementary treatment only is required. Examples: 3 16  4   =   8   ;   16    3  log 8 4 = ; log32  =  5log 2   ; (2 3   )   4   2   12   −   =   −   .    Laws of exponents; laws of logarithms.  1.2 Change of base. log log log  c b  c  a a  b  =.  1.3 The binomial theorem: expansion of    (//a  //  +  //b  //  )  //n  // ,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">n  //  ∈   􀁠. On examination papers: students may determine the binomial coefficients, <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> n r  ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ,  by using Pascal’s triangle, or by using a GDC. The formula ! !! <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> n n r r n r  ⎛ ⎞ ⎜ ⎟ = −   ⎝ ⎠    and consideration of combinations. Topic 2—Functions and equations 24 hrs Aims The aims of this section are to explore the notion of function as a unifying theme in mathematics, and to apply functional methods to a variety of mathematical situations. It is expected that extensive use will be made of a GDC in both the development and the application of this topic. Details © International Baccalaureate Organization 2006 13 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Concept of function //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f //  :  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  􀀶  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ) : domain, range; image (value). On examination papers: if the domain is the set of real numbers then the statement “ //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  ∈   􀁜   ” will be omitted. Formal definition of a function; the terms “one-to-one”, “many-to-one” and “codomain”. Composite functions //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f //  􀁄  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">g  //  ; identity function. The composite function ( //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  􀁄  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">g  //  )(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ) is defined as   <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> f (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">g  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )). <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 2.1 Inverse function //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  −   1. On examination papers: if an inverse function is   to be found, the given function will be defined with a domain that ensures it is one-to-one. Domain restriction. The graph of a function; its equation //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ). On examination papers: questions may be set requiring the graphing of functions that do not explicitly appear on the syllabus. The linear function //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">ax //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  //  is now in the presumed knowledge section. Function graphing skills: use of a GDC to graph a variety of functions; investigation of key features of graphs. Identification of horizontal and vertical asymptotes. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 2.2 Solution of equations graphically. May be referred to as roots of equations, or zeros of functions. Topic 2—Functions and equations (continued) © International Baccalaureate Organization 2006 14 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions 2.3 Transformations of graphs: translations; stretches; reflections in the axes. Translations: //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )   +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  //  ;  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  −  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //  ). Stretches: //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">pf  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ) ;  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x q  //  ). Reflections (in both axes): <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> y = −  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ) ;  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  (   −  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ). Examples: //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  2   used to obtain  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  =   3  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  2   +   2 by    a stretch of scale factor 3 in the //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  -direction followed by a translation of 0 2 ⎛ ⎞  ⎜ ⎟  ⎝ ⎠  . <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> y =   sin  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  used to obtain  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  =   3sin 2  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  by a stretch of scale factor 3 in the //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  -direction and a stretch of scale factor 1 2 in the //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  -direction. The graph of //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  −   1   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ) as the reflection in the line //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  of the graph of  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ). <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 2.4 The reciprocal function //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  1 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> x 􀀶 ,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ≠   0 : its graph; its self-inverse nature. Topic 2—Functions and equations (continued) © International Baccalaureate Organization 2006 15 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions The quadratic function //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  􀀶  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">ax  //  2   +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">bx  //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">c  //  : its graph, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  -intercept (0,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">c  //  ). Rational coefficients only. Axis of symmetry 2 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> x b a  = − //. // The form //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  􀀶  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  −  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">h  //  )   2   +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">k  //  : vertex (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">h  // ,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">k  //  ). <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 2.5 The form //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  􀀶  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  −  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">p  //  )(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  −  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">q  //  ) : <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> x -intercepts (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">p  //  ,0) and (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">q  //  ,0). The solution of //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">ax //  2   +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">bx  //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">c  //  =   0,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //  ≠   0. On examination papers: questions demanding elaborate factorization techniques will not be set. The quadratic formula. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 2.6 Use of the discriminant  Δ =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  //  2   −   4  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">ac  //. The function: //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  􀀶  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  // ,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //  >   0. The inverse function log//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  􀀶  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  // ,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  >   0. log //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  // a<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ;  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //  log  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  // ,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  >   0. Graphs of //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  and log  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 2.7 Solution of //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  //  using logarithms. The exponential function //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  􀀶   e  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //. The logarithmic function //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  􀀶   ln  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  // ,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  >   0. //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">e  <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ln  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 2.8 Examples of applications: compound interest, growth and decay. Topic 3—Circular functions and trigonometry 16 hrs Aims The aims of this section are to explore the circular functions and to solve triangles using trigonometry. Details © International Baccalaureate Organization 2006 16 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions 3.1 The circle: radian measure of angles; length of    an arc; area of a sector. Radian measure may be expressed as multiples of π  , or decimals. Definition of cos θ   and sin   θ   in terms of the unit circle. Given sin θ  , finding possible values of    cos  θ   without finding   θ. The reciprocal trigonometric functions sec θ   , csc θ   and cot   θ. Definition of tan θ   as sin cos θ θ . Lines through the origin can be expressed as <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> y =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  tan   θ  , with gradient tan   θ. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 3.2 The identity cos  2   θ   +   sin   2   θ   =   1. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 3.3 Double angle formulae: sin 2   θ   =   2sin   θ   cos   θ   ; cos 2 θ   =   cos   2   θ   −   sin   2   θ. Double angle formulae can be established by simple geometrical diagrams and/or by use of a  GDC. Compound angle formulae. The circular functions sin //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x // , cos  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  and tan  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  : their domains and ranges; their periodic nature; and their graphs. On examination papers: radian measure should be assumed unless otherwise indicated by, for example, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  􀀶   sin  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ｰ. The inverse trigonometric functions: arcsin //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  , arccos //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  and arctan  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //. Composite functions of the form <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> f (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )   =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //  sin(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">c  //  ))   +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">d  //. Example: //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )   =   2cos(3(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  −   4))   +   1. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 3.4 Examples of applications: height of tide, Ferris wheel. Topic 3—Circular functions and trigonometry (continued) © International Baccalaureate Organization 2006 17 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Solution of trigonometric equations in a finite interval. Examples: 2sin //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  =   3cos  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  // , 0   ≤  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ≤   2   π. 2sin 2//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  =   3cos  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  // , 0   o   ≤  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ≤   180   o. 2sin //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  =   cos2  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  // ,   −   π   ≤  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ≤   π. Both analytical and graphical methods required. The general solution of trigonometric equations. Equations of the type //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a //  sin(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">c  //  ))   =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">k  //. Equations leading to quadratic equations in, for example, sin //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 3.5 Graphical interpretation of the above. Solution of triangles. The cosine rule: //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">c //  2   =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a  //  2   +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  //  2   −   2  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">ab  //  cos  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">C  //. Appreciation of Pythagoras’ theorem as a   special case of the cosine rule. The sine rule: sin sin sin <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> a b c A B C  = =. The ambiguous case of the sine rule. Area of a triangle as 1 sin 2 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> ab C. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 3.6 Applications to problems in real-life situations, such as navigation. Topic 4—Matrices 10 hrs Aims The aim of this section is to provide an elementary introduction to matrices, a fundamental concept of linear algebra. Details © International Baccalaureate Organization 2006 18 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions 4.1 Definition of a matrix: the terms “element”, “row”, “column” and “order”. Use of matrices to store data. Use of matrices to represent transformations. Algebra of matrices: equality; addition; subtraction; multiplication by a scalar. Matrix operations to handle or process information. Multiplication of matrices. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 4.2 Identity and zero matrices. Determinant of a square matrix. Elementary treatment only. Calculation of 2   ﾗ   2   and 3   ﾗ   3determinants. Cofactors and minors. Inverse of a 2 ﾗ   2 matrix. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 4.3 Conditions for the existence of the inverse of a  matrix. Obtaining the inverse of a 3 ﾗ   3 matrix using a    GDC. Other methods for finding the inverse of a 3 ﾗ   3 matrix. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 4.4 Solution of systems of linear equations using inverse matrices (a maximum of three equations in three unknowns). Only systems with a unique solution need be considered. Topic 5—Vectors 16 hrs Aims The aim of this section is to provide an elementary introduction to vectors. This includes both algebraic and geometric approaches. Details © International Baccalaureate Organization 2006 19 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Vectors as displacements in the plane and in three dimensions. Distance between points in three dimensions. Components of a vector; column representation. 1 2 1 2 3  3 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> v v v v v  v  ⎛ ⎞ = ⎜ ⎟   = + +   ⎜ ⎟    ⎜ ⎟  ⎝ ⎠ <span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;"> v i j k. Components are with respect to the unit vectors <span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;"> i,  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">j  //**  , and  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">k  //**  (standard basis). Algebraic and geometric approaches to the following topics: the sum and difference of two vectors; the zero vector, the vector −  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v  //**  ; The difference of **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v //**  and  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">w  //**  is  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v  //**  −  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">w  //**  =  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v  //**  +   (   −  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">w  //**  ). multiplication by a scalar, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">k **<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v  **//  ; magnitude of a vector, **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v //**  ; unit vectors; base vectors **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">i //** ,  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">j  //**  , and  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">k  //**  ; <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 5.1 position vectors OA  → = **// a //**. AB OB OA  → → → = − = **// b //** −  **// a //**. Topic 5—Vectors (continued) © International Baccalaureate Organization 2006 20 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions The scalar product of two vectors <span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;"> v ⋅  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">w  //**  =  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v w  //**  cos   θ   ;   1 1 2 2 3 3  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v  //**  ⋅  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">w  //**  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">v w  //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">v w  //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">v w  //. The scalar product is also known as the “dot product” or “inner product”. Projections. Perpendicular vectors; parallel vectors. For non-zero perpendicular vectors **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v //**  ⋅  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">w  //**  =   0 ; for non-zero parallel vectors **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v //**  ⋅  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">w  //**  = ｱ  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">v w  //**. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 5.2 The angle between two vectors. Representation of a line as **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">r //**  =  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">a  //**  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">t  **<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">b  **//. Lines in the plane and in three-dimensional space. Examples of applications: interpretation of //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">t //  as time and  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">b  //**  as velocity, with  **//<span style="font-family: TimesNewRoman,BoldItalic;"><span style="font-family: TimesNewRoman,BoldItalic;">b  //** representing speed. Cartesian form of the equation of a line: 0 0 0 //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x x y y z z // l m n − = − = −. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 5.3 The angle between two lines. Distinguishing between coincident and parallel lines. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 5.4 Finding points where lines intersect. Awareness that non-parallel lines may not intersect. Topic 6—Statistics and probability 30 hrs Aims The aim of this section is to introduce basic concepts. It may be considered as three parts: descriptive statistics (6.1–6.4), basic probability (6.5–6.8), and modelling data (6.9–6.11). It is expected that most of the calculations required will be done on a GDC. The emphasis is on understanding and interpreting the results obtained. Details © International Baccalaureate Organization 2006 21 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions 6.1 Concepts of population, sample, random sample and frequency distribution of discrete and continuous data. Elementary treatment only. Presentation of data: frequency tables and diagrams, box and whisker plots. Treatment of both continuous and discrete data. Grouped data: mid-interval values, interval width, upper and lower interval boundaries, <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.2 frequency histograms. A frequency histogram uses equal class intervals. Histograms based on unequal class intervals. Topic 6—Statistics and probability (continued) © International Baccalaureate Organization 2006 22 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Mean, median, mode; quartiles, percentiles. Awareness that the population mean, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">μ // , is    generally unknown, and that the sample mean, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  , serves as an estimate of this quantity. Estimation of the mode from a histogram. Range; interquartile range; variance; standard deviation. Awareness of the concept of dispersion and an understanding of the significance of the numerical value of the standard deviation. Obtaining the standard deviation (and indirectly the variance) from a GDC is expected. Other methods for finding the standard deviation or variance. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.3 Awareness that the population standard deviation, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">σ // , is generally unknown, and that the standard deviation of the sample, //<span style="font-family: TimesNewRoman,Italic;">s <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">n // , serves as    an estimate of this quantity. Discussion of bias of 2 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> n <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">s  as an estimate of  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">σ  // 2 .   <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.4 Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles. Concepts of trial, outcome, equally likely outcomes, sample space (//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">U //  ) and event. The probability of an event //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A //  as P    <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> A n A n U  =. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.5 The complementary events //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  and  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  ′   (not  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  ); P(//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A //  )   +   P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  ′   )   =   1. Topic 6—Statistics and probability (continued) © International Baccalaureate Organization 2006 23 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Combined events, the formula: P(//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A //  ∪  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  )   =   P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  )   +   P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  )   −   P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  ∩  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  ). <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.6 Appreciation of the non-exclusivity of “or”. P (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  ∩  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  )   =   0 for mutually exclusive events. Use of P( //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  ∪  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  )   =   P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  )   +   P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  ) for mutually exclusive events. Conditional probability; the definition P (   |   )   P    P <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> A B A B B  = ∩. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.7 Independent events; the definition P (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  |  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  )   =   P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  )   =   P   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  |  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  ′   ). The term “independent” is equivalent to “statistically independent”. Use of P(//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  ∩  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  )   =   P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">A  //  )P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">B  //  ) for independent events. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.8 Use of Venn diagrams, tree diagrams and tables of outcomes to solve problems. Concept of discrete random variables and their probability distributions. Simple examples only, such as: P 1 (4 ) 18 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> X =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  = +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  for  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ∈   {   1,2,3   }   ; P 5, 6 , 7 18 18 18 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> X =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  =. Formal treatment of random variables and probability density functions; formal treatment of cumulative frequency distributions and their formulae. Expected value (mean), E(//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">X //  ) for discrete data. Knowledge and use of the formula E(//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">X //  )   =   Σ   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  P(  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">X  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )   ). <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.9 Applications of expectation, for example, games of chance. Topic 6—Statistics and probability (continued) © International Baccalaureate Organization 2006 24 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Binomial distribution. The formula ! !! <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> n n r r n r  ⎛ ⎞ ⎜ ⎟ = −   ⎝ ⎠    and consideration of combinations. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.10 Mean of the binomial distribution. Formal proof of mean. Normal distribution. Normal approximation to the binomial distribution. Properties of the normal distribution. Appreciation that the standardized value ( //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">z //  ) gives the number of standard deviations from the mean. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 6.11 Standardization of normal variables. Use of calculator (or tables) to find normal probabilities; the reverse process. Topic 7—Calculus 36 hrs Aims The aim of this section is to introduce students to the basic concepts and techniques of differential and integral calculus and their application. Details © International Baccalaureate Organization 2006 25 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Informal ideas of limit and convergence. Only an informal treatment of limit and convergence, for example, 0.3, 0.33, 0.333, ... converges to 1 3 .  Definition of derivative as  0 lim <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> h <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> f x f x h f x → // h // ′ = ⎛⎜   + −   ⎞⎟    ⎝ ⎠  .  Use of this definition for derivatives of  polynomial functions only. Other derivatives can be justified by graphical considerations using a GDC. Familiarity with both forms of notation, d d <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> y x  and <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> f ′   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )  , for the first derivative. Derivative of //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">n  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">n  //  ∈   􀁟   ), sin  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  , cos  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  , tan  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  , e//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  and ln  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 7.1 Derivative interpreted as gradient function and as rate of change. Finding equations of tangents and normals. Identifying increasing and decreasing functions. Topic 7—Calculus (continued) © International Baccalaureate Organization 2006 26 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Differentiation of a sum and a real multiple of the functions in 7.1. The chain rule for composite functions. The product and quotient rules. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 7.2 The second derivative. Familiarity with both forms of notation, 2 2  d  d <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> y x  and //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  ′′   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ), for the second derivative. Local maximum and minimum points. Testing for maximum or minimum using change of sign of the first derivative and using sign of the second derivative. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 7.3 Use of the first and second derivative in  optimization problems. Examples of applications: profit, area, volume. Indefinite integration as anti-differentiation. Indefinite integral of //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">n  //  (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">n  //  ∈   􀁟   ), sin  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  , cos  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  , 1 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> x and e//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //. 1 d//<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  ln  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x C  // x ∫  = + , //x//  >  0. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 7.4 The composites of any of these with the linear function //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">ax //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  //. Example: cos(2 3) 1 sin(2 3) 2 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> f ′  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  +   ⇒  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f x  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  + +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">C  //. Topic 7—Calculus (continued) © International Baccalaureate Organization 2006 27 <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> Content Amplifications/inclusions Exclusions Anti-differentiation with a boundary condition to determine the constant term. Example: if d 3 2 d <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> y x x x  = + and //y//  =  10 when <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> x =   0, then   3   1   2   10 2 <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> y =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  +  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  +. Definite integrals. Areas under curves (between the curve and the <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">  x  -axis), areas between curves. Only the form d //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  // a ∫ // y x //. d //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> b  // a ∫ // x y //. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 7.5 Volumes of revolution. Revolution about the //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  -axis only,   π   2   d  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b  // a <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> V =   ∫  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y x. // Revolution about the  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  -axis;   π   2   d. //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">b // a <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> V =   ∫  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x y  // <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 7.6 Kinematic problems involving displacement,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">s  //  , velocity, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">v // , and acceleration,  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">a. // d  d <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> v s t  = , 2 2  d d  d d <span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;"> a v s t t  = =. Area under velocity–time graph represents distance. Graphical behaviour of functions: tangents and normals, behaviour for large //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x //  , Both “global” and “local” behaviour. horizontal and vertical asymptotes. Oblique asymptotes. The significance of the second derivative; distinction between maximum and minimum points. Use of the terms “concave-up” for //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f //  ′′   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )   >   0 , “concave-down” for //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f //  ′′   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )   <   0. <span style="font-family: TimesNewRoman,Bold;"><span style="font-family: TimesNewRoman,Bold;"> 7.7 Points of inflexion with zero and non-zero gradients. At a point of inflexion //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f //  ′′   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )   =   0 and  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f  //  ′′   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ) changes sign (concavity change). //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f //  ′′   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  )   =   0 is    not a sufficient condition for a point of  inflexion: for example, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y  //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  4   at (0,0). Points of inflexion where //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">f //  ′′   (  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  ) is not defined: for example, //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">y //  =  //<span style="font-family: TimesNewRoman,Italic;"><span style="font-family: TimesNewRoman,Italic;">x  //  1 3   at (0,0). 28 © International Baccalaureate Organization 2006 ASSESSMENT