0.0346 (b) 6mm a(t) =500 24,0 < < 250 225 minutes 935cm? (b) 2.04cm 8. The number of social media postings per week by students and business owners is presented below. 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. It follows that an angle of 180 has a radian measure of exactly 7. Using the symmetry shown in Figure 5.19, we can then evaluate the values for the other three angles: 140 sin (135) = Q sin (225) = 7 sin (315) = = cos(135) = -22 cos (2259 = Y22 cos (3157 = 422 tan(135) = 1 tan(225) = 1 tan (315) = 1 2 2 Note that whether the value ofa trigonometric function for a given angle is positive or negative depends on which quadrant the terminal side of theangle ~ isin, because the signs ofx; y, or- change in each quadrant. We know that neither of the assumptions in (1) is true. Find the most general antiderivative of each function. The average cost, Alx) = @ 1 A company produces fans with a maximum capacity of 500 units per day. Figure 20.3 The graph of the function y is called the logistic curve. To pay off the loan, Phil makes annual deposits of $P at the end of every year in a savings account, paying an annual interest rate of 2%. 55 Functions 16. Chapter 4 practice questions 1. (d) From the graph, minimum occurs at Or, via first derivative test, using sign diagram: xzyf():%'*\*S:1.48331. He makes his first deposit at the end of the first year after taking out the loan. The steel frame consists of a rectangular base, two semi-circular ends and two further support rods, as shown in the diagram. Yes, the 3271t term 8. They would argue that it is built into the structure of the cosmos - a fact that explains why the laws of the natural sciences lend themselves so readily to mathematical expression. If there were three groups being compared, perhaps girls, boys, and teachers, to see if their preferences were comparable, the first three values for the boys would also have to be entered into this table before the additional row of entries for teachers would be pre-determined. 34. When viewed from a distance of 50 m, the top of a tree has an angle of elevation of 22. (a) Sketch a graph of the function. However, with a little bit of practice, anyone can learn to solve them. Determine if 9803 is a term in this sequence. 1B mark awarded 7 6 5 4 3 2 1 Total Mathematics 4 6 3 5 3 2 2 25 Biology & 4 7 9 4 2 1 29 Chemistry 2 7 10 7 3 1 1 31 Physics 6 4 0] 10 3 3 0 35 Test her claim at the level of significance of @ = 0.05 773 Statistical tests and analyses 7 At the top of the league standings in La Liga and Bundesliga in 2017-18 were Barcelona and Bayern Munich respectively. It follows that the angle will be a central angle of a circle whose centre is at the origin as shown in Figure 5.12. Webconfidence. Emerge 7 years later. Broadly speaking, the constructivist views mathematics as a human invention. The French mathematician Pierre de Fermat wrote the conjecture in 1627 as a short observation in his copy of The Arithmetics of Diophantus. The great circle route from Dubai, UAE, to Seattle, USA, comes very close to the North Pole. The pipe makes one full revolution around the pivot point in a day. Prior to the latest syllabus which has 2 maths course options (Analysis and Approaches and Applications and Interpretatons) at 2 different difficulty levels (HL and SL), only one main maths course was offered at 3 different levels (Studies, Standard Level and Higher Level). When we enter the data into a GDC, we can use it to estimate the logistic model for this data as shown in the screenshot. The problem occurs when we make a subconscious switch to nine as an abstract noun in the sort of problems we deal with in Mathematics class. Now we have to use some symbolic rules. (c) Show that the length of string, Scm, required to tie up the parcel can be written as s=40+4w+3% o0 However, the function % is differentiable for x < 0 too. Speed (kmh-!) 12; If 2//2 is the imaginary component of a complex number also with a modulus of 10, what are the possible values of its real component? A woman, standing 3 m from a ( tan6=1 streetlight, casts a 2-metre shadow on level ground. (a) 75 9. The Platonist might want to try to find them somewhere in nature. For example, present computer programs would take longer than the 13.8 billion years since the big bang to find the two prime factors of the number: 25195 027 777 641 249 392 807 613 379 817 195 259 085 823 824 133 844 924 616 912 154 212010 908 137 515 287 859 118 141 281 143 515 831 397 475 836 082 776 095 746 865 198 603 723 438 122 657 043 189 735 700 121 462 163 833 350 167 822 893 662 298 971 097 515 043 815 904 778 899 120 494 020 559 418 330 172 576 010 414 707 885 720 027 707 149 347 459 654 798 674 952 749 040 357 183 595 176 270 748 632 423 810 634 817 445 240 556 184 261 808 282 387 451 432 125 364 048 264 502 896 428 216 184 660 190 772 023 398 018 808 375 401 869 774 377 114 467 527 571 525 489 014 797 987 447 306 657 962 381 429 880 120 971 429 549 920 056 544 926 951 282 784 072 824 100 182 739 201 454 386 378 126 406 844 691 642 422 934 619 178 356 636 204 918 992 165 458 433 236 676 424 373 564 032 290 687 077 691 637 584 256 020 289 391 But this number is indeed of the form of the product of two large primes. (a) 26 () 1 () 2. Many of the streets in Sun City, Arizona, USA, are formed by concentric circles. Webinfinite geometric series calculator symbolab lingering covid cough and sore throat lingering covid cough and sore throat (Think carefully about this.) How much money will Jane have on her 16th birthday? (b) 3,3.1,3.3 (e) 0.64 0475 (@1 14 @ de) _=202000 ) s =519+ 54 0.0549x(x + 95)dx + 2 (b) 233310 85 ~ 27449 (o) 134 barrels Z 95 ~233310m L V=mf 15 5 w5 womf -Saer 35),17() 0 + 2(0.087 + 0.226 + 0.405 + 0.633) + 1) L=1-2X0.3702 o 1.0041s 204 (b)b) =~ %= 166 13 () =6 h 166 &5 5 (b) 2725m () 10s (b) 10625 10. There is a sense in which the world of mathematics is one such virtual universe, containing all manner of exciting and weird things. (b) Find the domain of f~! (e) Interpret the coefficient of determination for the model. [RegEan [aw2ib'x o c . Please, consider disabling it to access contents from our website, Download All Biology International Baccalaureate (IB) SL Past Question papers 2021. The negation of a mathematical truth (like odd + odd = odd) is to utter a self-contradiction or absurdity. Given that sin 6 = %, and g < 0 < , find the values of cos and tan 6 without finding the size of angle 6. Model Answers. 26. oy 11 OF=E- %;% (@ g(lzu 25/15) ( 27(In2 ) +282+35+25+ . Weneed to find where ddvn = Qhence 6.227r 2mr? 5. (f) 6: The non-zero digits and the zeros in between are significant. It is difficult to believe that mathematics is just a mind game that we humans have invented. Use your graphing application to generate a graph of y = sin(x ). 305 Modelling real-life phenomena (e) The ball hits the ground when the height is zero. 55,667 1(3)= V94 V(2516 As before, we find that the rope should be anchored 8.75 m from the post, and that the minimum length of the rope is 32m. We can use a spreadsheet to calculate differences quickly and then average them. 3. (a) Calculate the distance along the Earths surface that an aeroplane must travel to get from Seattle to Dubai. Sum to infinity of met In this section we consider if it is possible to calculate the sum of infinitely many terms in a geometric series. Text extracts relating to the 1B syllabus and assessment have been reproduced from IBO documents. 2. each element of set X All the various ways of representing a mathematical function illustrate that its defining characteristic is that it is a rule by which each number in the domain determines a unique number in the range. == Solution (a) Each year the amount in the account increases by 4.5%. Find a formula for the total amount of silver that will be consumed within years from 2017 and estimate when the known world reserves of 533 thousand metric tons will be exhausted. ib mathematics analysis and approaches sl inertialearning web ib mathematics analysis (b) 4 (a) 4 (0 8 (d) 128 + 128/3i (b) 4, ~2+2/3i (a) (0 * (@) VZe 5 20. = u(wdx 5ltb0) = Jau=u+e ) %{sin(u)} = cos(u)u'(x) = d(sin(u)) = cos(u)u'(xdx f cosudu = sinu + %(cos(u)) = sinw)u'(x) = d(cos(u)) = sin(uu'(x)dx fsinudu = cosu+c e Alm) = T A wwn= -1- = 4] = wwds | fwdu= 00+ ane - -1 tan = L) d ey (o) == = dtanw = w()de euy e(n) = dle W) == e ety () de inlul) == 29 Juce = dlinful) ul) == ioful) () ) ok du=tanu+ U gy = et+c Jerdu=et dx [i du=1nlu| d=tulud++ Table 16.4 Integration formulae Example 16.6 Evaluate: 672 [(sx + 2)8dx (@) [V6x+ 11 dx () [t VX' 8x+13 (d) [xsint(3x?) (@) y = ce + (d/b) ) 14. Such a story is just not credible. 361.95 (4 points of intersection!) Figure 5.11 When the radius ris equal to the arc length s, the measure of the angle is 1 radian 131 Geometry and trigonometry 2 The unit circle When an angle is measured in radians it makes sense to draw it, or visualise it, so that it is in standard position. An influenza epidemic hits a large city and spreads at the rate of 12e%2 new cases per day, where # is the number of days since the epidemic began. 9.30 years Exercise 3.3 1. The conjecture is that the equation A"+ Br=Cr where A, B, C are positive integers and n > 2 has no solution. It would appear that the o sum to infinity, Z(Zl-) = (1\k-1 , equals 2. (a) dd: = kA =A(D) = A, k = 0.13863, A(0) = 50(45)=2250 mg A(1) = 2250 = Age 0601 anaesthetise the dog properly. (@) i* () i*! One way to think about math problems is to consider them as puzzles. (nzv,%e 02 5 184 w1 Euler:{x" ! (a) Convert 30 and 45 to radian measure and sketch the corresponding arc on the unit circle. >% s NRD 32? (a) V2 (e) 13 3. Is he right to suggest that if mathematics is just a meaningless set of formal exercises, then it will not be valued by society? Example 5.4 Great circle routes are widely used in transportations as they give the shortest distance between two points on Earth. The other chapters in this book are all about establishing and using this very special vocabulary and becoming fluent in the methods that connect mathematical concepts into meaningful mathematical sentences. 3 At present, it is not a triangle. The volume of the lobster trap is 0.75 m?. The difference in decimals is: 1000 000. (iv) 1344m 16. that at t = 1, production is greatest. Find the thickness when the brake pads should be replaced. 18371 7. Since the tangent ratio is _ v2 2 2 = -~ 2 1 opposite adjacent we see that it is equal to ; so that tan (45) = 1. Calculate the percentage error of your prediction. Therefore, a,=a;tn1Dd=2+9n-10=9n-7 (b) a5, = 9(50) 7 = 443 Example 3.4 (a) Find the recursive and the explicit forms of the sequence 13, 8,3, =2, (b) Calculate the value of the 25th term. They were deeply particular, they were interested in of the shorter part to the longer part whole. Conversion between degrees and radians Because 180 = 7rradians, 1 = %radians, and 1 radian = 180 An angle with a radian measure of 1 has a degree measure of approximately 57.3 (to 3 significant figures). Find the interest rate to the nearest tenth of a percent. To take care of the sign change, we use some power of (1) that will start with a positive value. (b) Let h(f) metres be the height of P above ground level after t minutes. The cost of a journey can be modelled by _[60+12d, C(d){xwd, 1sd5 where d is the distance travelled in kilometres. You should try to select u to be a part of the integrand whose differential is also included (except for the constant). A decreasing exponential function is said to decay exponentially. All courses, with the exception of Math, Environmental Systems and Societies and ab initio language, are taught at the HL level. Note that the ratio of consecutive terms equals a fixed constant. Does this experiment agree with that theoretical value? (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. $500 is invested in an account paying simple interest at a rate of 3.2% per year. (31 9.91.812 10. L T 4x 4 = Consider the relationship hx) = Vx 4,x=a (a) Find the least value of @ which makes 4 a function. When f"cx) < 0 forall xin some interval of the domain of f, the graph of fis concave down in the interval. of terminal velocity? (b) Find the cost of a journey of distance 9 km. 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. Therefore, the cubic model for this turbine is P =0.297AdV? 39. Algebraically, we can certainly solve the equation to express x in terms of y. (c) Write down a suitable domain for your model. Determine the critical t-values given the sample size 7 and the level of significance a. However, remember that the predictive power of your least-squares regression line is only as good as the strength of the correlation. Since the second derivative is the rate of change in the first derivative, if the second derivative is positive then the graph must be curving up (concave up) since the first derivative is increasing. Find the 95% confidence interval for the true mean, given n=230,X = 64,ands, ;= 12 . If it is, state which term. (a) y=x22x+5 (O y=x2+6/2x+20 10. His skills as a number theorist presented him with the opportunity of solving a very real scientific problem. Find the distance between each pair of points. (a) What is the angular velocity of the wheel in degrees per second? For example (2 3) 5 7 + (x y) a b, = (2+x 5+a 3+y) 7+, 205 6 7/ Matrix algebra We carry out subtraction in a similar way (2 3 1) (X y 8) (Z*X 3= J a b 2 5 7T-b 570 *7) 2 The operations of addition and subtraction of matrices obey all rules of algebraic addition and subtraction. Examples of many-to-one functions that we have already encountered are y = x%, x R andy = As Figure 2.19 illustrates fory = |, a horizontal line exists that intersects a many-to-one function at more than one point. Five minutes later, the angle of depression of the boat is 12. The range ofa function can be found graphically or by analysing the function algebraically. (a) ) 2X 5 4 g2 (b) %+%+%+2x 5 47+ sint+ cE+ & (e) 2sinf+ %cusz +e 3 (@) () 3sin62tan6 +c 21+ 11 20 (b) T 8/ (5)675+ 2) + (d) 3x* dx? Ox3)2 N (i) y = figw) e L s 2 2 e The outside function is fiu) = u ' = f'u) = u 2 The inside function is gix) = 2x 3 Using the chain rule (function notation form): = flgw)-g = (2x 3)72:2 &y dx 2 2x-237 559 Introduction to differential calculus @x=15):3=8x=2]-2 (0% =612 _11 @2x 57 (ii) Rewrite fix) as a product and apply the product rule (with chain rule embedded) =(3x 2)2x 5L fory =uv,u=3x2andv=(2x5)! By definition: sinm=y=0 tan 7 = cosar =% 1i 53 (d) Anarc oflength t = 3717 is equivalent to three-quarters of the circumference of the unit circle, so it terminates at the point (0, 1). Determine if 9803 is a sense in which the world of mathematics is just a game! A number theorist presented him with the opportunity of solving a very real scientific problem 20.3 the graph the. Invested in an account paying simple interest at a rate of 3.2 % per year be. 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