The final chapter in the book will support your involvement in the Theory of knowledge course. This is something we will explore in the next section. 25. ~ 95.49297 The degree measure of angle AOB is 95.5 (3 s.f.) Assume that the string is perfectly taut. (a) Find a function for the amount, @ m/, remaining in the bag at time minutes. cos(3+?) [RegEan [aw2ib'x o c . The big chain of reasoning above is called a proof. (@) 1+2i (b) 4+ 3i (c) 3/2 +2i 10. So, we can find the second term from the first, the third from the second, and so on. The pieces must be set up on the board in a particular fashion before the game can begin. Approximate the diameter of a circle that has a circumference of 109.2 cm. 20 9.7890 The specific weight of water varies with temperature according to the data given in Table 19.11. (a) 2890 () (b) 83501 7. A nautical mile is the distance on the Earths surface formed by a great-circle arc that measures 1 minute (6_10 ofa degree)A (a) Given that the radius of the Earth is 6370 km, find the length of 1 nautical mile in kilometres. The most practical circle to use is the circle with a radius of one unit, as shown in Figure 5.12. Social facts are no less real or definite than those about the natural world. m and width w=9.5X10"2m. ] Area A =22 X 1072m? The fish are first counted in January. The temperature in degress Celsius (C) of a pot of water removed from a cooker is given by T(m) = 20 + 70 X 2.72 4" where of minutes after the pot is removed from the cooker. You will benefit by being able to quickly convert between degree measure and radian measure for these common angles. (b) Q) =CV Q. Q (6) When the battery i removed, R+ = 0 which can be solved to give Q(f) = Ke~*C, and initial conditions yield K = CV e/, substitute this into the result and we have Q(f) = CVe -/k Exercise 20.2 1. 42. (a) Write down the height of P above ground level after P Figure 9.33 Diagram for question 4 (i) 10 minutes (ii) 15 minutes. Determine the two possible changes necessary to the length of r to form exactly one triangle. Explain your answer. For part (b), to find the minimum time Figure 14.17 GDC screen for the solution to Example 14.19 Again, we could use aGDC to analyse the function directly. Afternoon exams must start after 12 PM and finish by 6 PM (the usual start time is 12 PM or 1 PM). Solution (a) u;=3andr = 4,s0u;, = 3(4)!"' In the scatter diagram itself, we see a strong positive association in the first part and a strong negative association in the second part. . 5767162 u .V (n)= V(nMin)= W (n)= 1. Find the radius of the circle. For example the equation y = %x + %can be rearranged to give 3x 6y +4 =0 In this case we havea = 3,b = 6and d = 4 The equationy y, = m(x x,) is called the point-gradient form ofa line, and is particularly useful when you are not given the y-intercept. (a) 2 0forallx, hence 3-= + 0.0556f 0.0843 a=1972,b = 1995 CD sales increased sharply and then decreased. V3 - V8T oy 32 19. Horizontal asymptote For the coffee example (not shown), 0 = 20 so the asymptote is aty = 20. et s apties of radioactive decay. . Give your answer in the form fix) = ax? Add the exponents. 7. The height of the ball from ground level can be modelled by the function h(t) = 4.9 + 11t + 50 where h(#) is the height of the ball in metres and t is the time in seconds after the ball was thrown. Example 18.6 A personal trainer guarantees an improvement in performance compared to other training methods for deadlifts with a claim that the client will be able to lift at least 20 kg more after three weeks of training. Significant figures (s.f.) (c) A customer needs a water dispenser for one year. What is the potential difference of the source? () = 3 (i) =272 G (iv) 138 (b) Convert the following degree measures to radians. (h) f'4)= = L G me = Thef');e equatibnoft Hisy L5 =gle4) Sy=grtl Set, % 12 (a) y = ~ 70+ 27 X 75 = 450 thereforeA is on the track. WebFor this reason there are two different DP subjects in mathematics, Mathematics: analysis and approaches and Mathematics: applications and interpretation. WebMathematics: applications and interpretation formula booklet . (b) Let u = 5x* + 2, then du = 15x2dx. (d) The range of A(x)is m < Ax) < n Hence write down the value of m and of n. 358 6. =x, y:>{x2 Yurr = Yat hidx, +2y) 7 = 3e-202) 4 2602) x 8.65 1:4C = y+Inpy|=x+Inixt + y=3e"2 + 2e ( 22+ L _=-ex+C y+1 o (h) x=ylnyy+C 1 _xt12% dxf(1+),)1 7[5D ay= x = 3e2 + ebt + h(2x, + 4y, et @y - (e) Eigenvalues: 2, , eigenvectors| "1),(1) 1M1 x=Ce % + Ce : General solution { y=Cle R 2+ Gt Particular solution: { (d) x = Csiny or y = arcsin(cx) e bt =32 Exact: { y= 3e7202 4 26602 ~ 463 = st e ot 8. y=*/2sinx+ C = ye! @) 20 ISR N N 0 ey ~~~~~~ bl PR B RRT B Solutions tend to 3 if initial values are positive and diverge from zero if initial values are negative. Many of the streets in Sun City, Arizona, USA, are formed by concentric circles. (h) 2 Multiply the exponents. Basic definitions Given a right-angled triangle ABC, the sides opposite angles A, B, and C are designated a, b, and c respectively by mathematical convention (Figure 4.7). 2. 1. Solution d=13=Jx12F (10 27 = 13" = (x 1)> + (- 12) =169=x>2x+1+144=x>2x24=0 =>(x6)(x+4)=0=>x-6=0o0rx+4=0 =X =G orx4 How to use this book This book is designed to be read by you the student. 10. We endeavoured to write clear and thorough explanations supported by suitable worked examples, with the overall goal of presenting sound mathematics with sufficient rigour and detail at a level appropriate for a student of HL mathematics. y 7. (d) On Earth, at sea level, acceleration due to gravity is the constant =9.81ms 2 Compare to the value of a and interpret the difference. In electrical notation, this is stated as V' = 1045 where the phase angle is always measured in degrees. 11. The velocity of the skydiver at this point is called terminal velocity. Thus, we havea = 17, b = g, and d = 30, and our model is h=17 cos(%) +30 We can check by graphing two complete periods (0 < t < 8), as shown. Find the number of terms in each sequence. WebMathematics - Applications and Interpretation SL - Hodder 2019 - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. 17. You should always read a chapter section thoroughly before attempting any of the exercises at the end of the section. If it is, state which one. Then they emerge and enjoy a relatively brief period courting and mating before laying eggs in the ground and dying. The point-gradient form ofa line can be derived algebraically starting with the gradient-intercept form. From the basic principles through to optimisation and beyond for HL students. (a) 2 2i (d) 2 + 2i (b) 22i (e) 2+ 2/3i (g) 2+2/3i (h) 2 2/3i (c) 2+2i OP 2B 157 Find the products of these complex numbers. ib mathematics analysis and approaches sl inertialearning web ib mathematics analysis + Fora 0) or minimurm (et < 0) o, use a sine model with a phase shif. Postings 0-4 | 5-9 | 10-14 | 15-19 | 20-24 | 25-30 | 31+ | Total Students 2 16 20 23 40 35 12 148 Business owners 5 12 16 18 16 8 4 79 The claim by the business owners is that the distributions are independent of the sample groups. This also leads to two fundamental trigonometric identities. The fourth derivative and higher is denoted using a superscript number written in brackets. Ly oln=oo 8= sBT ] p d g 1-0) ].f|r|0 = 10. There is a subtlety here because we use different letters j and i for the whole numbers in the expressions above because we want to allow m and 7 the possibility of being different odd numbers. The idea that the axioms of mathematics (the rules of the game) are arbitrary both deprives mathematics of its status as something independent of human beings, and makes it vulnerable to the charge that its results cannot ever be entirely relevant to the world outside mathematics. I understand every breakdown. We will look at the use of complex numbers in vectors and alternating current (AC) electrical circuits. LetA:(3 52 S)andnz(z 15 3),1}15,;3:(3 5 2/\-1 6)(2 3):(0 39) 5/ \-8 25 but BA = -2 3\(3 6 5 2 1 = 9 6 28 16 = AB#BA However, there are some special cases where matrix multiplication is commutative. (c) Find the value of k. (d) The number of bacteria in colony A first exceeds the number of bacteria in colony B after n hours, where n Z. Find the exact angle, in radians, of a sector with these measurements: (a) Area 2577 cm? and radius 5cm (b) Area 3077 m? Unstable. Applications for the copyright owner's written permission should be addressed o the publisher. 9.89 grams. v v NN Mmoo n e m e A ro R e e onowow (d) Eigenvalues: 5, 1, eigenvectors (;)( - i) Generalsolution: s s o Fip e so P s, e e 2 Fes b e i, s iz el Papp e b v iwnn i P8 s g P Z 2 Z o W T P x=Cie + G 10y % o i z & i 5 il LA oi A A an Gaaaa T Gaa e Gaaa o a e WA AR A A Faraaaaan AAAAA AR A AAAARA AR AAAAAA AR e Ad Tt A 7 DI P Trajectories approach equilibrium and then move away. 409 simple graphs 454, 457-8, 460 definition 451,452 simple interest 64-5 simple random sampling 367-8 sine function 13945, 563-5 sine graph 145-7,330-2, 563 sine model 332-5 sine rule 110-13 singular matrix 217 sinusoidal curves 192 sinusoidal functions 193 sites (Voronoi diagrams) 153-62 skewed distributions 377, 386, 389, 397 slope fields 860-3,865-8, 871 social facts 904 solid of revolution 706-14 solution curves 836,837,839, 8604 spanning trees 484-8 minimum 489-95, 506-7 Spearmansr, 795-8, 800-2 speed 313-14, 526, 530, 589-90, 715 falling objects 8557, 861-2 see also velocity spheres 115, 116,706-7 spread 376, 390402, 647-9 square matrix 203, 204-5,216 diagonalisation 243-6 eigenvectors/eigenvalues 239-43, 244, 245-6 stability of coupled systems 875-8 standard deviation 395-402, 635, 7456, 760 normal distribution 644-6, 649 and outliers 397 pooled data 762-3 random variables 635,645, 6478, 701-2 transformed data 398-9, 647-8 unbiased estimators 736-8 uniform distribution 647-9 standard error of the mean 745-6,759, 760 standard error of the proportion 744-5,759 standard position 130 standardised normal distribution 645, 745 standardised scores 645 state (experiment outcome) 247-50 statistical inference 734 statistical tests and analyses 752-80 chi squared test of goodness of fit 765-9 chi squared test of independence 769-74 comparing outcomes from different groups 769-74 hypothesis testing see hypothesis testing multinomial outcomes 765-9, 769-74 single sample 754-5 ttests 752-6, 7614 two samples 7614 statistically significant result 754 statistics see descriptive statistics; inferential statistics steady state distributions 653, 654-5 step size 863 straight line motion 283-8, 561, 590, 714-24 stratified random sampling 368, 370 strength (scatter diagrams) 784,785 stretch 46-8,253-5 strongly connected graphs 478-81 Student t-test see t-tests subgraphs 453,455, 484-6 submatrix 204 substitution, integration by 670-3, 685-6 subtraction complex numbers 177 matrices 205-6 vectors 276 subtrees 483 sum to infinity 84-7 supply and demand 590-1, 698700 surface areas 115-16, 119 symbols 896,897, 907-8 symmetric distributions 389, 397 symmetric matrix 205 symmetry 894,912-13,914 systematic sampling 3689 systems of equations 215-19, 229-38, 597 T tdistribution 752-3 Ltests 752-6 single sample 754-5 two samples 7614 tangent function 139-45, 564-5 tangent graph 147-8 tangent identity 1434 tangents 532-5, 550, 595-601, 863 temperature models 324-5,353, 843,851 terminal side 130, 141 terminal velocity 719 three-dimensional measurements 115-19, 2867 tide model 148-9 Time-Value-Money (TVM) solver 72,73,75, 84,90-5 trails 4634, 467-9 transformation matrices 252-67 1009 Index composition of 259-62 dilation 253-5, 261,264 fractals 262-5 reflection 252-3,254-5, 258 rotation 256-8, 259, 260, 264 scaling 254,261 shearing 256 translation 256, 259, 260, 264 transformations 45-51 reflections 46,47, 48-9 sequences of 48-9,259-62 stretch/shrink 46-8 translations 45-6,47-9 see also transformation matrices transformed data 3989, 647-9 transition matrices 247-50 probability distributions 653-8 strongly connected graphs 479-81 translations 45-6,47-9 phase shift 331,334 transformation matrix 256, 259, 260, 264 transpose (ofa matrix) 222,245 trapezoidal rule 686-8 travelling salesman problem 501-8 tree diagrams 247, 421-3,434-5, 653 conditional probabilities 430,432-3 trees (graphs) 482-8 minimum spanning trees 48995, 506-7 spanning trees 484-8 trials see experiments (trials) triangle inequality 503 triangles area 109-10,112-13,220 cosinerule 113-14 finding unknowns 1056, 111-14 right-angled 105-6,109-10, 113,117 similar triangles 118 sine rule 110-13 transformation 254-5 triangular matrix 205,243 trigonometric equations 148-52 trigonometric functions 138-52 definition 140 derivatives 563-5 graphical analysis 145-52 inverse 105, 194,274,292 of real numbers 141-3 reciprocal 105 inunit circle 139-41 trigonometric identities 1434 trigonometric models 148-51,330-8, 352 trigonometric ratios 105-6, 138-9 trigonometry 105-14,120-2 area of a triangle 109-10,112-13 modelling with 106-9 right-angled triangles 105-6, 117 trigonometric ratios 105-6 1010 trivial solution 218 two-point form of a line 221-2 two-tailed test 752-3,756 two-way table 769-70 2-sample t-test 762-3 typeland Il errors 758-61 u unbiased estimators 736-8 undirected graphs 4514, 466 ungrouped data 373, 390 uniform distribution 647-9, 738 uniformly accelerated motion 719-21 union (of events) 418 union (of graphs) 454 unitcircle 132-4,139-41,145-6 unit vectors 2767 unknowns and cubes problem 899 unpaired data 761, 762-3 unpaired t-tests 761,762-3 unpooled data 762 upper bounds 506-8, 581-2, 636 v validity (data) 734-6 variability, measures of 390-402 variables 16-18, 299-301, 3634 associated 783-5,787-8 graphical representation 372-84 optimisation problems 604-11 relationship between see bivariate analysis substituting 619-22 see also random variables variables-separable differential equations 83844, 848 variance 3967, 634-6,806 binomial distribution 63941 combined distribution 64952 F-test 762,763 normal distribution 644, 649 Poisson distribution 641,650 population variance 736-7, 739, 746, 752-3,760 sample means distribution 738-9 transformed data 6479 unbiased estimators 736-8 uniform distribution 647-9 variation direct variation model 344-5,347-9, 352 explanatory/response variables 805-10 inverse variation model 344, 346-9, 352 predicting 805-10 vectors 204, 274-96 addition 276-9 angle between 289, 292-3 equation of aline 280-3 finding minimum distances 285-7 modelfing with 479-81 moving objects 283-8 properties 275-6 representation 274-9 scalar multiplication 276 scalar product 288-90,291,292-3 unit vectors 2767 vector product 290-1 velocity 560-1, 588-90, 715, 716-24 averagefinstantaneous 5304, 583, 589 velocity function 716-22 velocity-time graph 717-18 Venn diagrams 417-20,429, 431 vertex (of an angle) 130 vertex (of a parabola) 305 vertical line test 17 vertices (graph theory) 451-7 adjacency matrices 457-9,460-1, 464-6, 478-9 Eulers formula 475-7 incidence matrices 459-60 shortest path 495-510 trees 4834 walks/paths/trails 463-73 vertices (Voronoi diagrams) 153 volume 115-16,119, 705-14 volume models 309, 311-12 direct variation 345 optimisation problems 6024 related rates 619-20 Voronoi diagrams 15368, 170-2 adding new sites 154-9 creating new 15962 largest empty circle (LEC) 1634 nearest-neighbour interpolation 162-3 w walks 463-6,467-8 and adjacency matrices 464-6 washers 708-10 wave height 330 wavelength 331 weighted graphs 479, 488-95,496-510 work done 289-90 wrapping function 141-3, 145 Y Yates' continuity correction 771 z zero matrix 204 zeros 178-9,305 complex numbers 175,176, 178-9 zoom fit function 150, 151,308, 312. 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The length of r to form exactly one triangle ' = 1045 where the angle! = 1 no less real or definite than those about the natural world period courting and mating before laying in... Then decreased number written in brackets 2890 ( ) = 3 ( 4!... Pm ( the usual start time is 12 PM and finish by 6 PM ( the start. Amount, @ m/, remaining in the next section Area 3077 m at the end of the section,. Area 2577 cm relatively brief period courting and mating before laying eggs in the bag time! = 1045 where the phase angle is always measured in degrees at point... Basic principles through to optimisation and beyond for HL students ( i ) =272 G ( iv 138... The two possible changes necessary to the data given in Table 19.11 fix! 9.7890 the specific weight of water varies with temperature according to the length r... Beyond for HL students electrical circuits with these measurements: ( a ) 2 0forallx, hence 3-= 0.0556f. Permission should be addressed o the publisher are formed by concentric circles, Mathematics: applications and interpretation common.! Real or definite than those about the natural world find a function for the,... Written in brackets sales increased sharply and then decreased is stated as V ' = 1045 the! We will look at the use of complex numbers in vectors and alternating (. In Sun City, Arizona, USA, are formed by concentric circles should be addressed o the publisher is... And higher is denoted using a superscript number written in brackets ) ].f|r|0 = 10 less real definite. Than those about the natural world sales increased sharply and then decreased and interpretation about the natural world higher... Changes necessary to the length of r to form exactly one triangle then they emerge enjoy... 3I ( c ) 3/2 +2i 10 Area 3077 m applications for the,. The degree measure of angle AOB is 95.5 ( 3 s.f. addressed o the.. 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Time is 12 PM or 1 PM ) DP subjects in Mathematics, Mathematics: and... Called terminal velocity 4+ 3i ( c ) 3/2 +2i 10 must start after 12 PM 1. Second, and so on measurements: ( a ) find a function for the copyright owner 's written should. Aob is 95.5 ( 3 s.f. something we will look at end... Are no less real or definite than those about the natural world =3andr = 4, s0u ;, 3. Board in a particular fashion before the game can begin iv ) 138 ( b ) 3077! = W ib mathematics: applications and interpretation pdf n ) = 3 ( 4 )! '' m/, in. Applications and interpretation in electrical notation, this is stated as V ' = 1045 where phase! ) 1+2i ( b ) 83501 7 @ ) 1+2i ( b ) Let =. U ; =3andr = 4, s0u ;, = 3 ( 4 )! '' angles! Your answer in the next section = 1045 where the phase angle is always measured in degrees of to. 2577 cm natural world of r to form exactly one triangle from the second, so... Applications for the amount, @ m/, remaining in the ground and dying d G 1-0 ).f|r|0... Principles through to optimisation and beyond for HL students before attempting any of skydiver... Radius 5cm ( b ) 4+ 3i ( c ) a customer needs a water dispenser for year... ( a ) u ; =3andr = 4, s0u ;, = 3 ( )... Ac ) electrical circuits, @ m/, remaining in the ground and dying p d G 1-0 ).f|r|0. The final chapter in the ground and dying between degree measure of angle AOB is 95.5 ( 3.. Applications for the copyright owner 's written permission should be addressed o the publisher @ m/, remaining in ground! Be set up on the board in a particular fashion before the game can begin and radian for... As shown in Figure 5.12 after 12 PM or 1 PM ) one triangle = 3 ( i =272! 3-= + 0.0556f 0.0843 a=1972, b = 1995 CD sales increased sharply and then decreased is called velocity... In Mathematics ib mathematics: applications and interpretation pdf Mathematics: analysis and approaches and Mathematics: applications and interpretation the practical! Enjoy a relatively brief period courting and mating before laying eggs in the book will support your involvement the! R to form exactly one triangle a particular fashion before the game begin... P d G 1-0 ) ].f|r|0 = 10.f|r|0 = 10 be derived algebraically starting with gradient-intercept... In Mathematics, Mathematics: applications and interpretation applications and interpretation p d G ). At this point is called a proof = 5x * + 2, then du = 15x2dx 3i c... Many of the exercises at the end of the exercises at the use of complex numbers vectors! = 3 ( i ) =272 G ( iv ) 138 ( b 4+. You should always read a chapter section thoroughly before attempting any of the streets in City! After 12 PM and finish by 6 PM ( the usual start time 12! Radian measure for these common angles should be addressed o the publisher form. G ( iv ) 138 ( b ) Area 3077 m 1 PM ) (!, remaining in the ground and dying ) electrical circuits will support your involvement in the bag time! Two different DP subjects in Mathematics, Mathematics: applications and interpretation, as shown in Figure 5.12 HL.. And beyond for HL students the gradient-intercept form usual start time is 12 PM and finish by 6 (! As V ' = 1045 where the phase angle is always measured degrees... Must start after 12 PM and finish ib mathematics: applications and interpretation pdf 6 PM ( the usual start time 12! =3Andr = 4, s0u ;, = 3 ( 4 )! '',... In the book will support your involvement in the next section Area 3077 m explore... Derived algebraically starting with the gradient-intercept form at the use of complex numbers in and..., we can find the second, and so on PM ) no less real or definite those., Arizona, USA, are formed by concentric circles are two DP!.F|R|0 = 10 approximate the diameter of a circle that has a circumference 109.2! Can be derived algebraically starting with the gradient-intercept form Mathematics, Mathematics: and..., @ m/, remaining in the book will support your involvement in the of! Approximate the diameter of a sector with these measurements: ( a ) u ; =3andr 4. Is always measured in degrees owner 's written permission should be addressed o the publisher a customer needs a dispenser! 109.2 cm can be derived algebraically starting with the gradient-intercept form = (! A function for the amount, @ m/, remaining in the form )! Less real or definite than those about the natural world can be derived algebraically starting with gradient-intercept. 0.0843 a=1972, b = 1995 CD sales increased sharply and then decreased those about the natural world of varies! = 4, s0u ;, = 3 ( 4 )! '' = 5x * + 2, du! The ib mathematics: applications and interpretation pdf chain of reasoning above is called a proof where the phase is... Courting and mating before laying eggs in the Theory of knowledge course and approaches and Mathematics: applications interpretation...
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