Snnrpsgymnsiet Lott Hydén Mthemtics, Stndrd Level Curriculum SCHEME OF WORK FOR IB MATHS STANDARD LEVEL Min resource: Mthemtics for the interntionl student, Mthemtics SL, Hese PART 1 Sequences nd Series nd the Binomil Theorem (syllbus ref 11,13, 8) Arithmetic sequences nd series, sum of finite rithmetic series Geometric sequences nd series; sum of finite nd infinite geometric series Applictions of grphing skills nd solving equtions tht relte to rel life situtions Sigm nottion Emples of pplictions, compound interest nd popultion growth n The binomil theorem, epnsion of ( + b), n N n n Clculting the binomil coefficients using nd Pscl s is found using both the r r formul nd GDC (p 175 05 Supplemented by Mthemtics for the interntionl student, Mthemtics SL, Hese Chpter 6) Eponents nd Logrithms (syllbus ref 1, 6, prt of 7) Elementry tretment of eponents nd logrithms; lws of eponents; lws of logrithms logc including chnge of bse, log b = logc b Solving eponentil equtions = b The number e, definition Emples of pplictions: Compound interest, growth nd decy Eponentil nd logrithmic functions nd their grphs: The function y = nd the function y = e The function y = log nd the function y = ln Reltionships between the functions: ln = e log = log = (p79-93 Supplemented by Mthemtics for the interntionl student, Mthemtics SL, Hese Chpter 3 nd 4)
Algebr (syllbus ref 4, prt of 7) The qudrtic function + b + c, its grph, y-intercept (0, c) Ais of symmetry The form y = ( p)( q), the -intercepts (zeros) (p, 0) nd (q, 0) The form y = ( h) + k, verte (h, k) The solution of + b + c = 0, 0 The qudrtic formul Use of the discriminnt (p17-49) Test (Appro beginning of Oct yer 1) PART Functions nd Equtions (syllbus ref 1,, 3, 5) Nottion f : f (), domin, rnge, imge (vlue) Composite functions, ( f g)( ) Inverse functions, f 1 ( ), the grph of y = f 1 ( ) s the reflection in the line y = of the grph y = f () (The rnge of f () becomes the domin of f 1 ( ), nd vice vers) 1 Identity function, ( f f )( ) = Function grphing skills, including the use of the GDC Investigtion of key fetures of grphs, such s mimum nd minimum vlues, intercepts, zeros (roots), turning points, horizontl nd verticl symptotes, symmetry nd considertion of domin nd rnge Grphicl solution of equtions 1 The reciprocl function y =, its grph nd self-inverse nture + b The rtionl function y = nd its grph c + d Trnsformtions of grphs Trnsltions: y = f ( ) + b; y = f ( ) Reflections in both es: y = f ( ); y = f ( ) Verticl stretch with scle fctor p: y = pf() Stretch in the -direction with scle fctor q 1 : y = f(q) Composite trnsformtions Differentition 1 (syllbus ref 61, prts of 6, 63) Informl ides of limit nd convergence Limit nottion
Definition of derivtives s f ( + h) f ( ) f ( ) = lim h 0 h (Only to be used for derivtives of polynomil functions) Derivtive interpreted s grdient function nd s rte of chnge Tngent nd norml, nd their equtions dy Fmilirity with both forms of nottion, f () nd for the first derivtive d d y Fmilirity with both forms of nottion, f () nd for the second derivtive d Derivtives of n (n is rtionl) nd sums nd liner multiples of such functions The chin rule, product rule nd quotient rule Identifying incresing nd decresing functions Locl mim nd minim Testing for m nd min using chnge of sign of the first derivtive nd using the second derivtive Use of the term concve-up for f () > 0 (minimum point) Use of the term concve-down for f () < 0 (mimum point) Points of infleion with zero nd non-zeros grdients Horizontl infleion if f ( ) = 0 nd f () = 0, non-horizontl infleion if f ( ) 0 nd f () = 0 At point of infleion f () = 0 nd f () chnges sign Grphicl behviour of functions, including the reltionship between the grphs of f, f nd f, both globl nd locl behviour Use the first nd second derivtive in optimiztion problems (P 107-145, 09-33, hnd out concve-up/down, points of infleion) (p 80-106 Supplemented by Mthemtics for the interntionl student, Mthemtics SL, Hese Chpter 14) Test (Appro mid Dec yer 1) PART 3 Circulr Functions nd Trigonometry (syllbus ref 31-36) Sine- nd cosine rules nd re of tringle The mbiguous cse of the sine rule Applictions Rdin mesure, rc length nd sector re Convert degrees to rdins nd rdins to degrees
Definition of sinθ nd cosθ in terms of the unit circle sinθ Definition of tnθ s cosθ π π π π The ect vlues trigonometric rtios of 0,,,, (0, 30, 45, 60, 90 ) nd their 6 4 3 multiples The Pythgoren identity sin θ + cos θ = 1 Double ngle identities for sine nd cosine Reltionships between trigonometric rtios E: Given sin θ, finding the vlues of cosθ without findingθ The circulr functions sin, cos nd tn, their periodic nture, domins nd rnges, mplitude nd grphs The generl sine function: y = Asin B( C) + D The generl cosine function: y = Acos B( C) + D Trnsformtions Applictions to rel-life situtions Lines through the origin cn be epressed s y = tnθ, with grdient tn θ Solving trigonometric equtions in finite intervl, both grphiclly nd nlyticlly Equtions leding to qudrtic equtions (p 50-78, 34-6 Supplemented by Mthemtics for the interntionl student, Mthemtics SL, Hese Chpter 8-11) Test (Appro begin Feb yer 1) PART 4 Sttistics (syllbus ref 51-54) Concepts of popultion, smpling, rndom smpling, discrete nd continuous dt Presenttion of dt: frequency distributions (tbles), frequency histogrm with equl clss intervls, bo- nd whiskers plots nd outliers Grouped dt: mid intervl vlues, intervl width, upper nd lower intervl boundries, modl clss Averges: Men, medin, mode Qurtiles, percentiles Awreness tht the popultion men, μ, is generlly unknown, nd the smple men serves s n estimte of this quntity Dispersion: Rnge nd interqurtile rnge, vrince, stndrd devition Awreness of the concept of dispersion nd n understnding of the significnce of the numericl vlue of the stndrd devition Obtining the stndrd devition/vrince from the GDC
Awreness tht the popultion stndrd devition, σ, is generlly unknown, nd tht the stndrd devition of the smple, sn, serves s n estimte of this quntity Effect of constnt chnges to the originl dt E: If 5 is subtrcted from ll the dt items, then the men is decresed by 5, but the stndrd devition is unchnged Cumultive frequency, cumultive frequency grphs use to find medin, qurtiles nd percentiles Liner correltion of bivrite dt Independent () nd dependent (y) vrible Person s product-moment correltion coefficient, r Positive, zero, negtive nd strong, wek nd no correltion Sctter digrms, lines of best fit The line of best fit psses through the men point Eqution of the regression line nd use of the eqution for prediction purposes Interpoltion nd etrpoltion INTERNAL ASSESSMENT Introduction, Mrch yer 1 Probbility (Syllbus ref: 55-59) Concepts of trils, outcome, eqully likely outcomes, smple spce (U) nd event n( The probbility of n event A is P ( = n( U ) The complementry events A nd A (not P ( + P( A ) = 1 Use of Venn digrms, tree digrms nd tble of outcomes to solve problems Combined events, P( A the non-eclusivity of or Combined events: P( A = P( + P( P( A Mutully eclusive events: ( ) = 0 P A B nd the use of P ( A = P( + P( P( A Conditionl probbility: The definition P( A B ) = P( Independent events: The definition P ( A B ) = P( = P( A B ) nd P ( A = P( P( Probbilities with nd without replcement Concepts of discrete rndom vribles nd their probbility distributions Epected vlue (men), E(X) for discrete dt Applictions Binomil distribution Men nd vrince of binomil distribution Norml distribution nd curves Probbilities nd vlues must be found using GDC Stndrdiztion of norml vribles Apprecition tht the stndrdized vlue (z) gives the number of stndrd devitions from the men (P 410 43), (P 367 383), (P 384 410) Test (ppro beg My yer 1)
PART 5 Further clculus (syllbus ref 6, 64-66) Simple differentition nd integrtion of sin, cos, tn, e nd ln, nd the composites of these with ny liner function Chin rule, Product nd Quotient rules for ll bove Applictions Indefinite integrtion s nti-differentition Indefinite integrl of n nd 1, nd liner functions of these Anti-differentition with boundry condition to determine the constnt term Definite integrls Are under curves (between the curve nd the -is) nd res between two curves Volumes of revolution (bout the -is) Kinemtic problems involving displcement, s, velocity, v, nd ccelertion, (P 63 78, 300 315), (P 146 174 PART 6 Vectors, (syllbus ref 41-44) Vectors s displcements in the plne nd in three dimensions Components of vector column representtion nd with respect to the unit vectors i, j nd k v1 v = v = v1i + v j + v3k v 3 Algebric nd geometric pproches to: - the sum nd difference of two vectors - the zero vector - the vector v - multipliction by sclr, kv, prllel vectors - mgnitude of vector, v - unit vectors, bse vectors i, j nd k - position vectors, OA = - AB = OB OA = b Distnce between points A nd B is the mgnitude of AB The sclr product (dot product/inner product), v w = v1w1 + vw + v3w3 v w = v w cosθ Perpendiculr vectors: v w = 0 Prllel vectors: v w = ± v w
The ngle between two vectors Vector eqution of line s r = + tb Interprettion of t s time nd b s velocity, with b representing the speed Coincident (the sme line) nd prllel lines (never meet) Finding points where lines intersect Determining whether two lines intersect (P 341 366, supplemented by Hese, chpter 1,13) Revision