b DIPLOMA PROGRAMME MATHEMATICS SL INFORMATION BOOKLET For use by teachers ad studets, durig the course ad i the examiatios First examiatios 006 Iteratioal Baccalaureate Orgaizatio Bueos Aires Cardiff Geeva New York Sigapore
Diploma Programme Mathematics SL Iformatio Booklet Iteratioal Baccalaureate Orgaizatio, Geeva, CH-8, Switzerlad First published i November 004 by the Iteratioal Baccalaureate Orgaizatio Peterso House, Malthouse Aveue, Cardiff Gate Cardiff, Wales GB CF3 8GL UNITED KINGDOM Tel: + 44 9 054 7777 Fax: + 44 9 054 7778 Web site: www.ibo.org Iteratioal Baccalaureate Orgaizatio 004 The IBO is grateful for permissio to reproduce ad/or traslate ay copyright material used i this publicatio. Ackowledgmets are icluded, where appropriate, ad, if otified, the IBO will be pleased to rectify ay errors or omissios at the earliest opportuity. IBO merchadise ad publicatios i its official ad workig laguages ca be purchased through the olie catalogue at www.ibo.org, foud by selectig Publicatios from the shortcuts box. Geeral orderig queries should be directed to the sales departmet i Cardiff. Tel: +44 9 054 7746 Fax: +44 9 054 7779 E-mail: sales@ibo.org Prited i the Uited Kigdom by Atoy Rowe Ltd, Chippeham, Wiltshire. 565b
CONTENTS Formulae Presumed kowledge Topic Algebra Topic Fuctios ad equatios Topic 3 Circular fuctios ad trigoometry 3 Topic 4 Matrices 3 Topic 5 Vectors 4 Topic 6 Statistics ad probability 5 Topic 7 Calculus 6 Area uder the stadard ormal curve (topic 6.) 7 Iverse ormal probabilities (topic 6.) 8
Formulae Presumed kowledge Area of a parallelogram A= ( b h ), where b is the base, h is the height Area of a triagle Area of a trapezium Area of a circle A= ( b h ), where b is the base, h is the height A= ( a+ b ) h, where a ad b are the parallel sides, h is the height A=π r, where r is the radius Circumferece of a circle C = πr, where r is the radius Volume of a pyramid V = (area of base vertical height) 3 Volume of a cuboid V = l w h, where l is the legth, w is the width, h is the height Volume of a cylider Area of the curved surface of a cylider Volume of a sphere Volume of a coe Distace betwee two poits ( x, y) ad ( x, y ) Coordiates of the midpoit of a lie segmet with edpoits ( x, y) ad ( x, y ) V =π r h, where r is the radius, h is the height A= πrh, where r is the radius, h is the height V V 4 = π 3 = π 3 3 r, where r is the radius r h, where r is the radius, h is the height d = ( x x ) + ( y y ) x + x y, + y Iteratioal Baccalaureate Orgaizatio 004
Topic Algebra. The th term of a arithmetic sequece The sum of terms of a arithmetic sequece The th term of a geometric sequece u = u + ( ) d S = ( u + ( ) d) = ( u + u ) u = u r The sum of terms of a fiite geometric sequece The sum of a ifiite geometric sequece. Expoets ad logarithms S u( r ) u( r ) = = r r u S =, r < r x a = b x=log b a = e x xl a log log x a = = x log a a x a b a log a log b = c c a, r.3 Biomial theorem ( a b) a a b a b b r + = + + + r r + + Topic Fuctios ad equatios.5 Axis of symmetry of graph of a quadratic fuctio f( x) ax bx c axis of symmetry x = + + = b a.6 Solutio of a quadratic equatio Discrimiat b± b 4ac ax + bx + c = 0 x =, a 0 a = b 4 ac Iteratioal Baccalaureate Orgaizatio 004
Topic 3 Circular fuctios ad trigoometry 3. 3. Legth of a arc Area of a sector Idetities l = θ r, where θ is the agle measured i radias, r is the radius θ A = r, where θ is the agle measured i radias, r is the radius siθ taθ = cosθ cos θ + si θ = 3.3 Double agle formulae si θ = siθ cosθ cos cos si cos si θ = θ θ = θ = θ 3.6 Cosie rule Sie rule Area of a triagle + = + = a b c ab c a b abcos C; cosc a b c = = si A si B sic A= absi C, where a ad b are adjacet sides, C is the icluded agle Topic 4 Matrices 4.3 Determiat of a matrix Iverse of a matrix Determiat of a 3 3 matrix A a b = det = ad bc c d A a b d b A= A =, ad bc c d ad bc c a a b c e f d f d e A= d e f det A = a b + c h k g k g h g h k Iteratioal Baccalaureate Orgaizatio 004 3
Topic 5 Vectors 5. Magitude of a vector v 3 = v + v + v, where v v = v v3 Distace betwee two poits ( x, y, z) ad ( x, y, z ) d = ( x x ) + ( y y ) + ( z z ) Coordiates of the midpoit of a lie segmet with edpoits ( x, y, z ), ( x, y, z ) x + x y, + y z, + z 5. Scalar product v w = v w cosθ, where θ is the agle betwee v ad w v w = vw + vw + vw, where 3 3 v v = v, v3 w w = w w3 Agle betwee two vectors 5.3 Vector represetatio (equatio) of a lie vw + vw + vw cosθ = v w r = a+tb 3 3 4 Iteratioal Baccalaureate Orgaizatio 004
Topic 6 Statistics ad probability 6.3 6.5 Populatio parameters Mea µ Variace σ Stadard deviatio σ Sample statistics Mea x Variace s Stadard deviatio s Probability of a evet A Let = k f i. i= k fx i i i= µ = k fi i i= σ = ( x µ ) k fi i i= σ = x = s s = i= i= = k k fx i i= ( x µ ) f ( x x) k i i i ( ) P( A) = A U ( ) f ( x x) i i Complemetary evets P( A) + P( A ) = 6.6 Combied evets P( A B) = P( A) + P( B) P( A B ) Mutually exclusive evets P( A B) = P( A) + P( B ) 6.7 Coditioal probability P( AB) P( A B) = P( B) Idepedet evets P( A B) = P( A) P( B ) 6.9 Expected value of a discrete radom variable X 6.0 Biomial distributio Mea 6. Stadardized ormal variable x E( X ) = µ = xp( X = x ) r r X ~B(, p) P( X = r) = p ( p), r = 0,,, r E( X ) = p µ z = x σ Iteratioal Baccalaureate Orgaizatio 004 5
Topic 7 Calculus 7. Derivative of f( x ) Derivative of x d y f( x+ h) f( x) y = f( x) = f ( x) = lim dx h 0 h f ( x) = x f ( x) = x Derivative of si x f ( x) = si x f ( x) = cosx Derivative of cos x f ( x) = cos x f ( x) = six Derivative of ta x f( x) = ta x f ( x) = cos x Derivative of e x x f( x) = e f ( x ) = e x Derivative of l x f( x) = l x f ( x) = x 7. Chai rule y = gu ( ), where dy dy du u = f( x) = dx du dx Product rule Quotiet rule 7.4 Stadard itegrals dy dv du y = uv = u + v dx dx dx du dv v u u dy y = = dx dx v dx v + x x d x= + C, + 7.5 Area uder a curve d x = l x + C, x > 0 x sixdx= cos x+ C cosxdx= si x+ C x e dx = e x + C A= b a ydx Volume of revolutio (rotatio) V πy dx = b a 6 Iteratioal Baccalaureate Orgaizatio 004
Iteratioal Baccalaureate Orgaizatio 004 7 Area uder the stadard ormal curve (topic 6.) z 0 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0...3.4.5.6.7.8.9.0...3.4.5.6.7.8.9 3.0 3. 3. 3.3 3.4 3.5 0.5000 0.5398 0.5793 0.679 0.6554 0.695 0.757 0.7580 0.788 0.859 0.843 0.8643 0.8849 0.903 0.99 0.933 0.945 0.9554 0.964 0.973 0.9773 0.98 0.986 0.989 0.998 0.9938 0.9953 0.9965 0.9974 0.998 0.9987 0.9990 0.9993 0.9995 0.5040 0.5438 0.583 0.67 0.659 0.6950 0.79 0.76 0.790 0.886 0.8438 0.8665 0.8869 0.9049 0.907 0.9345 0.9463 0.9564 0.9649 0.979 0.9778 0.986 0.9864 0.9896 0.990 0.9940 0.9955 0.9966 0.9975 0.998 0.9987 0.999 0.9993 0.9995 0.5080 0.5478 0.587 0.655 0.668 0.6985 0.734 0.764 0.7939 0.8 0.846 0.8686 0.8888 0.9066 0.9 0.9357 0.9474 0.9573 0.9656 0.976 0.9783 0.9830 0.9868 0.9898 0.99 0.994 0.9956 0.9967 0.9976 0.9983 0.9988 0.999 0.9994 0.9996 0.50 0.557 0.590 0.693 0.6664 0.709 0.7357 0.7673 0.7967 0.838 0.8485 0.8708 0.8907 0.908 0.936 0.9370 0.9484 0.958 0.9664 0.973 0.9788 0.9834 0.987 0.990 0.995 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988 0.999 0.9994 0.9996 0.560 0.5557 0.5948 0.633 0.6700 0.7054 0.7389 0.7704 0.7995 0.864 0.8508 0.879 0.895 0.9099 0.95 0.938 0.9495 0.959 0.967 0.9738 0.9793 0.9838 0.9875 0.9904 0.997 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988 0.999 0.9994 0.9996 0.599 0.5596 0.5987 0.6368 0.6736 0.7088 0.74 0.7734 0.803 0.889 0.853 0.8749 0.8944 0.95 0.965 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.984 0.9878 0.9906 0.999 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989 0.999 0.9994 0.9996 0.539 0.5636 0.606 0.6406 0.677 0.73 0.7454 0.7764 0.805 0.835 0.8554 0.8770 0.896 0.93 0.979 0.9406 0.955 0.9608 0.9686 0.9750 0.9803 0.9846 0.988 0.9909 0.993 0.9948 0.996 0.997 0.9979 0.9985 0.9989 0.999 0.9994 0.9996 0.579 0.5675 0.6064 0.6443 0.6808 0.757 0.7486 0.7794 0.8079 0.8340 0.8577 0.8790 0.8980 0.947 0.99 0.948 0.955 0.966 0.9693 0.9756 0.9808 0.9850 0.9884 0.99 0.993 0.9949 0.996 0.997 0.9979 0.9985 0.9989 0.999 0.9995 0.9996 0.539 0.574 0.603 0.6480 0.6844 0.790 0.757 0.783 0.806 0.8365 0.8599 0.880 0.8997 0.96 0.9306 0.949 0.9535 0.965 0.9699 0.976 0.98 0.9854 0.9887 0.993 0.9934 0.995 0.9963 0.9973 0.9980 0.9986 0.9990 0.9993 0.9995 0.9996 0.5359 0.5753 0.64 0.657 0.6879 0.74 0.7549 0.785 0.833 0.8389 0.86 0.8830 0.905 0.977 0.939 0.944 0.9545 0.9633 0.9706 0.9767 0.987 0.9857 0.9890 0.996 0.9936 0.995 0.9964 0.9974 0.998 0.9986 0.9990 0.9993 0.9995 0 z p P( ) p Z z =
Iverse ormal probabilities (topic 6.) p = P( Z z) p 0 z p 0 0.00 0.00 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.50 0.0000 0.005 0.0050 0.0075 0.000 0.05 0.050 0.076 0.00 0.06 0.5 0.05 0.076 0.030 0.036 0.035 0.0376 0.040 0.046 0.045 0.0476 0.5 0.050 0.057 0.055 0.0577 0.060 0.067 0.065 0.0677 0.070 0.078 0.53 0.0753 0.0778 0.0803 0.088 0.0853 0.0878 0.0904 0.099 0.0954 0.0979 0.54 0.004 0.030 0.055 0.080 0.05 0.30 0.56 0.8 0.06 0.3 0.55 0.57 0.8 0.307 0.33 0.358 0.383 0.408 0.434 0.459 0.484 0.56 0.50 0.535 0.560 0.586 0.6 0.637 0.66 0.687 0.73 0.738 0.57 0.764 0.789 0.85 0.840 0.866 0.89 0.97 0.94 0.968 0.993 0.58 0.09 0.045 0.070 0.096 0. 0.47 0.73 0.98 0.4 0.50 0.59 0.75 0.30 0.37 0.353 0.379 0.404 0.430 0.456 0.48 0.508 0.60 0.534 0.559 0.585 0.6 0.637 0.663 0.689 0.75 0.74 0.767 0.6 0.793 0.89 0.845 0.87 0.898 0.94 0.950 0.976 0.300 0.309 0.6 0.3055 0.308 0.307 0.334 0.360 0.386 0.33 0.339 0.366 0.39 0.63 0.339 0.3345 0.337 0.3398 0.345 0.345 0.3478 0.3505 0.353 0.3558 0.64 0.3585 0.36 0.3638 0.3665 0.369 0.379 0.3745 0.377 0.3799 0.386 0.65 0.3853 0.3880 0.3907 0.3934 0.396 0.3989 0.406 0.4043 0.4070 0.4097 0.66 0.45 0.45 0.479 0.407 0.434 0.46 0.489 0.436 0.4344 0.437 0.67 0.4399 0.447 0.4454 0.448 0.450 0.4538 0.4565 0.4593 0.46 0.4649 0.68 0.4677 0.4705 0.4733 0.476 0.4789 0.487 0.4845 0.4874 0.490 0.4930 0.69 0.4959 0.4987 0.505 0.5044 0.507 0.50 0.59 0.558 0.587 0.55 0.70 0.544 0.573 0.530 0.533 0.5359 0.5388 0.547 0.5446 0.5476 0.5505 0.7 0.5534 0.5563 0.559 0.56 0.565 0.568 0.570 0.5740 0.5769 0.5799 0.7 0.588 0.5858 0.5888 0.598 0.5948 0.5978 0.6008 0.6038 0.6068 0.6098 0.73 0.68 0.658 0.689 0.69 0.650 0.680 0.63 0.634 0.637 0.6403 0.74 0.6434 0.6464 0.6495 0.656 0.6557 0.6588 0.660 0.665 0.668 0.674 0.75 0.6745 0.6776 0.6808 0.6840 0.687 0.6903 0.6935 0.6967 0.6999 0.703 8 Iteratioal Baccalaureate Orgaizatio 004
Iverse ormal probabilities (topic 6., cotiued) p 0 0.00 0.00 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.76 0.7063 0.7095 0.78 0.760 0.79 0.75 0.757 0.790 0.733 0.7356 0.77 0.7389 0.74 0.7455 0.7488 0.75 0.7554 0.7588 0.76 0.7655 0.7688 0.78 0.77 0.7756 0.7790 0.784 0.7858 0.789 0.796 0.796 0.7995 0.8030 0.79 0.8064 0.8099 0.834 0.869 0.804 0.839 0.874 0.830 0.8345 0.838 0.80 0.846 0.845 0.8488 0.854 0.8560 0.8596 0.8633 0.8669 0.8706 0.874 0.8 0.8779 0.886 0.8853 0.8890 0.897 0.8965 0.900 0.9040 0.9078 0.96 0.8 0.954 0.99 0.930 0.969 0.9307 0.9346 0.9385 0.944 0.9463 0.950 0.83 0.954 0.958 0.96 0.966 0.970 0.974 0.978 0.98 0.9863 0.9904 0.84 0.9945 0.9986.007.0069.00.05.094.037.079.03 0.85.0364.0407.045.0494.0537.058.065.0669.074.0758 0.86.0803.0848.0894.0939.0985.03.077.3.70.7 0.87.64.3.359.407.455.504.55.60.65.700 0.88.750.800.850.90.95.004.055.07.60. 0.89.65.39.37.46.48.536.59.646.70.759 0.90.86.873.930.988.3047.306.365.35.385.3346 0.9.3408.3469.353.3595.3658.37.3787.385.397.3984 0.9.405.48.487.455.435.4395.4466.4538.46.4684 0.93.4758.4833.4909.4985.5063.54.50.530.538.5464 0.94.5548.563.578.5805.5893.598.6073.664.658.635 0.95.6449.6546.6646.6747.6849.6954.7060.769.779.739 0.96.7507.764.7744.7866.799.89.850.8384.85.8663 0.97.8808.8957.90.968.943.9600.9774.9954.04.0335 0.98.0538.0749.0969.0.444.70.973.6.57.904 0.99.364.3656.4089.4573.5.5758.65.7478.878 3.090 Iteratioal Baccalaureate Orgaizatio 004 9