On this view, mathematics is just like the glass bead game. 11. this will take, use a GDC to evaluate the function for total time at x = 281 to obtain 91.1s. The sequence G,is defined by G, = G, = 1and G, = 2G,, + G5, n>2 A second sequence b, is defined by b, = G G, n=1 (a) Use a spreadsheet to find the first 30 terms of both G, and b, (b) Hence, write down the limit of b, as n tends to infinity, 1}9:27 b, Arithmetic sequences Here are three sequences and the most likely recursive formula for each one. The first term of an arithmetic sequence is 2 and the 30th term is 147. Solution (a) This is an arithmetic sequence whose first term is 2 and common difference is 9. 12-518 15 -o3sarsanion 10718675296 0-53B1s50161 (wHB)/C 12. For each function below: (i) use your GDC to graph the derivative of the function (ii) hence, find the locations of all local extrema and justify each using the first derivative test. V = lwh = x(30 2x)21 2x) =4x3 102x> + 630x (b) Since the paper is 21 cm wide, we need to cut less than %(21) = 10.5cm from the edge in order to make an open box. If we were to make the modelling assumptions more realistic, the mathematics in the model would become too complicated to solve easily. Use the rules of exponents to write each of the following in the form 3, wheren Z. In electrical notation, Z represents impedance, and Z = R + j(X; X,). The humanistic philosophy brings mathematics down to earth, makes it accessible psychologically, and increases the likelihood that someone can learn it, because its Jjust one of the things that people do. in context. We will consider impedance in AC circuits here. = 0.297(80)(1.225)V?3 = 29.1V? A language act would be a registry officer saying I pronounce you married The use of language in a performative manner creates social facts. As such, the value of the annuity is the sum of 30 terms of the geometric series with first term $1200(1.02) and common ratio 1.02. Mathematics Applications and Interpretation for the IB DiplomaHigher Level provides comprehensive coverage of the new c, Mathematics Applications and Interpretation for the IB Diploma Standard Level provides comprehensive coverage of the new, Worked solutions for the Mathematics Analysis and Approaches for the IB Diploma Higher Level Pearson book, Mathematics Analysis and Approaches for the IB Diploma Higher Level provides comprehensive coverage of the new curriculu, Mathematics Analysis and Approaches for the IB Diploma Standard Level provides comprehensive coverage of the new curricu, R ==} ] Mathematics Applications and Interpretation for the IB Dlploma (UYL R ety IBRAHIM WAZIR GAI 3338 JIM NAKAMOTO EEEEEEEEEEEEEE e Published by Pearson Education Limited, 80 Strand, London, WC2R ORL wwwpearsonglobalschaols.com Text Pearson Education Limited 2019 Theory of Knowledge chapter authored by Ric Sims Development edited by Jim Newall Copy edited by Linnet Bruce Proofread by Eric Pradel and Linnet Bruce Indexed by Georgie Bowden Designed by Pearson Education Limited 2019 Typeset by Tech-Set Ltd, Gateshead, UK Original llustrations Pearson Education Limited 2019 lustrated by Tech-Set Ltd, Gateshead, UK Cover design by Pearson Education Limited 2019 Coverimages: Front: Getty Images: Busi Photography Inside front cover: Shutterstock.com: Dimitry Lobanov Weare grateful to the following for permission to reproduce copyright material: Text pages 904-905, Edge Foundation Inc.: What Kind of Thing Is a Number? (a) 1216 ( w,=2u, 31 () 893871739 0T 19 =220, 2187 andu, = 6 andu, =95 __ og _ u=6(3) (b) 4, =952 19\ ) u, _= 100(20) andu, =100 32 @ Tag576 u,,:%u,, candug=2 33. T M 17 Geometry and trigonometry 1 Solution If the mound is considered to be at M(0, 0, 0), then the starting point up the tree is T(30, 60, 20). Those who can do it often find themselves labelled as nerds or as people who are, in some sense, socially deficient. Example 14.18 can also be solved without using calculus. Note that value 5E+4 for the y-coordinate of the vertex is the GDC's version of scientific notation (or standard form). Generally it is also possible to identify this point on a graph. 92 (c) For the interval 1 < < 3, find the time at which the velocity is a minimum and find its value. )3) = x* = C(1 (+ 2)v2) 3 Substitute v = % into this equation and simplify SR x:C(1+2 F) 848 2\ = x(l o ZF) =C= (x2+2y%)%=Cx? Solution The gradient of AB is 76;225 = 1, so the gradient of the perpendicular to - _ [AB] is 1. (a) -85 (b) u, =35 15n () thy =ty 15anduy =2 2. Exercise 1.4 1. The purchase price of the car is $16,500. 5 Figure 3.1 Triangular pattern of dots The pattern can also be described, for example, in function notation: f) =1f2)=3, f(3) = 6, etc., where the domain is Z* The first unit represents one dot, the second represents three dots, etc. That a particular piece of paper is money is a social fact that does make things happen. (d) Find AB and BA (e) Find xand y such that C= D 2. () t,=3u, andu, =4 2. (@) a,=2n3 (b) b,=n+2 () c,=cp-1+2,andc; = 1 (d) 2,5,7,12,19, &) 2.-512-15.., 2. Like other areas of knowledge, it possesses a specialised vocabulary naming important concepts to build this map. There are 3 times zones (A, B and C) for the first time this year (May 2023 exams). Now, instead of merely keeping a tally, after rolling the dice 6 times, find the mean of the outcomes. 2. _ Solution (a) A graph of fproduced on a GDC reveals that it is not monotonic over its domain ] o0 , co[. (b) Give a reason why linearisation using logarithms should not be used for this data. h(t) B(12.5, 1.5 ( ) 15 0 A(6.25,0.6) 0 24 48 Figure 9.32 Diagram for question 1 72 How much time is there between the first low tide and the next high tide? Let (a,) be an arithmetic sequence with first term @, and a common difference d. We can construct the series in two ways: forwards, by adding d to a, repeatedly and backwards by subtracting d from a, repeatedly. 100 X T -| T =l 1 == I = ! (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. Letgix) =2x>3x25x+ 1, x+2x23 (a) Sketch the graph of y = fix) for 2 < x < 3, labelling the coordinates of any axes intercepts. + YB? I ) _ . EINSTEIN'S L FIELD EQUATION i) LR Figure 5 Einsteins field equation 1. The Maths HL class surveyed random samples of the groups, and their findings are given in the table below. So, of the whole population tested there would be a total of 99999 + 99 = 100 098 positive results. In the circle in this diagram, the value of the area of the shaded sector is equal to the value of the length of arc / (ignoring units). 23. Consider the potential difference given as a sinusoidal function, V= 10cos (mt + %T) whose potential difference (amplitude) is 10, with a phase angle of % These characteristics can be illustrated by the Argand diagram shown. When a skydiver is falling towards the Earth, she will accelerate until the force due to gravity becomes equal to the force due to air resistance. 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. 3.32 2,34.32 ] 5. (b) Indicate the degrees of freedom, v, in this analysis. Answers will differ. (a) ) \5 ks - 2 Figure 9.31 GDC screen for solution to Example 9.15 (f) We do not consider (g) Although this model seems reasonable, it assumes that the rotation speed of the wheel is constant. Colour Teachers 12th grade blue | green | beige | pink | yellow | Total frequency 9 8 11 6 6 40 12 7 5 6 9 39 11th gmde 11 10 8 7 9 45 10th grade 10 9 5 8 9 41 Test the claim that the colour preferences are independent of the school grade or group at the level of significance of & = 0.05 (a) State the null and alternative hypotheses. (a) Express the impedance of the circuit as a complex number in the forma + bi (b) Express the impedance in Euler form, with 6 given in radians to 3 significant figures. 10392 (20-25) - B12x) Remember that LI4E + 3 = 1.14 X 10 = 1140cm? Z, + Z, = 92 + 55j Next convert Z, Z, and Z+2Z, with your GDC into Euler form: Change the output mode to Euler form, then enter the impedances. Is the flying speed of animals related to their overall body length? The data in the table shows a sample of the data collected. Mathematics applications and interpretation International Baccalaureate past questions Standard Level and Higher Level plus Marking Scheme. (a) (b) 2 7E ) 13 () 2 (8) 29 % 3] 24 il () 2 (h) 6/3 | o5 {n | | 5. The resulting initial-value problem leading to the population growth can now be written as & & (b doyly where by and d, are positive constants representing specific birth and death rates. B) V2mrt 11. Determine if 995 is a term in this sequence. Figure 5.4 Anarcand its We use the same logic we used for the area of a sector: we need the size of the central angle for the arc, and then we calculate the portion of the circumference using a ratio. 209 Matrix algebra 1. iy 4xland = e+ x Lan e*+ linto (b) y(1) ~ 0.32768 () y(1) ~ 0.348678 4401 (d) Actual value is 0.367 879 4412. Theres no math without people. P(X=10) = 1 P(X = 9) ~0.0139 (a) 0.817 (b) 1 (c) 0.0162 (a) 0.0338 (b) 0.0245 (c) 0.783 (@ 3 (b) 0.101 (c) 0.000215 (a) 0.0025 (b) 0.9826 (c) 0.9999 (a) 0.1396 (b) 0.1912 (c) 0.9576 (a) 0.8187 (b) 0.5488 (a) 0.9877 (b) 0.999998 (c) 0.0000244 (a) 0.0334 (b) 646 (c) 98" P(X=7) =1~ P(X < 7)~0.212 (Note: In a continuous 2.00004 5. Figure 9.30 GDC screen for solution to Example 9.15 (c) 334 We can see that a local maximum is at t = 5 minutes. A stone is dropped down a 30 m well. 114 Determine the areas of the two possible triangles produced in question 4. Also, the Pythagorean Identity allows us to find the value of the sine function given the cosine function, or vice-versa. Hence, the second derivative of displacement is acceleration. (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. =9 Solution (a) Since the diameter of the Riesenrad is 200 feet, the wave height of the model must be 200, so the amplitude a = 100 (b) Since the minimum value of the function is 12 feet, the principal axis must be located at 12 + 100 = 112. Solution Since the arc measures 150, we use that fraction of the circumference: Figure 5.5 Diagram for Example 5.3 L 150 =0Qq 360(2 57 X 3) 3)i= = > =7 7.85m (3 s.f) 1. 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. In this case, the acceleration due to gravity at the Earths surface. The relationship between two quantities how the value of one quantity depends on the value of another quantity - is the key behind the concept of a function. (a) The Voronoi diagram has i 3_4, -32-% (c) 25 (0) 36 4,8 L (b) y = x2+ 14x + 50 ) y=xx1 (d)y:x2~3x+% (f) y=x2+43x+15 +1) () y=x7-6/2i -8 260 = (291 = (8)n = ( 1) (8)21 = g 2% 4i 2% = (222" = (4i)> = (16i2)" = (~16)* 2% 13. 11. derivative changes sign atx = c, then (c, fio) is apoint of inflection. The methods and concepts of mathematics, therefore, are quite unlike anything to be found in the sciences, although they do seem to bear a strong resemblance to the arts in terms of the setting of the rules of the game and the use of the imagination. Afternoon exams must start after 12 PM and finish by 6 PM (the usual start time is 12 PM or 1 PM). = (25 % + 132 I=Vx2+49 m=y(25 x? Simple interest is interest calculated only on the initial investment. So all are install to solve our maths problems, thank you for this life saver. 26. Find the value of d. (c) Calculate the number of minutes until a person riding the Riesenrad reaches the top of the wheel. Determine the equation of the line perpendicular to y = 2x 6 that passes through the point (4, 2) 8. :wo Exercise 7.5 rA = L. () y-axis reflection. (b) An alarm sounds when the amount of fluid remaining is 50 ml. Give your answer correct to the nearest minute. 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? Therefore a reasonable domain for this model is 0 < t =< 4.51(s) The range indicates the possible heights of the ball. Therefore, given L, we can determine a relationship between C and k. Since the curve must pass through (x, %) we can substitute and simplify as follows: fo = 1+ Ie"x - LZ: 1+ ek" 2=1+Ce* 1ew - The equation x = lnTC also gives us some insight into the location of the point of inflection. Also, at the end of each chapter there is a set of practice questions, which are designed to expose you to questions that are more exam-like. Find the quotient zl State your answer in a + bi form. (a) 12,20 TilneeT - 3\z) (54 (f) z(? ) m(zmm+2) 256 W) fzmxx)dXN%(z_z) +2(3-f3) + 4+ fd) + - + 9-9 + 10 f(10)) )+ ) :%(2-0+2(3-1.5+4~z+m +9:3)+10-0)~ 127 10, (a) i 35 8] S d) 2/3 - 66273 17 166 . () = 3 (i) =272 G (iv) 138 (b) Convert the following degree measures to radians. We can use the logic from section 5.1 and write the area of a sector as 0 1 A=sdmr) =26r Inacircle of radius r, the area of asector witha central angle measured T Sl 2 Again, be careful! In 99% of these cases the test does its job and records a negative result. His data are Temp | Solubility C)_| (gper100mD) 0.0 66.7 4.0 71.0 10 76.3 15 80.6 21 85.7 29 929 (d) Use your least-squares regression line to predict the solubility of NaNO; at 25C. = 2u,,3+ 1 andu; =0 (h) a,=a,;+2anda; =1 Find a recursive definition for each sequence. 2k 3] = g = 1 )T 5= X0 =AYy =02 13 Since the domain of gis x ] oc, 0], then the range of g~! 609 Further differential calculus Example 14 A supply of four metres of wire is to be used to form a square and a circle. 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. The solution lies in the chain rule. i1T j 0 0 0 320 | Graz Vienna salzburg 191 Graz Salzburg Innsbruck Linz 298 478 185 282 461 220 188 Innsbruck 135 320 Table 7.3 Distance (in km) between Austrian cities. (b) Calculate the value of Pearsons r and interpret it in context. Vectors are covered in chapter 8. (a) Find a linear function to model the cost of model A. The distance between the two posts is 25m. @@ B=4"'C 10 (b) DA = (o 00 11. Such a story is just not credible. Solution (a) Substituting into the formula gives: 232 X110 10712 = 10log,(3.2 X 107) = 75.1dB n(3.2 X 1079= = 1olog,,,( 5 . WebMathematics Applications and Interpretation for the IB Diploma Higher Level 9780435193447. much? But this means that mathematics is really radically different from other areas of knowledge, including the natural sciences. This avoids having too many negative numbers in calculations. Find the value of n. 3. (a) ) L @) fin il= glinx + R = e w 0 Exercise 20.1 (@) y = cos~1(31) . Interest is paid at the end of each year. Example 9.5 The demand for a certain style of jeans is modelled by the function D = 1000 5p where D is the number of items sold and p is price of the jeans in euros. Trigonometric models Trigonometric models are well-suited to describing repetitive phenomena: tides, seasonal temperatures, the motion of wheels, etc. where Pis the power in watts A is the area swept by the turbine blades (swept area), in m? For example, the approximate distance from the Earth to the sun is 149600000 km. (a) Formulate a function to model the length L of a Gila Monster years after birth. Our aim is to support genuine inquiry into mathematical concepts while maintaining a coherent and engaging approach. (b) = a. Likewise, In C = 0 when C = 1, so the x-coordinate of the point of inflection will be zero. () sinR (b) cosR (c) tanR (d) arcsm(;) (e) arctan G) f) s 3. identity is an equation thatis true for all values of the variables. 9432 8. 10. _eeeeeeeeeee Solution (a) MCis the rate of change of cost, thus, cost itself is an antiderivative of MC. @ 55 XL By the Net s Change Theorem, the increase in velocity is (d) 49ms' (e) 12465 ) 73.08ms"! This makes sense. (d) In words, accurately describe the motion of the object during the interval -1 0 so production costs are increasing forallx >0 (d) Here we need a new model. 625 Further differential calculus (a) Write down an expression for T in terms of r, [ and 7. Chapter 4 practice questions 1. Inflation means an overall rise in the cost of goods and services. Information about the values of the derivatives of fis given in the table. These materials include: * Interactive GeoGebra applets demonstrating key concepts * Worked solutions for all exercises and practice questions * Graphical display calculator (GDC) support To access the eBook, please follow the instructions located on the inside cover. 30 20. (a) 2,-1%3 (b :(%Ar%i) V2i, v2 2i - & (b) 43es o 6. Figure 5.16 Diagram for Example 5.9 Solution (a) s:r(?#S:Q(z?v) =6m The length of the arc is exactly 677cm. In mathematics, concavity is whether a graph curves up (concave up) or curves down (concave down), as shown in Figure 14.9. The prism has a height of . This is a position taken by surprisingly many mathematicians. 10. Mathematical objects such as perfect circles and numbers existed in this real world; circles on Earth were mere inferior shadows. There is a common misconception that November exams are more difficult. 3 o7 o IR 1.27m 14. oy o 7 2 12,2 25 15,275 17 157 . Determine if 9803 is a term in this sequence. Technically, this comes from the fact that the whole numbers are closed under the operation of addition because they form an important structure called a group. made at the beginning of each year. The population growth rate in China was 0.5% and the population was 1.379 billion (source: World Bank). If it is, state which term. Then the two variables P and are related by the formula P = P;(1.02)", where P, is the population at time = 0. (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. (a) a=4 ) (i) d=-5 6. Find the distance between each pair of points. 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. Speed (kmh-!) What is the difference between Mass and Weight? 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. A sample of size n = 23 from a normally distributed population shows X =100 and s,_; = 12. Figure 5.10 shows three circles with radii of different lengths (r, < r, < r;) and the same central angle 0 subtending (intercepting) the arc lengths s;, 5, and s;. Xpry1= =5, X, By,(y!{xz =3.337 i Yurr = Yu h(6x, = yi) Bt x =302 4 2202353 | iy Y2 =429 Particular solutiof (b) Eigenvalues: 2, 5, eigenvectors ()() 2 Trajectories approach equilibrium and then move away. The distinction between pure and applied mathematics becomes blurred in the hands of someone like the great Carl Friedrich Gauss (1777-1855). (a) 197.1 million 41. The variance is unknown. This type of truth is independent of place and time. chain rule. (a) C=08d+4 (h) x = 3log,y (i) x=2lny o -1 () 5505C @ F= %K 4504 () () 4594F (i) 273C 3. Find the measure of angle AOB in degrees to 3 significant figures. = (AB')* + (z, z)* =@ x)+ () (g 2) =>AB=[tc,x)+(,~y) + (& 2) Given two points in space, A(x,, y,, z,) and B(x,, y,, z,), the distance AB is found by taking the square root of the sum of the squares of the differences in each dimension. Enter the observed values into a 2 X 4 matrix A. 2 o ) go=b--3 (c)}ZbVb:5 (d) Point of tangency is at g(1) = 3 = (1, 3), gradient is 3 (given), equation of Tisy 3 = 3(x )= y = 3x. (d) Use your model to predict the depth 500 m from the shore and 1.5 km from the shore. 15. Ly oln=oo 8= sBT ] p d g 1-0) ].f|r|0 = 10. 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 Hence, the bird is at B(30 - %, 60 - %, 20 118 %) = B(ZO, 40, %) i A cuboid measuring 10 cm x 12 cm x 20.cm is cut into a rectangular pyramid, keeping the face that is 12 cm x 20 cm as its base and 10 cm as its height. Figure 5.19 Calculating values of trigonometric functions from coordinates of points on the unit circle Symmetry is crucial to finding the values of the trigonometric ratios in the unit circle. (a) V= )(zf 2 = 0.3702 11. (@) flry=x+2 (b) fiy=322t+1 g =%-20 (d) fiH) = (t VRt +3) (&) gu) = u 4u? Potential difference across a resistor Vy is in phase with the current; however, potential difference across a capacitor V lags, and potential difference across an inductor V; leads. What problems does it solve or what questions does it answer? (@ () P(X=24)~0.105 (i) P(X = 22) ~ 0.0925, P(X = 26) ~ 0.0902 (iif) P(X = 20) ~ 0.0616, P(X = 28) ~ 0.0595 (b) As the probability is the highest at the expected value and those adjacent gradually decrease, the distribution will likely fit a normal curve. Diracs own equation for the electron must rate as one of the most profoundly beautiful of all. Predict the temperature of this solution, or give a reason why you cannot. (a) The value of Pearson correlation coefficient for the time it takes an athlete to run 5km and the time for her to cycle 30 km is r = 1.21 (b) The fuel economy of passenger cars decreases linearly with the mass of the car with Pearsons r = 0.78 (c) Among mammals, those with greater average body mass have longer life expectancy; the rank correlation coefficient is r, = 0.85 (d) For a set of (x, y) data, the least-squares regression line is y=253.52x with r = 0.64 2. 197kmh! () x-intercept is at (0.439, 0). 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. Figure 5.10 The ratio of arc length to radius remains constant The ratio ; indicates how many radius lengths, 7, fit into the length of the arc s. For example, if % = 2, then the length of s is equal to two radius lengths. Either one of the players has achieved checkmate, or a stalemate (a draw) has been agreed. (@) 2,2,6,10, (b) 10.07,9.95,9.83,9.71, 3 1 7 W B (c) 100,97,94,91, 3. 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. | @x=0x=-2 x="% V3 (e) 2. (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. Find the sums of the complex numbers. Time (years) Amount in the account () 0 2000 1 2000 + 2000 2 2000 + 2000 X 0.05 + 2000 X 0.05 = 2000 + 2000 X 0.05 X 2 3 2000 + 2000 X 0.05 X 2 + 2000 4 2000 + 2000 X 0.05 X 3 + 2000 X 0.05 = 2000 + 2000 X 0.05 X 4 X 0.05 > 0.05 = 2000 + 2000 X 0.05 X 3 Table 3.1 Simple interest calculations This appears to be an arithmetic sequence with five terms (as both the beginning and the end of the first year are counted). Figure 5.8 Angle in standard position Two angles in standard position that have the same terminal sides - regardless of the direction or number of rotations - are called coterminal angles (Figure 5.9). By continuing to use this website, you consent to our use of these cookies. + -3+ i (d) 1 18(3x2 + 5) =4 vx + 3) 36 T 5 (2x + 3)8 g -FFIS L @ S, +c 18x (h) %ln[costz ~D+3)+ (@) tan(s +e ) Lsinim +3) + R Leen e W %zu itc () Inlln2z] + (m)%ln) fe 4. 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. Our normal approach would be to draw a right-angled triangle. They also identify common errors and pitfalls. A typical example of a problem in this category is how to solve a particular type of equation. 133 Geometry and trigonometry 2 3 radians ] 6 radians 4 radlab\ y)f/ 5 radians Figure 5.13 Arcs with length of the radius placed along the circumference ofa circle Figure 5.14 Degree measure and radian measure for common angles Figure 5.14 shows all of the angles between 0 and 360 that are multiples of 30 or 45, and their equivalent radian measure. Each course is designed to meet the needs of a particular group of students. Assume that only 100 people in the whole country have the disease. Find the areas of the triangles in question 17. (b) 0.439cmss! Inverses and reciprocals IfsinA = %, its inverse is expressed as A = a.rcsin(%), often noted on a GDC as si.nl(%). For instance, should a relation be expressed as y = a*, then x is the exponent of the base a which yields the quantity y, written simply as x = log, y When the base is 10, we typically do not write it. + 322 2 - 20 - 32 cos 60 = V [ 1 /400 + 1024 1280 - = D = AB =784 1. If the impedance is X, = 4 + 2j Q, what is the magnitude of the potential difference? Read more about cookies, Mathematics: applications and interpretation SL, Mathematics: applications and interpretation HL, develop mathematical knowledge, concepts and principles, develop logical, critical and creative thinking. At Grande Anse Beach the height of the water, in metres, is modelled by \(9@ the next period, since the question asked only above the first ten minutes. athlete performed worse than we would expect? This book is designed to complete the course in conjunction with the Mathematics: Core Topics SL textbook. Your teacher will advise you on the timeline for completing your exploration and will provide critical support during the process of choosing your topic and writing the draft and final versions of your exploration. (a) 3910 (b) 3901 (c) 8200 (d) 8200.0 (e) 100.3 6. First convert 42 to 24, then add the exponents. That is, the angular velocity of each wheel is 1.5 revolutions per second. 0.3 316 0.4 d35 0.5 =500 (c) Find a least-squares regression line to predict velocity v(f) based on time . We can use our knowledge of the function to choose more appropriate settings. French A Language and Literature IB past papers May 2019, Social and cultural anthropology IB past papers 2021. Nevertheless, it often happens that pure mathematics created for no other purpose than solving internal mathematical problems turns out to have some extraordinary and very practical applications. Identify the claim. (b) Coincident: the second is a scalar multiple of the first. 45 8. (d) Since the first maximum is reached 5 minutes after the first minimum (when the person board the Riesenrad, we can conclude that one complete rotation takes 10 minutes. 10. (c) i (d i . The boundary XY between the two types of soil is a straight line running from east to west and point A lies 50 km directly north of it. (c) Calculate how long it takes for the temperature of the water to reach 56C. Therefore, the percentage change per minute in the difference between the coffee temperature and room temperature is 21.3% g 75e-020s aziye 5(0.787)* Since1 0213 = 0.787, we confirm our result st the cois temperature difference is decseasing by a fxed rate of21.3%. Express each value in simplest form. 19. (b) After the initial 4 years, the employee has the option to buy the car from his company by paying 50% of the purchase price. = /702 + d* = /4900 + d* Since XY = 100, then PY = 100 d Using this in triangle PYB gives PB = /PY? The simplest Japanese kokeshi (wooden doll) is formed from a sphere (head) on top of a cylinder (body). IB Applications and Interpretation Formula Sheet. Consider a taxi company that has the following structure for charging customers. And to ask extra questions about the question requires the premium which students like me can't spend that much on an app. (b) How much interest has been earned on the investment in 30 years? With 6, = 40 and fit) = = (ksint? Sequences formed in this manner are called arithmetic sequences. Measurements Volumes and in 3 dimensions gEloc] You should already be familiar with these formulae for volume and surface area. correspondence (mapping) between two sets X and Y in which corresponds to (maps to) exactly one element of set Y. This is best done application such as GeoGebra that allows you to observe each parameter has on the shape of the graph. 7 1 dy = f3x2dx:>f(;+ 4y)dy: f3x2dx ln|y| T2y =ik The solution is exact, but implicit, as it cannot be written in an explicit form y = fix). Analysis and Approaches is u, = @) and r :g 0+ 8 8 =25+0 =125 Since V(10, 16) and W(15, 15) are both /65 units from 30 best location for a new bank branch is at L(48, 24). Therefore, a reasonable domain is 0 < x < 10.5 Remember, when usinga GDC to analyse a model, it is important to choose the viewing window carefully. The resistance of three resistors connected in parallel is given by: % = Ril F Riz k= i So, in AC circuits, impedance is defined as 11Xl 7z T T % 195 Consider when there are two impedances in a parallel circuit, Z; and Z,: 1 %72 1 _ 4 Zy+Z 2 Z,Z; 142 Z= zz | n+a which is the product of two complex numbers divided by their sum. We are also trying to minimise the total length of the rope, so we will assign variables to each part of it as well, labelling them [ and m. Finally, since the posts are 25m apart, we can label the horizontal distance from R to the taller post as 25 x il Q x R ~2m B5-x S Now, use the diagram to try to find a general model. 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. Ask extra questions about the question requires the premium which students like me ca n't spend that much an. Instead of merely keeping a tally, after rolling the dice 6 times, find the areas knowledge... Of this solution, or a stalemate ( a ) this is antiderivative... 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Is a position taken by surprisingly many mathematicians use our knowledge of the function to model length... In this analysis in Figure 5.12 consent to our use of these cookies more difficult on. X T -| T =l 1 == i = population was 1.379 billion ( source world... Papers 2021 _ ; = 12 @ x=0x=-2 x= '' % V3 ( ). Speed of animals related to their overall body length a total of 99999 + 99 100. The sine function given the cosine function, or a stalemate ( a ) V= ) ( i LR... Vocabulary naming important concepts to build this map: the second is a term this! Were mere inferior shadows and Literature IB past papers 2021 c = 0 when c =,! = 40 and fit ) = 3 ( i ) d=-5 6 someone like the great Carl Friedrich Gauss 1777-1855... This map a linear function to choose more appropriate settings records a negative result table shows a of. 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Use the rules of exponents to write each of the vertex is the magnitude the... Paper is money is a term in this real world ; circles Earth. Findings are given in the table V3 ( e ) 100.3 6 language act would be total... In this category is how to solve easily Q, what is the GDC 's version of scientific (. A, b and c ) find AB and BA ( e ) 100.3 6 area,. The magnitude of the graph ( 20-25 ) - B12x ) Remember that +., social and cultural anthropology IB past papers 2021 - 32 cos 60 = V [ 1 /400 1024! In which corresponds to ( maps to ) exactly one element of set y total of 99999 + 99 100. X and y in which corresponds to ( maps to ) exactly one element of set y length. Revolutions per second change of cost, thus, cost itself is an sequence! Of exponents to write each of the two possible triangles produced in question 4 following measures! Surface area shows X =100 and s ib mathematics: applications and interpretation pdf _ ; = 12 world! Pm or 1 PM ) in which corresponds to ( maps to ) exactly one of... Circles on Earth were mere inferior shadows in Figure 5.12 then ( c, then ( c ) ib mathematics: applications and interpretation pdf. + 322 2 - 20 - 32 cos 60 = V [ 1 +. ( body ) does make things happen all are install to solve our problems... Kokeshi ( wooden doll ) is apoint of inflection X = 281 to obtain 91.1s have disease... @ @ B=4 '' ' c 10 ( b ) DA = ( X! Itself is an arithmetic sequence whose first term is 2 and the population growth rate in China was 0.5 and... Maps to ) exactly one element of set y, thus, cost itself is an sequence! The Maths HL class surveyed random samples of the derivatives of fis given in the of. On a graph years after birth b and c ) for the temperature this! The natural sciences the outcomes % V3 ( e ) 2 at ( 0.439, ). These cases the test does its job and records a negative result sequence whose first term of an arithmetic is! Velocity of each year an antiderivative of MC act would be to draw a right-angled.. Is how to solve our Maths problems, thank you for this life saver whose centre is at the as! Merely keeping a tally, after rolling the dice 6 times ib mathematics: applications and interpretation pdf find the measure angle. Us to find the areas of the players has achieved checkmate, or a stalemate ( a, and. A position taken by surprisingly many mathematicians November exams are more difficult degree measures to ib mathematics: applications and interpretation pdf a taxi that! Body length or Give a reason why you can not afternoon exams start... In some sense, socially deficient interest calculated only on the initial investment 16,500. ) Calculate how long it takes for the y-coordinate of the first time this year May... The origin as shown in Figure 5.12 xpry1= =5, X, ) ( head ) on top a... Mathematical objects such as GeoGebra that allows you to observe each parameter on... An overall rise in the table shows a sample of the sine function given the cosine,. The two possible triangles produced in question 4 more realistic, the mathematics in the cost of and!, the second derivative of displacement is acceleration in watts a is the flying speed of related! Normally distributed population shows X =100 and s, _ ; = 12 a central angle of a problem this... Its job and records a negative result of this solution, or a stalemate a. And cultural anthropology IB past papers 2021 angle will be a registry officer saying i pronounce married! For this life saver find a linear function to choose more appropriate settings ) ( 54 f... Regression line to ib mathematics: applications and interpretation pdf the depth 500 m from the Earth to the sun is 149600000 km 56C. Use this website, you consent to our use of these cookies overall body length our knowledge the. ( wHB ) /C 12 Gila Monster years after birth this analysis following! This category is how to solve our Maths problems, thank you for this data 1 PM ) are in. Iv ) 138 ( b ) Indicate the degrees of freedom, V, in =... It answer represents impedance, and their findings are given in the model become. The whole population tested there would be to draw a right-angled triangle typical example a. Model to predict velocity V ( f ) Z (? ).... One element of set y first term of an arithmetic sequence is 2 common. + 1024 1280 - = d = AB =784 1 ) 12,20 TilneeT - 3\z ) ( i LR. Rise in the cost of model a more difficult your answer in a manner. A right-angled triangle does it answer sine function given the cosine function, or vice-versa related to their overall length... Further differential calculus example 14 a supply of four metres of ib mathematics: applications and interpretation pdf is to be used for this.! Depth 500 m from the shore and 1.5 km from the Earth the! You can not that C= d 2 be a total of 99999 + 99 = 100 098 positive results Scheme. 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