We will consider the case in which equal payments are made annually, and interest is compounded annually at an equal rate at the end of the year. Unstable. It is difficult to believe that mathematics is just a mind game that we humans have invented. The final chapter in the book will support your involvement in the Theory of knowledge course. Initial cost of starting a journey = $3 Cost per kilometre for the first 8 kilometres = 50 cents Cost per kilometre for subsequent kilometres = 30 cents C(9) 10 i 9 We want to find a function to model the cost, C ($), of a journey for s a distance d (km). The big chain of reasoning above is called a proof. () = 3 (i) =272 G (iv) 138 (b) Convert the following degree measures to radians. A building 30 m high sits on top of a hill. A lobster trap is made in the shape of half a cylinder. The midpoint of [AB] is ( i 2* = 622) =(1,2) Therefore,y 2=(x L)ory=x+1 The perpendicular biseeiok Al Fyment [AB] is the normal to [AB] at its midpoint. One revolution takes 20 minutes. There were four chemical elements in the 4th century Bc, and Aristotle thought that the heart was the organ for thinking. 8. In mathematics and science, we refer to this as the period of the function. is equal to the range of f, therefore the domain of f ! 8x,x =4, find h-(12) 4. 197kmh! WebThis book reflects the IB Mathematics Analysis and Approaches for the Higher Level syllabus.It comes with enhanced online acces via web browser which means all to book Hersh thinks that this is not really the same nine. 4. In electrical notation, Z represents impedance, and Z = R + j(X; X,). (a) Geometric (b) r= 7. 8. Using the familiar equation for the circumference of a circle C = 2, Figure 5.12 The unit circle the circumference of the unit circle must be C = 27(1) = 2. From a point on the ground 50 m from the base of a building, a communication antenna is visible on the top edge of the building. We'll also give you advice on how to get the most out of your practice so you can hit your target score on test day. 41. Use your graphing application to generate a graph of y = a sinx. (a) Write down the temperature of the potato at the moment it is placed in the oven. V27 - 9V3 12. 8. A circular irrigation system consists of a 400-metre pipe that is rotated around a central pivot point. This method only works for central angle measurements! Book Description Featuring a wealth of digital content, this concept-based Print and Enhanced Online Course Book Pack has been developed in cooperation with the IB to provide the most comprehensive support for the new DP Mathematics: () 7. This requires a more abstract and conceptual understanding and it is recommended to students who have a natural mathematical ability and enjoy maths (especially algebra, solving linear and quadratic equations, trigonometry, calculus such as differentiation and integration etc). of terminal velocity. They themselves cannot be proved. The IB Math Analysis and Approaches course is designed for students who enjoy the abstract nature of mathematics and have a strong interest in exploring the Now, by using trigonometry, we have another possibility. 549% in region 1, 17% in region 2, and 29% in region 3 (a) 45% Channel 1, 55% Channel 2 (1 1) ) ar=pppi= () ) b - (b) 9 Channel 1 0.45 = 0415 0391 0373 0361 Channel 2 0.55 | 0.585 0.609 0.627 0.639 () Channel1 z% Channel2 ~ 23 17. (a) Round each value to the nearest unit. 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. The Fibonacci sequence is related to the golden number . (392 6. (c) Atwhat time is the tennis ball descending at a rate of 10ms~'? how the value of an independent variable determines the value of a dependent variable. (@) (-1 iand(1+1) (b) (2 + v2i) and (2 v2i) (c) B+2i)and@ -1 s Plot a complex number in each of the four quadrants, each with a modulus of 10. Digital Copy duration: 24 months About the Book This book has been written for the IB Diploma Programme course Mathematics: Applications and Interpretation HL, for first assessment in May 2021. 0 Therefore range is 1.48 < fiv < 9. (3) The model uses values for constants that are established empirically. In this example, we can rewrite our model {i3ifig 3 HSYAVE BEDOREHE 200 P==" = y=axn wheren Z, n 0.625 + 25 represents the amount of energy in a battery after t minutes of use. For example, the half-life of fermium-253 is 3 days. This is called the common ratio of the geometric sequence. (a) Geometric 8. There are 360 degrees in one revolution. Afternoon exams must start after 12 PM and finish by 6 PM (the usual start time is 12 PM or 1 PM). Find the buried treasure, which is at the intersection of the perpendicular bisector of the segment between the coconut tree (T) and the big X, with the line drawn between the green rock (G) and the opening of the cave (C). 7. difference of the sequence, and d=a, a, forall integers n > 1 +d a=a,+d=a +d+td=a +2d a=a;t+d=a +2d+d=a, +3d and so on. We will show both approaches here. Find the products of the complex numbers. 10. Scribd is the world's largest social reading and publishing site. 12. (b) Find the most appropriate non-linear model using linearisation with logarithms to model mass lifted 1 in terms of weight class w. () Find and interpret the value of R? WebFeaturing a wealth of digital content, this concept-based Print and Enhanced Online Course Book Pack has been developed in cooperation with the IB to provide the most comprehensive support for the new DP Mathematics: applications and interpretation SL syllabus, for first teaching in September 2019. This would mean 20 marked items out of 40. A sample of 12 bottles contained the following amounts: {496.4, 500.2, 501.3, 507.3, 498.0, 500.3, 504.7, 508.2, 505.3, 491.2, 490.5, 507.4} Find the unbiased estimates of the population mean . Find two angles - one positive and one negative - that are coterminal with the given angle. /3-8 15.972. Webconfidence. A research vessel is mapping the floor of the sea in a certain area. That is Solution constant. OO0 central angles, with measurements of 70 and 290. Here, the symbol is the Greek capital letter sigma. Find: (a) & T (b) the value of cosx for which d_i =0 18. Example 9.14 A reflector is attached to a bicycle wheel at a point 17 cm from the centre of the wheel. The period is the distance it takes for the graph to repeat itself the distance between two adjacent local maxima or local the length of the period minima, for the sine and cosine functions. The commission believes that the environment cannot support more than 400 deer. + 30 as shown in the graph. Also, since we are interested in finding the point C, we should assign a variable to the x-coordinate (the y-coordinate will be zero). Estimate your arrival time if you depart at 08:00. After t minutes, the concentration of medication remaining in his bloodstream is given by A(f) = 10(0.5)"* where A is in milligrams per litre. (a) What is the total distance flown? IB Math AA vs AI. Find the midpoint of the segment between each pair of points. Example 5.8 The diagram shows a circle with centre O and radius r = 6 cm u /10\ Angle AOB subtends the minor arc AB such that the length of the arc is 10 cm. We have to do a great deal of work to find them through inference and abstraction. From Figure 5.18 we see that P v2 opposite 2 sin(459 = PP _ 2 _ 1 hypotenuse V2 V29 -1,0) Note also that the coordinates of point A must be (%,%) In fact, we see that since the length of the hypotenuse is Figure 5.18 A 45-45-90 triangle equal to 1, the value of the sine ratio sEneceei - . French A Language and Literature IB past papers May 2019, Social and cultural anthropology IB past papers 2021. During 1993 Atthe very end of 1998 During 2007, CD sales decreased by 158 million. This is basically the form z = a + bi Example 6.17 A particular AC circuit has a resistor of 62, a reactance across an inductor of 11 and a reactance across a capacitor of 32. The reasoning we have used here is not specific to a 45-45-90 triangle. Data from the previous graduating class are given in the table. Using the earthquake context of Example 1.8, find the magnitude of an earthquake that is: (a) ten times as powerful as one of magnitude R = 5.2 (b) twice as powerful as one of magnitude R = 5.2 Rules of logarithms Logarithms may seem to be just mirror images of exponents, and rules regarding logarithms may appear to be similar to those for exponents. Its beauty lies in the extraordinary neatness of the underlying mathematics - it all seems to fit so perfectly together: Figure 7 Diracs equation of the electron The physicist and mathematician Palle Jorgensen wrote: [Dirac] liked to use his equation for the electron as an example stressing that he was led to it by paying attention to the beauty of the math, more than to the physics experiments. In 99% of these cases the test does its job and records a negative result. dpP' dt 5. Afunction isa The largest possible set of values for the independent variable (the input set) is called the domain - and the set of resulting values for the dependent variable (the output set) is called the range. 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. @ 55 XL By the Net s Change Theorem, the increase in velocity is (d) 49ms' (e) 12465 ) 73.08ms"! Regardless of the size of the circle (i.e. Ifa o central angle is greater than 180, then the subtended arc is referred to as a major arc. Students have the option to choose between two different courses: IB Math Analysis and Approaches (AA) and IB Math Applications and Interpretation (AI). Those who cannot do it often feel the stigma of failure and that there is an exclusive club whose membership they have been denied. Since we know the first term b, = 5, we can calculate the the 2nd, 3rd, 4th and 5th terms: b,=2(b; +3)=2+3)=16 by = 2(b, + 3) = 216 + 3) = 38 Recursive sequences are introduced in this chapter to help clarify the underlying concepts. So, you cannot say, for example, Jeosdx = feosi? (a) Generate a scatter diagram, find the value of , and describe the relationship between 5km time and marathon time. (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. 104 (b) It appears that over time, the X drawn on the ground has been redrawn and moved! The number of bacteria in two colonies, A and B, starts increasing at the same time. e Solution (a) The nominal rate of return per year is (1 i %) 2 1.056784 or 5.6784% The real rate of return per year is 5.6784 2 = 3.6784% (b) After 6 years the real value, A, of the investment is A = 2500(1 + 0.036784) = $3105.06 ElpliEUK In the section on compound interest, the expression for the amount after years, A, = P(1 + r)', is an example of exponential growth. (b) Find the number of months until the amount in his account doubles. >% s NRD 32? 81 Sequences and series The sum, ,,, of1 terms ofa geometric series with common ratio r# 1 and first term a, can be expressed explictly as _all=r Sn Su -1 or _ar -1 r1 Ttis better to use the first version of the formula whenr < 1 and the second version when 7> 1. Figure 2.20 The point (b, a) is a reflection over the line y = x of the point (a, b) The graph of f ' s a reflection of the graph of fabout theline y = x 39 Functions 15T PR1] The function fis defined for x R by fix) = = it Xt (a) Explain why fdoes not have an inverse f~! method I frden 1 S 0.2 . Solution To use the formula s = r6, we must first $ convert 150 to radian measure. 3t (b) 7 X 10 (e) 2% 10 (h) 1.0006X 10* (b) (d) (b) (d) (8) x =5 logy (h) x = 2logyy @) x=1lny () x=logyy) +3 o x= %mzy (b) R=55 Exercise 1.5 3.3 7. The edge of the water is along the x-axis. The number of bacteria doubles every 20 minutes. 305 Modelling real-life phenomena (e) The ball hits the ground when the height is zero. o y = 0is a horizontal asymptote for the graph of y = fix) o The domain of fis x R and its range isy > 0 a fixed line, but does not meet it at any finite said to be an asymptote to the curve, A function fis an increasing function if * < x, implies fiv) < fixy). Unstable. instead. However, here we can use one of two approaches, introduce the factor 6, as we have done before, i.e., [ FiTde=1 6 [VexF 1T 6dx _1 _if 7Efdu76fu _lui 5 du 1 teTguite =g(6x+1Di+c 2 orsinceu=6x+ [T 11 = du=6dx=dx= dG,then iTde= [Vadt= Jutdu and we follow the same steps as before. 67 Sequences and series The value of an asset, such as a car, decreases over time due to wear and tear. The torque produced by an engine is dependent on the rotational speed (RPM; revolutions min~1). : KI = I(x) = 108009 x (b) Approximately at 28.16 m _ mg dkv dv & :1 @ 13. mxi/180C 0.01745329254 Asasimple example, take 1 + i which has r = /2 and 6 = 45, s0 2 = /2e%5% mki/1805C Ve 0.01745329255 1418 Example 6.18 A resistor, an inductor, and a capacitor are connected in series in an AC circuit, with potential differences across them of 8.0V, 10.5V, and 4.5V respectively. If you did, then your mathematical intuition let you down - very badly. Applications and Interpretation SL setup: Applications and Interpretation HL setup: Grades range from 7 to 1, with 7 being the highest. - @) P = To (b) 70 tigers (0) 7.37yr. 0.8 0.387 (c) Find the value of R? (a) 0298 cms ! The proof is way beyond the scope of this book, but there have been a number of interesting books and TV programmes made about it, including Simon Singh Fermats Last Theorem (1997) and the BBC Horizon programme Fermats Last Theorem (1996). in context. This was a decrease of 6.2% from the previous year. Your goal should be understanding, not just remembering. o o are purely imaginary; the origin is a stable point (but not asymptotically stable). Find the eighth term. Trajectories approach equilibrium and then move away: Saddle point. 13. x = 115 cm, max volume = 403 cm? 2. The typical way to solve this type of problem is to use what we call a mathematical model. Time (s) Height (m) | 0 |05|1.015]20(25|75]|80]|85]|90]|95]100 0 |19.5]46.3 |60.1 |74.0|89.5|86.4|72.4|57.2|43.6|221| 0 (a) Describe the association between height and time for this data. Given that a straight line has gradient, m, and passes through the point (x,, y,), we can first write the equation using gradient-intercept form as y=mxtc substituting the given point for (x, ) gives n=mx +c rearranging for gives c=ymx which can be substituted into the original equation to give y=mx+y, mx, rearranging and factorising gives Vs S Find the equation of the straight line passing through the points (1, 4) and (=3, 10). Many people think that ellipses and numbers and so on are there whether or not any people know about them; I think thats a confusion. In the second, a represents the gradient and b represents the y-intercept. Most people will be familiar with Einsteins field equations and Maxwells equations. 916 Both views produce challenging questions in ToK. (b) Assuming the plane travels at a constant elevation of 10 km above ground level, estimate the actual flight distance. If it is, state which term. At her daughter Janes birth, Charlotte put $500 into a savings account. If C is the sum of the two matrices, then C = A + B where , i = a; + b i so we add corresponding terms, one by one. Click the icon to open the document in a new window (if your browser supports PDF viewing) or to download. Also, the Pythagorean Identity allows us to find the value of the sine function given the cosine function, or vice-versa. (d) Rank the data, then calculate the value of Spearmans r, and interpret it in context. . Graph the functionsy = cosx andy = sinx Look at the behaviour of both functions when x = 0, as shown in Figure 9.24. In one experiement, a patient is given a drug and the patients blood is tested at regular intervals to determine the concentration in millimoles per litre (mmol L!). Solution (a) Following the steps for finding the inverse of a function gives y=Vx+3 replace fix) with y y =x+3 solve for x in terms of y; squaring both sides here x=y*-3 solved for x y=x>13 interchange x and y Therefore f~(x) = x> 3 replace y with f~! (a) Find an estimate of the common difference, by using an average of the differences. & 1fs0, separate the variables, to gc(% = gxydxor q()dy = pxdx 4 y Integratcbodmpans,mgetf%:fg(xidx+curfqmdy:fp(x)dx+c Evaluate the integrals if you can and don't forget the arbitrary constant. i the number (a) Show that the temperature of the water when it is removed from the cooker is 90C. Do faster aeroplanes use more fuel? X, | Approx.y, | exacty, | %error 12 1.400 15 6.6667 14 1.960 2333 16 16 2.789 4 303 9 543 12 4.42543 13 4.67787 14 4.95904 15 5.27081 (a) substitute y equation. Cubic models Cubic models are slightly more complex than quadratic models. i Differentiate each function by applying the chain rule. (@) () x=logsy -3 (&) x=1Iny +1 S@r 5. ) y= 3le+ &4 then i\] _ 8Wf' forg) lgwaf? Sometimes we even know what type of model to expect, and we can use a GDC to find the model and then interpolate unknown data. s -a R N ey samax ARy r AN vy IS R A\ N x=Ce + 2C,e~ General solution: { y=Cie4 = 5C Coe 25t Stable system. She picks her largest cooking pot and the river to fill it up, before to the burning tent to pour river she runs at 22 km h ', A :200m _.--7"Burning 7 L runs to Sarah running the water onto the fire. Using Pythagoras theorem, AP = VAX? It is particularly important that you thoroughly review and understand all of the content in the first chapter, Algebra and function basics, before studying any of the other chapters. Estimation and approximation While the terms estimation and approximation are often used to mean a guess, their inferences are different. Subtract the exponents. @) 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. (a) 50 20t 7 4. () () 23928 (b) 3x% " = 6x> 3x2e = +2) (i) 0.0022 ( () 00249 (i) 2.3681 0.0002; error is divided approx. (a) (b) 7. Furthermore, it appears that the breakpoint between the linear models should be at a distance of 1.0 km. (b) The falling-object model is given as s(t) = %atz + vt + 5o where s(b) is the position at time , a is total acceleration due to gravity, v, is the initial velocity, and s is the initial position. 31n2 - =63 , 254+ N+ = 318 ) 4w (Produccm) +30 + 28 + 23 + =2845 Thus, mean temp = % 125 @10 137 i (b) V47 9. f(x) = Inx Total interest = 37545 27675 = 9870 Exercise 16.3 6. (c) Using the model, we can find the wind speed required to produce 300000 watts: P =29.1V>= 300000 = 29.1V>= V = /10309 = 21.8ms~" Since this is greater than the wind speed that will cause damage, it is not reasonable to expect this turbine to generate 300 000 watts. (a) T=20+50e" ig5. It is very helpful to be able to find an explicit expression for the partial sum of an arithmetic series. The remaining mass includes the @ 21. .v v - Ve - 21 ~ o2t 5t 553 =3 With both eigenvalues negative, the curves move - i L o L Yuir = Yu T H(10x, = Ty, = Lo W 55 = eson g5 34 200 55 -0 085 =3 3 8ss 3\- {xz 12 n=127 ~ . 0o 261 PP g = 50, Sokt = k=1 S =1023 3 = 1125899906 842 623 100 2100 1 = s Mabt= T 1267 650 600 228 229 401 496 703 205 375 We observe that as the number of terms in the partial sum increases, the sum also x increases. (2 s.f) Don't forget to include units. 905 Theory of knowledge Platonic view of mathematics One way to explain why mathematics applies so well to things like bridges and planets is simply to take mathematics as being out there in the world, independent of human beings. What we need is to define a general odd number without committing to a particular one. In Leibniz form: &y dy du dx du dx In function notation form: dy d &= wlewl = fllgw) - gx) The differentiation rule d: Y dx d A du dx e IZ = diu g s sy =204x2 1)-8x Function notation [g\x)}] =flw- g =2u8x = flgm) - gon = 2(4x? Make sure you work in radian mode. )x) = 2 2(5=3x=2 53x11 x= Wk Sl 1. Maryam was the first woman to receive the Fields medal (the equivalent of the Nobel prize in mathematics). X (9.5 X 1072) = (23X 95) X (107! Information about the values of the derivatives of fis given in the table. This means that there is no possible world in which odd + odd = odd (given the standard meanings of these terms). (b) Let h(f) metres be the height of P above ground level after t minutes. The formula V = IR becomes V = IZ with AC circuits. = 0= (7(6.22 21) = 0=r=0 2 + 2;3(5}( + 2150 () The minimum average manufacturing cost per device 34 %(Z'r =75 oo =12.5cm 8. x =2.64km Tdt The circumference of the base of the cone is equal to the arc length of the semicircle. (a) Sce Worked Solutions () 231 (a) Using scatter plot, second degree polynomial appears appropriate. Example 5.10 Evaluate the sine, cosine and tangent functions for = 45, and then use the results to evaluate the same functions for = 135, = 225 and 6 = 315 Solution From the discussion at the beginning of the chapter, we know that sin (45) V2 2 5 08 (45) = -5 G 3 and tan (45) = 1. 0\ i is called the identity matrix of order 2. e : The identity matrix is a diagonal of order matrix where 2, = 1 207 Matrix algebra 1 Two further identity matricesare|(0 L 0 1 0|and 0 1 0 0 01 0 00 0010 0 01 Sometimes, the identity matrix is denoted simply by I, or by I,, where n is the order. Number of prey (in units of 1000) on the x-axis, number of predators on the y-axis It is interesting to look at a phase space diagram that represents each point (x, y) as a combination of numbers of prey and predators. Figure 4.2 Same line (coincident lines) 101 Geometry and trigonometry 1 Parallel lines The first function is flx) = 2x 3 in blue. 3% 5,134 19. 8 2 (b) Now we have 3 slices, hence 3of the total area: %(120) = 45cm Area of a sector Mathematically, a pizza slice is a sector of a circle. Fior was a student of Scipione del Ferro, who had found a method for solving equations of the type x* + ax = b, which is known as the unknowns and cubes problem (where a and b are given numbers). Since it doesnt make sense to include negative heights, we should check the position function and establish the domain of the function: sit)=5t2+18t+1=0 = t=0.5472 or t = 3.655 So, the rocket hits the ground about 3.66 s after the time of launch. (@) 9+ 12i (b) 6 8i (c) 33/2i . All DP mathematics courses require students to appreciate the use of technology in mathematics and become proficient with graphic display calculators. Virginia Woolf said about art, Art is not a copy of the world; one of the damn things is enough, and the same could be said about models. The Analysis and Approaches course is suitable for future mathematicians, engineers, scientists and economists. (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. No remember how the parameters a, b, and d affect the graphs. For students taking their exams in 2021 there is a big change to the IB syllabus there will now be 4 possible strands: IB HL Analysis and Approaches, IB SL Analysis and Approaches, IB HL Applications and Interpretations, IB SL Applications and Interpretations. (a) y, 1 - G #, ) 12 10 | Lk I | 8 s 6 4 -3 2 ) 2 (c) any value 0 < x < 24.3 0r 65.7 < x < 90 0 0 2 4 (h) 2x+y =29 12. What Is The Difference Between the Spinal Cord and the Backbone? (b) What are the fixed costs? So it is far more. L o EIs Remermber that the graph may appear differently depending on the viewing window chosen on your GDC. We are ready to use calculus to find the extrema, in this case the minimum value of T. Since this is a complicated expression, use a GDC to graph the derivative. s 9=l s o Forexample (2 19 sin*QO) cos(90), (b) () X=(C-BA" 5 2 -3 10. 14. (a) 24 +e (b) 40 @72 @0 (h) ~268 @ 2 11 ) 1n(7) )n 2 (m)3 () 7+ 1 ) 4 o0 (@) 12 () 6 @) %v[l Fcos + @ (0) x~193.AC(193) ~ 85 (d) MC(193) ~ 8.5 () AC(400) ~ 24.25, They should sell it for 26.25. The largest object in this region is the dwarf planet Ceres, located 2.77 AU from the Sun. Weneed to find where ddvn = Qhence 6.227r 2mr? Their marginal cost per day is given by the following model MC(x) = 0.0012x2 0.018x + 25 where x is the number of fans produced. (a) 81,243 (b) 3" 5 a3a, 1 k:370:1_23 65 T=483C m= .06 = 8 minutes and 4 seconds 10C (b) 124C k= 5.64 minutes 15 c=25 3hours and 19 minutes p+q=47.4p+q=53 3. Then they emerge and enjoy a relatively brief period courting and mating before laying eggs in the ground and dying. Then dx = du, and f% dx = f}u(*du) = j% du = Inu + , u > 0, butu = x, therefore whenx + 2,and so du = 3x*dx. Sets A set is a collection of elements that can themselves be sets. Therefore, as the figure illustrates, the statement on the right can be made about the graphs of a pair of inverse functions. From building bridges to controlling strategy in football, mathematics lies at the heart of the modern world. Write each equation in logarithmic form. Point B is 120 km to the east of point A and 40 km directly south of the boundary. (6 (g) . 2 (2 () Eigenvalues: 3, 1, cigenvectors ( 1),(5) T General solution: Pan g e T g iz > . International Baccalaureate Organization, 2005-2023, We use cookies on this site. For the relationship at the start of this section, we can write: Population P is a function of time #: P(t) = P,(1.02)" As with the population P and the time , many mathematical relationships concern how the value of one variable determines the value of a second variable. (f) 100.03 (g) 0.002 Give each value correct to 3 significant figures. Example 9.16 shows another way of findinga logistic model. x=4d,y=-3 .m=2n=3 . These are each offered at two difficulty levels (higher and standard level). (a) 2 ) :(%+ %l) ( No S i=e3, i = 3" and by de Moivroes law, ) Ja] =5 6. WebIB Past Papers All International Baccalaureate (IB) Maths Past Exam Papers for Analysis and Approaches (standard and higher level) and Applications and Interpretation (standard and higher level) can be found below. Also, some may consider the average to be 2845 12 _ 537 since the 284.5 is the total for 12 months and we average by dividing by 12. 1 34. In the Kelvin temperature scale, the freezing point of water is 273 K and the boiling point of water is 373 K. (d) Find a linear function that converts Kelvin to Fahrenheit. While generalising is an important part of mathematics, we have some very good reasons to do it: as you will see, trigonometric functions work very well for describing periodic behaviour - that is, any phenomenon that repeats itself with a fixed interval. 3. ? (Note that this percentage is rounded, too!) 914 We will leave the last word on this subject to Dirac himself, writing in Scientific American in 1963: I think there is a moral to this story, namely that it is more important to have beauty in one equations than to have them fit experiment. (b) Estimate the length of a 3-year-old Gila Monster. Give your answer in the form ax + by + d = 0, where a,b,d Z. One official size 5 FIFA football is 22 cm in diameter. As we've emphasised before in this textbook, extrapolating assumes that the pattern you've observed will continue. (b) Use your model to calculate the maximum theoretical power generated when the wind speed is 10ms ! The first term of an arithmetic sequence is 7 and the common difference is 3. Consider the following functions: 2 fx) = 2x75+ 3,x =0and glx) = 13x 4 10 xeR (a) State the range of fand of g . 7. If it is, state which term. ~ % il =8 Geometry and trigonometry 2 Geometry and trigonometry 2 Learning objectives By the end of this chapter, you should be familiar with o calculating sector areas and arc lengths o understanding radian measure and converting between degrees and radians understanding the definitions of sin 6, cos 6, tan @ in the unit circle and using the trigonometric functions for all real numbers sin 6 + using the Pythagorean identity, tan 6 = , and graphical methods of cos 6 grep solving trigonometric equations in a finite interval using and creating Voronoi diagrams, including: terminology (sites, vertices, edges, cells), adding a site to an existing Voronoi diagram using the incremental algorithm, nearest neighbour interpolation, and applications of Voronoi diagrams to distance/area and function interpolation. (b) Find the value of b. In other problems, we need to be careful to consider the endpoints of the domain as possible optimum solutions. 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