Features 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. One of the most important ideas in the whole of mathematics is that ofa mapping. () () 23928 (b) 3x% " = 6x> 3x2e = +2) (i) 0.0022 ( () 00249 (i) 2.3681 0.0002; error is divided approx. In 1% of the cases however it gets it wrong and produces a positive result. (a) Write down the temperature of the potato at the moment it is placed in the oven. (b) Find the average cost per unit produced and graph AC(x) and MC(x) on the same set of axes. n 1 2 Extension (cm) 24 4.8 7.0 9.5 11.8 | 142 | 164 | 189 Difference (cm) = 2.4 22 25 2.3 2.4 22 25 The average difference, d = 3 4 5 24522425523 7 6 7 8 +24F22+25 = 2136 (b) Since u; = 2.4, applying the formula gives u, = 2.4 + 2.36(n 1) =2.36n + 0.04 (c) (i) For100g, u;p =236 X 10 + 0.04 = 23.6cm (if) For 150g, u,s = 2.36 X 15 + 0.04 = 35.4cm (d) Itis possible that this size mass has caused the spring to become almost fully extended, hence the model is no longer appropriate and cannot be extrapolated to this size mass. Find the value of the car after 11 years. As mathematician Roger Penrose remarked, QED: how to solve the greatest mathematical puzzle of your age. 5. This result is called a theorem. But this is precisely the definition of an even number that we started with. (e) Eigenvalues: . Person X takes the test and it shows positive. The rate of salt going out is (4/120)Q g/min. Either way, the mathematics is the same and these do not really count as different mathematical cultures. The key can be made public because in order for it to work it has to be split up into its component prime factors. 12. 10. s The mean monthly temperatures for Vienna, Austria, are given in the tables below. (f) Describe a reasonable domain and range for this model. Determine the equation of the line parallel to y = 7%){ + 4 that passes through the point (6, 1) 7. 16. 1356 Geometry and trigonometry 2 In questions 1-9, convert each angle to radians. For the snowboarder in Figure 5.6, we can think of the snowboard as the ray. Therefore, the cubic model for this turbine is P =0.297AdV? is equal to the range of f, therefore the domain of f ! (b) Calculate the percentage decrease in UK total greenhouse gas emissions from 2016 to 2030. embeddedin the unit circle tennhoes will always be equal to the y coordinate when we draw a right-angled triangle in this manner. 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. 0.999999 really does look very different to 1 but if the proof works then we are forced to believe that they are the same. 13. That's what math is. 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. Students have the option to choose between two different courses: IB Math Analysis and Approaches (AA) and IB Math Applications and Interpretation (AI). (a) (CPl) = [ Hoear value/ %o Wells) _= [ 10 240000-05 006t dtf ~ 51804753 (b) (capi( alvale) " s 16. (@) Using the TVM solver, we make the following entries and solve for the future value, FV, PMT: END The number of payments, N, is 6, because interest is compounded once per year for six years. At the end of 6 years, the company estimates it can sell the machine for $8000. Free Exam Papers Ib Math Studies Pdf Pdf is available in our digital library an online access to it is set as public so you can download it instantly. In 2016, the UK government estimated the total greenhouse gas emissions in the UK at 483.0 million tonnes carbon dioxide equivalent. Example 3.9 (a) Find the 12th term in the sequence 3, 12, 48, 192, 768, (b) The third term of a geometric sequence is 5 and the seventh term is 405. Area = 4x? Five minutes later, the angle of depression of the boat is 12. The domain is set X (independent variable) and the range is set Y (dependent variable). 972, 324,108, 36, ORiEo ol749" 30195, 1938, 56,855 31.0100595,90.25,55 27 3 9 32. (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. Give your answer correct to the nearest minute. State the null and alternative hypotheses and test this claim at a level of significance of a = 0.05 using this data set in kilograms. A B = AB mXn nXp mXp T el 1 L otrotan | This definition means that each entry with an address ij in the product AB is obtained by multiplying the entries in the ith row of A by the corresponding entries in the jth column of B and then adding the results: b, A e Ay, by =apby + aub +apby y, + b Example ) 7.1 Find C = ABwhen A = 3 . cA by factoringc out of entry in A. The most practical circle to use is the circle with a radius of one unit, as shown in Figure 5.12. Let C have coordinates (c, 0); this is our independent variable. The recursive definitionis a, = 13 ,1 = 5n> @y, =a,The explicit definition is (b) a5 = 18 5(25) = 107 62 a,=135n1=185n Find the 20th term of the arithmetic sequence with first term 5 and fifth term 11. On this view, mathematics is what might be called a social fact. You will see later, however, that this definition of the product of two matrices has many practical applications. 2(x)=-3+x+2 Figure 4.4 Intersecting lines Example 4.3 By inspection, determine if each pair of lines are parallel, coincident, or intersecting. Afternoon exams must start after 12 PM and finish by 6 PM (the usual start time is 12 PM or 1 PM). (e) 3: The non-zero digits and the zero in between are significant. Winner (country) Halil Mutlu (Turkey) Zhiyong Shi (China) Guozheng Zhang (China) Taner Sagir (Turkey) George Asanidze (Georgia) Milen Dobrev (Bulgaria) Ditry Berestov (Russia) Weight class (kg) 56 62 69 77 85 94 105 Mass lifted (kg) 295 325 347.5 375 3825 407.5 425 (a) Generate a scatter diagram for these data and describe it. Statistics and Probability 5. They thought that in mathematics they could glimpse the very framework on which the world and its myriad processes rested. (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. Solution (a) This is clearly an arithmetic sequence, since 5 is the common difference. The number of cells in the culture at 12:00 on Monday is 47. (b) 180 (c) 35.3 933 10. instead. The IB is excited to welcome educators to explore Education for an inclusive future at the IB Global Conference, Toronto 2023 held from 27-30 July 2023. The hypotenuse will be Figure 5.17 The unit circle with a 45 angle a radius of the unit circle and the terminal side of the angle; one leg will be a segment on the x axis; the second leg will be perpendicular to the x axis. 308 (251.5,118) (503, 0) Note that most GDCs have a quadratic regression feature that can also find the model. Since the vertex must be halfway along the length of the bridge, its x-coordinate must be 251.5 (251.5, 118) (503,0) Then, we can use the general quadratic equation y = ax? What is the difference between speed and velocity? Suppose $2000 is invested in an account paying simple interest at a rate of 5% per year. made at the beginning of each year. We have to do a great deal of work to find them through inference and abstraction. The first series, 3251, does not have a limit and we say it diverges. Example 3.6 An experiment was undertaken to investigate the relationship between the length of a spring and the mass hanging from it, as shown in the diagram. The general (nth) term ofa geometric sequence, , with common ratio r and first term u; may be expressed explicitly as Uy =yt This result is useful in finding any term of the sequence without knowing all the previous terms. 9 Theory of knowledge Ma matics and personal qualities There are undoubtedly special qualities well-suited to doing mathematics. The point-gradient form ofa line can be derived algebraically starting with the gradient-intercept form. The final chapter in the book will support your involvement in the Theory of knowledge course. How long is the arc (L)? More worrying is that an expression such as the smallest real number larger than 1 doesnt actually mean anything either. A quadratic function y = ax? Multiplying a 2 X 3 matrix by a 3 X 2 matrix results in a 2 X 2 product matrix. The International Baccalaureate, formerly known as the International Baccalaureate Organization, is a nonprofit foundation headquartered in Geneva, Switzerland, and founded in 1968. Here we will look at the effect of the parameters and learn how to solve for them in simplified examples or using a GDC (check your GDC for logistic regression). This situation can be modelled by the exponential 100 function T = a + b(k~"), where T is the temperature of the water, in C, and m is the number of minutes for which the cup has been placed in the room. b S Find the distance between each pair of points. (a) 1.389 billion (b) 1.407 billion (c) 5.86 years, so towards the end of 2021 45. (@) #=30%r=10cm (b) 6=145r=8m (c) 6=52%r=180mm (d) =n%r=15cm 3. For example, find the partial sum for the first 50 terms of the series: 3k 8F 1318 We can express S5, in two different ways and then add them together: S S = 3+ 8+ 13+..+248 =248+243+238+ ..+ 3 285 =251 + 251 + 251 + + 251 There are 50 terms in this sum, and hence 285 = 50 X 251 = Sy, = 6275 This reasoning can be extended to any arithmetic series in order to develop a formula for the nth partial sum ,. The payment is made at the beginning of each year, so set the PmtAt = BEGIN. The thickness (f mm) ofa brake pad on a car, which has been driven k kilometres, can be modelled by the function t(k) = a/120000 k, where0 < k < 120000 and a R (a) Find the value of a, given that the initial thickness is 12 mm. (b) What is the minimum length of rope necessary? (@) F:C+32 (b) 98.6F ) 5 (d) P=100 (c) 161 years (e) increases towards 100 x 919 Answers 12. 123.45 has 5 s.f. He explored the formation of shells and the wings of dragonflies, and examined the skeletons of dinosaurs through the eyes of a civil engineer constructing bridges and wondered about the formation of bee cells and the arrangement of sunflower seeds. 0 56 5 234 | 10 15 979 | 410 | 20 L71 (a) Develop an exponential model for this data. Make sure to set a to at least the following four values: a = 1, %, 2,2 You should see that |a] is equal to the amplitude of the graph, which is half of the wave height, as shown in Figure 9.20. Answers will differ. () Interpret your model and R? (c) Write down the value of c. (d) Write down the interval of time during which the temperature is increasing from 4C to 2C. is critical. (a) Find a linear function to model the cost of model A. What is the difference between heat and temperature? Show that an expression for the minimum value of Q is 5000 + 1.028 5000 +o+ 1.0282 5000 1.028" (ii) Hence or otherwise, find the minimum value of Q that would permit David to withdraw annual amounts of $5000 indefinitely. Solution The value of the machine can be modelled by a geometric sequence with u; = 60000 and u; = 8000. 10. (a) (f) (e) % (f) 1 (b) 26 (g 2 W2 @2 @2 4. 3 At present, it is not a triangle. If runners ran at the same pace for marathons as they do for 5km, the gradient of the leastsquares model for marathon time based on 5 km time would be k. (i) 5km time | Marathon (min) time (min) 15 144 17 163 19 182 21 201 23 220 25 238 27 256 29 274 Table 19.14 Data for question 4 Find the value of k. (ii) Find the percentage difference between k and the gradient from part (b). The table below lists the gold medal winners from the 2004 Olympics in Mens weightlifting. (a) \0.21 Now | 1year | 2years | 3 years | 4 years | 5 years City [200000 190750 | 182240 | 174411 | 167208 | 160581 Suburbs| 25000 | 34250 | 42760 | 50589 | 57792| 64419 (b) City population will approach 84 375 and the suburbs population will approach 140 625. .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 ~ . (a) 2y=4x2 () y=2x3 2Xi=yi= 2 x=2y3 () y=2x4 3x*%y:6 Determine the coordinates of the points of intersection of each pair of lines. 31. We can also have log-log and lin-log graphs, in the same way that we might chose to transform the x or y or both variables when we linearise data. (b) Verify the result by showing that (fo f~! Ina + biform, V ~ 3.83 + 8.83i N _..9.622784407 s6sselesa 193 Complex numbers Impedance j1g[cTelyY] Physicists usej to represent the imaginary unit i to avoid confusing iwith [, the current ina circuit. 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