The interest is compounded half-yearly. when RPM increases by a factor of m, the (d) p(x) = 2x* + 12x* 15x () p'oo = 6x* + 24x 15 (f) poo increases for 0.775 < x < 3.22, decreasing elsewhere (g) Profit of 9350 THB for producing 322 baseball caps. Therefore, when x = 8.75m, the length of the rope is T = 32.0m ormeicen Figure 14.16 GDC screen for the solution to Example 14.18 By usinga GDC to calculate and graph the first derivative, we were able to perform the first derivative test graphically. The activity is the number of nuclei that decay per second or becquerel (Bq). )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? IB Math AA vs AI. thenitisaW dilationi followed 3. 31 17. Some conventions represent the number % by the decimal 0.3, others by 0,3. 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. 209 Matrix algebra 1. Wells) value, = A1) = [ o 1000 0d =500(1 e"!) Solve My Task Do math )98 = 0.0989% or about 1 in 1000. Sign analysis shows that 77"5 is positive to the left ofr = 3.1 and negative to the right, so the volume is maximized at r = 3.11 m. (i) V=3117G.117 %m&m! 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. The domain of f " s equal to the range of f; and the range of ~ is equal to the domain off Consider the function fix) = Vx +3,x= -3 (a) Determine the inverse function f~! (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? (b) Calculate the temperature of the water after 10 minutes. (b) Calculate the amount of energy available after 20 minutes. Find the quotient zl of these complex numbers and give the answer in a + biform. The interest is calculated at the end of each year and added to the amount outstanding. = $49,655.33 (b) The value of the payments made is $36,000, thus the interest earned is $13,655.33 Example 3.22 (a) can also be solved using the built-in financial package on a graphing calculator. . Model B costs $150 to install and has a monthly fee of $80. 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). State each value in scientific notation, correct to 3 significant figures. (iii) Find a least-squares regression line for CO, concentration C based on y, years since 1960. The points A, B, and D are on the same line. Model Answers. One of the main applications of recursive sequences in this course is to solve differential b, =2(b; +3) =238 +3) =82 equations using Eulers bs= 2(b, + 3) = 2(82+ 3) = 170 study in chapter 20. method, which you will 59 Sequences and series Be aware that not all sequences have formulae, So, the first five terms of this sequence are 5, 16, 38, 82, and 170. The first type of soil is made of clay and the cost of building the pipeline through this type of soil is $7 million per kilometre. The cost of a journey can be modelled by _[60+12d, C(d){xwd, 1sd5 where d is the distance travelled in kilometres. They thought that in mathematics they could glimpse the very framework on which the world and its myriad processes rested. The graph of T is shown in the diagram. (ii) After 20 months, Sue inherits some money and she decides to repay the loan completely at that time. In order to model this situation mathematically, we amend the differential equation leading to the modest exponential growth law we introduced by addition of a term that slows the growth down as the population increases. 4. According to the model, the population will exceed 7 million after 17.3 years. In 99% of these cases the test does its job and records a negative result. Theechain chain ruleruleisis casy easy tot remember Bexiiatini in Leibniz B, form, ==oDl = == By =, but you . The position of an orbit around the attractor depends on the initial numbers of prey and predator. (392 6. In this case, we can construct a piecewise linear model by partitioning the data set into two or more subsets. 150 = 150"( 17)715017,517 180) ~ 180 6 Given that the radius, r, is 10 cm, substituting into the formula gives The units of the product 1 are equal to the units of rbecause in radian measure has no units. Nevertheless, in the 20th century there were a number of projects that were designed to do just that: reduce the whole of mathematics to set theory. (a) == @ 167255 37 1| () 4 G) )1+ 1 () = () V-V w2 2 i (k) 5 4 3 [C] % o o< 44 0 -85 ()6 T 1 () In(2) () % @4 o m () 2cos(1) +2 973 Answers 531 3@ 2 ) = . Compound interest is interest calculated both on the initial investment and also on interest already paid. Again, since the sum is much easier in a + bi form, start with the denominator. Use the rules of exponents to write each of the following in the form 27, wheren Z. Here are some examples of estimation and approximation. (a) V(4, 3), W(=2,5) (b) K(3, 1), L(1,7) (c) T(6,7), G(2, 8) 3. 1. log,(m-n) =log,m + log,n 2 logb(%) = log,m log,n 3. log,(m") = n-log,m 4. log,1=0 5. (0)) Show that T'= (@ = Or =5~ a7 (@) Find 47 i dr The lobster trap is designed so that the length of steel used in its frame is a minimum. Costs are decreasing for 0 < x < 120 and increasing for 120 < x < 1000, so there must be a local minimum at x = 120. Example 20.11 The rate of decay of a substance y at any time is directly proportional to the amount of y and also directly proportional to the amount of another substance x. (d) Find the total time needed for the water to reach a temperature of 35C. Using sets and mappings we can show that there are many different types of infinity. Describe the feature of the data that makes linearisation difficult. A rope of length 81 metres is cut into n pieces of increasing lengths that form an arithmetic sequence with a common difference of d metres. Digital compact Digital standard DSLR Other Venice 153 211 82 308 Rome | Budapest | 98 74 120 57 31 12 242 183 Prague 56 29 5 107 TThe average selling price for each type of camera is as follows: Digital compact 1200; Digital standard 1100; DSLR 900; Other 600 Solution We set up a matrix multiplication in which the individual camera sales are multiplied by the corresponding price. If the world of mathematics is populated by some rather esoteric objects that are literally like nothing on Earth, then it is very important that these objects are precisely specified. Why is mathematics important when it comes to building bridges, doing science and medicine, economics and even playing basketball? The second type of soil is sandy and the cost of building the pipeline is $2 million per km. and variance o2 737 17 Inferential statistics 5. Interpret the slope and intercept in context. (a) Find a function for the amount, @ m/, remaining in the bag at time minutes. With a GDC, the problem is easier to visualise: Insert the points on a graph and (optionally) draw the segment [AB] a3, 61 -6.67 Draw the perpendicular bisector of the segment, then find its equation. (a) @ cosx ylnx= 7y =1 Y }Tl = Qul 2=y y+1 2 Chnptec; 20 Practice questions Ly=get 1 3+D 33T ) F =wu =29+ wy= 13,182, = %e (02) =55 gy B e Bt T e Y= =y+ouy=jesge (0.2) = 2.374 () -2 v .. (@ 19. The IB Math Applications and Interpretation (AI) HL Questionbank is a comprehensive set of IB Mathematics exam style questions, categorised into syllabus topic and concept, and sorted by difficulty of question. (a) Write down the amount of energy held by the battery immediately before it was used. Bivariate analysis 9. 4. From a 50-metre lighthouse on the shoreline, a boat is sighted at an angle of depression of 4 moving directly towards the lighthouse at a constant speed. Five minutes later, the angle of depression of the boat is 12. It turns out that there is an infinity of different types of infinity a whole hierarchy of infinities, in fact and this probably does not surprise you anymore, there are more infinities than finite cardinal numbers. (a) () () (g) -3 35cm 15em 4seconds t=0.595 () 12 hours 10. Find how long it will take for Nanako to repay the loan. Finally, the chess game ends. We will consider impedance in AC circuits here. It may or may not be the expected value of 3.5. (a) 4 (b) @) 5 ) y=0 (i) 7 3. 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. (0, 0), (3, 0), (3, 1) reflection iny = x. For Hersh, numbers and other mathematical objects are social facts. rahim, thank you foryour confidence in me and thinkingI am worthy of this ask. @ @) o =w+w+w) =w+w) +wh) since w' = we, (w2) = w, (wh) = w?, & Sa =w+w+w @) b=1,c=2 5 32.a=-4b=13 0 14. _ Solution (a) A graph of fproduced on a GDC reveals that it is not monotonic over its domain ] o0 , co[. | is usually considered constant. What does it tell us? (a)Fyexaa =1 dy (b) P (c) e7dy = xdx dy=x2xy*+1 (e) (xylnx)y = (y + 1) () o () (1 +tany)y' =x* +1 ) %: (i) ysecOdy = esin0 G) dy i 1427 s % =efl+y?) Hence, if the ordered pair (a, b) is a point on the graph of y = f{x), then the reversed ordered pair (b, a) must be on the graph ofy = f~!(x). Let P be an arbitrary point on the road GH and let GP = x (a) Find a function T(x) to model the total time taken for the farmer to get from Fto H. (b) Find the minimum time for his journey. . The reason is that they want to make generalised statements. Her data is shown in Table 19.17. The parameters ofa TVM solver are as follows. employ and refine their powers of abstraction and generalization. 5. The expected lifetime of the machine is 6 years. Ifx < 0, then we can writex = u where u > 0. (@ y=4x+1 nte (b) y=-2x+3 4. )x) = 2 Solution replace g(x) with y (a) interchange x and y 3x=1y y=13x Therefore, g~! Unfortunately, this situation makes the y statistic unreliable. Since it tells us how the first derivative is changing, it tells us how the gradient is changing, which means it can tell us about concavity. This description gives us the recursive definition of the arithmetic sequence. For the angle 6 in Figure 5.10, Rl it Note that this ratio is an arc length divided by another length (radius), so it is a number without units. 2k 3] = g = 1 )T 5= X0 =AYy =02 13 Since the domain of gis x ] oc, 0], then the range of g~! The Internal assessment chapter provides thorough information and advice on the required mathematical exploration component. V27 - 9V3 12. Our experience with the central limit theorem would suggest that repeated sampling or larger sample sizes should help. (b) Atwhat speed is the bicycle travelling along the ground, in kmh~'? (a) 9= @ 300(0.01 105 q)iing (b) g 104 no (9) qity= 1041 e~0003), (i) Afier one year the amount of pollutant wil be approximately 9278 g i 4 000031)) (d") lim g(1041 = e700%) = 10 104 (@ 5 = 300(~10%); g0 = 9278 dt () qit) = 9278(e~0%9%), affer one year there will be 670 g approximately () 2.6 years 4 s gZ &2 g 3 3 v kv, where k = V367 and is a constant dt of proportionality that depends on several factors like 25, eigenvectors H3) temperatutre and friction. Y -min ine lity If f,0x and f,;, represent the maximum and minimum values of a nonnegative continuous differentiable function f(x) over an interval [g, b], then the area under the curve lies between the area of the rectangle with base [a, b] and f,;, as height and the rectangle with f,,,, as height. =xx"=dx3wu=de u" o ' 2e7%(0.2) =226 = u=7e "+ 6% (0.2 ) = (e) y'=uu"=~5y6usy 271 d x-2 y __dx 3x2 3x+1 (x =2y, 1) oy = a2 X + & 4 12. What's the problem? (b) What are its coordinates when it has covered three-quarters of the total distance? We are often interested in calculating the area of sectors, and we can use the method of Example 5.1 to calculate the area of any sector. As the value of x increases, the value of y also increases and there is no limit on how largey can become. 6. Therefore, the height of the building is 50 m. (c) The time when the ball reaches its maximum height is the t-coordinate of the vertex of the parabola. 18, 12 12,8, 1632 rE LS8, 20,07 5=81,00 13035 6,12, 18 21527 = 1 10. (b) Find the least-squares regression line for length (L) based on width (W). First, chess is played on a special board with pieces that can move in a particular way. 360), consider a central angle and its associated arc or the circle. In this world, mathematicians can explore the patterns they encounter. A family of isosceles triangles is built upon the line segment [PQ] as the base where the coordinates of the endpoints are P(3, 6) and Q(5, 10). The domain of f! 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. As soon as we think of mathematics as a social construction then the exact arrangements by which this comes about - the institutions that build and maintain it - become important. 3. What percentage of the original volume is wasted? 0 Therefore range is 1.48 < fiv < 9. Shop A: 18.77 2 4 . 36. 13 Number and algebra basics 9 Write down the value of: (a) 2logg,8 (b) logsy8 (c) logys3/3 (d) 103VEi (e) logs8* (f) logs:\5 (g) logy93 + log;9* (h) log 522 + logg4* (i) @) log,34 log 52 log3 log; V3 10. 14. (b) x>+ dxy 3y 1 13. I function is either increasing or decreasing, it is said to be monotonic. Determine if 9803 is a term in this sequence. With 6, = 40 and fit) = = (ksint? Any angle which is a reflection across the x or y axis toa known angle can be found using this method. (a) Use these values to find a linear function that converts Celsius to Fahrenheit. (b) Find the common difference for the arithmetic sequences and the common ratio for the geometric sequences. The proportions of some of the rectangles in this painting is eauty in numbers Keats also put it the other way around: the beautiful is the true. In the meantime, Fior had made little progress with Tartaglias problems and it was obvious who was the winner. The angles shown above are not central angles. A mapping is a rule that associates every member of a set with a member of a second set. 903 Theory of knowledge Constructivist view of mathematics Having thought a little about what the purpose of mathematics could be, let's move on to the question of whether it is best thought of as an invention or as something out there in the world. In ToK we might want to ask: If mathematics is out there in the world, where is it? We do not see circles, triangles, V2, i, e, and other mathematical objects obviously floating around in the world. English Pages 1016 Year 2019. 999 Answers () Eigenvalues: 0.1 * i. For students taking their exams in 2021 there is a big change to the IB syllabus there will now be 4 (c) Find the minimum average cost, and the number of units produced at that level. 16. Solution Using the cosine rule, AB = y20? This time, consider Vi (a) How many roots should there be? The problem is with the limited sample size of 20. She decides to ask her parents for the extra money that she needs. This textbook has been designed so that the chapters proceed in a manner that supports effective learning of the course content. Given that & can be expressed in the form h(t) = a sin(blt d)) + c, find the value of d. / Bucket " Figure 9.34 Water level Diagram for question 5 . Series The word series in common language implies much the same thing as sequence. Solution Vi=8,V,=105and Vo= 4.5 Since V.= Vi +j(V, V) =80+ (105 4.5)j = 8 + 6] The magnitude of the source potential difference is [Vl = V82 + 62 = 10, that is, 10 V. Example 6.19 The current in a given AC circuit is 4.1 5.3j A and the impedance is 6.2 + 2.3j Q. The payment is made at the beginning of each year, so set the PmtAt = BEGIN. = /702 + d* = /4900 + d* Since XY = 100, then PY = 100 d Using this in triangle PYB gives PB = /PY? She i my rock, and | dedicate this book 10 hr: British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 9780435193447 My thanks go 10 allthe students and teachers who used the earlcr editions and sent us their comments Ibrahim Wazir Copyright notice Al rights reserved. 972, 324,108, 36, ORiEo ol749" 30195, 1938, 56,855 31.0100595,90.25,55 27 3 9 32. The Analysis and Approaches course is suitable for future mathematicians, engineers, scientists and economists. 10392 (20-25) - B12x) Remember that LI4E + 3 = 1.14 X 10 = 1140cm? Find the size of angle 6 in the diagram. The solution lies in the chain rule. The signs of the terms alternate and there are 99 terms. If we substitute a, + (n 1)d for a, then we arrive at an alternative formula for the sum. =2 ,.2 650 " 65 15. Given that the lengths of the shortest and longest pieces are 1.5 metres and 7.5 metres respectively, find the values of n and d. 12. A geometric sequence has a third term of 7 and a common ratio of 3 Find the first term. (0, 0), (~3,0), (=3, 1) (b) dilation in both directions of magnitude 2. 7. 244* (@) @92 (h) @% e Solution (a) 26 The bases are the same. 5. For example, the function y = 8x+ 6 = 2(4x + 3) is the composite of the functionsy = 2u and u = 4x + 3. The formula V = IR becomes V = IZ with AC circuits. The wheel rotates at a constant rate in an anticlockwise direction. (c) Interpret the gradients in your piecewise model. 25, 27, 29. Interchange xand y. (ii) Use algebra to split the numerator: xS 5yR x) = S [ SP ] == =x"lx 3 3 3x 3x2 3x2 3x2 32 Now, differentiate term-by-term using the power rule. The trick now, as you know; is to factor out (x 1) from equation (1) to give: @ G-DF-x2)=0 We can now try to find values ofx that make the second bracket in (2) equal to 0. The chain rule needs to be applied carefully. Thank you foryour acceptance andI hape you forgive me. This is shown in the diagram along with a possible route APB, where P is x km east of X. Substituting into u, = u;r""! Mathematicians discuss 11-dimensional hypercubes, infinite sets of numbers, infinite numbers, surfaces that turn you from being right-handed to left-handed as you traverse them, spaces where the angles of a triangle add up to more than 180 degrees, spaces where parallel lines diverge, systems where the order of the operation matters (where A * B is not the same as B * A), vectors in infinite-dimensional space, series that go on forever, and geometric figures that are self-similar called fractals (where you can take a small piece of the original figure then enlarge it and it looks identical - truly identical - to the original). The French mathematician Pierre de Fermat wrote the conjecture in 1627 as a short observation in his copy of The Arithmetics of Diophantus. For certain applications taking the derivative of the 2 derivative - that is, the second derivative denoted by f"(x) or E); - provides us with useful information. :wo Exercise 7.5 rA = L. () y-axis reflection. B A Figure 4.12 Cuboid room A room is a cuboid with every pair of opposite faces parallel, and intersecting faces, perpendicular. Find the time taken until the alarm sounds. 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. 998 AR R NN N N[N x=Ce Cet General solution: { y=2Ce%+ Gt Trajectories approach equilibrium and then move away. 11. mtn=Qi+1)+Qi+1) We can remove the brackets and rearrange to give: meAtn=2j+2i+2 Finally, we can use the fact that 2 is a common factor of all terms in the expression to place it outside a bracket. 17. statistic, we can use the formula However, after calculating the y? (a) (0,7) %(3)2 = % (b) (-2,3) (e) 1-2,0[ 967 Answers 3x2 6x = f3) = () Sce graph. Applying the recursive definition repeatedly helps us to see the expression: a=a Asequence 4y, ay, ay, is an arithmetic sequence if there is a constant d for which 4y =ay+d forallintegers n > 1, where dis the common. 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. Figure 2.20 shows that the point (b, @) can be found by reflecting the point (a, b) about the liney = x. The whole proof is a chain of such moves. 42. (b) Itis reccommended to replace the brake pads after the car has driven 90000 km. Another way of defining a sequence is the recursive (or inductive) definition. (a) i 4. (f) Find and interpret the value of R2 (g) Use your model to predict the blood concentration of this drug 4 hours after it is given. (a) B8, 1), N(4,4) (b) C(12,6), P( 3, 2) () E(/3,5),R(33,3) Solution (@ BN=/@+4+(-1-47=13 (b) CP=+(12+3)2F (6 +2)2=17 () ER=\(3-3/3)>+(5-372=4 100 and midpoints The coordinates of the midpoint C(x, y) of [AB] are the mean values of the abscissae (x-coordinates) and the mean values of the ordinates (y-coordinates) of the points A and B. There are two big assumptions made: that the stone will not experience air resistance, which will act as a significant drag force, and that the acceleration due to gravity is constant. Basic definitions relate to the use of a right-angled triangle, but any triangle will suffice. Therefore, we choose the cosine function, and we keep both a and b positive since no reflection across the x or y-axis is needed. In the equation m represents the gradient and (x,, y;) represents a point on the line. )dx + 3x2v(vdx + xdv) = 0 = xX1 + 2v)dx + 3x3vdv =0 Divide by x2 simplify and integrate dx 2 Gy (1 + 29 dx +3wdv=0= [T = vdy =3[0 Inx| = %m(l +202) + C = 41n|x| = 31n(1 + 202 + 4C, Now let 4C; = InC and simplify Inx* = In(C(1 + 2v? cm 5 (b) 28 9. (a) y~532 4 (b) Less i actial valie: Ey > 05 solition ciEves curving upward; short segments from Eulers method to approximate solution curve will be below the actual solution curve. @ Ll 31248192 " a 2a 8a> 32a7 W5 =g s () 4. a 5k,2a 4k,3a 3k, 4a 2k,5a k, Find an explicit formula that gives the nth term of each sequence. () x-intercept is at (0.439, 0). The epidemic started with 4 cases. A sample of data for some flying species are given in the table below. (b) For the reflector to start at the bottommost position, the model must have a minimum as an initial value. Time since administeation () | |025/05| (Concentration (mmol L) [38|80|49(32[17|13|06[03]02]01 0 6 |12 |18 24 | 30 | 36 | 42 | 48 (a) Generate a scatter diagram for the data. (c) Find the dimensions of the open box with the largest volume that can be created using this method. 5. Mathematics and the knower English poet John Keats said, Beauty is truth, truth beauty - that is all/ Ye know on earth, and all ye need to know. In this section we will see how mathematics impinges on our personal thinking about the world. , + ( n 1 ) d for a, + ( 1. Represents a point on the required mathematical exploration component width ( W ) she... Is no limit on how largey can become in 99 % of these cases test... Pmtat = BEGIN been designed so that the chapters proceed in a +.... As an initial value completely at that time @ m/, remaining the! ), ( 3, 0 ), ( 3, 1 ) reflection iny x. = A1 ) = [ o 1000 0d =500 ( 1 e ''! the loan the and. 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