170 =3 (f) Uy nr! Before you can use the trigonometric functions to model real-life phenomena, we need to evaluate sine, cosine, and tangent for any angle. =8-=2and du =14 ax Cdu M How are the derivatives of these three functions related? (a) () (b) u, = 307 27n 12 14. Then they emerge and enjoy a relatively brief period courting and mating before laying eggs in the ground and dying. The hero of The Glass Bead Game, a novel by the German writer Hermann Hesse, must learn music, mathematics, and cultural history to play the game. (c) 27 The bases are different. 780002 has 6 s.f. Calculate the percentage increase by multiplying by 1.045 for each year. Our normal approach would be to draw a right-angled triangle. (c) Given that Q(10) = 19.05, find the average amount of energy lost per minute for the interval 10 < < 20 (d) Calculate the number of minutes it takes for the energy to reach zero. Find the solution to the initial value problem. We say that fis an increasing function. This proportion features significantly in the design of the Parthenon and many other buildings of the period. (a) 26 () 1 () 2. (0, 0), (3, 0), (3, 3) Composition: reflection in y = x and reflection in y-axis since -1 0)(0 1)_(0 - ll)) (or rotation 90 degrees ( 0 1)(1 o= (1 anticlockwise about (0, 0)), (0, 0, (0, 3), (~ 1, 3) by reflection iny = x (@ 22 22-3a=tora=3 () (1) 2 () (d) () () (6,2) 2. =3X(-2)-5X8+2X10=26 and s 1 apfb,|=2 an =Y k=1 auby=0@n =220 by 7)|-4 bsy 1 = 1 XA 765 =53 Repeat the operation for each entry in the solution matrix to get: 34 C=AB= 26 33 21 Figure 7.1 Usinga GDC to fin = We can also use our GDC to find the product. 4 4 ' ' 4 4 \ \ Vo oo AR o e B ~ N 2. 972, 324,108, 36, ORiEo ol749" 30195, 1938, 56,855 31.0100595,90.25,55 27 3 9 32. But this means that mathematics is really radically different from other areas of knowledge, including the natural sciences. (a) 111 (d 1 1 () Iv 2@ 1 s q g(0) = 0. and in hours, ); in gram 13. 17 Functions A company wants to find the minimum cost of building an oil pipeline from point A to point B across two types of soil. There are photocopiable resource for general and credit students. The number of bacteria in colonyA after hours is modelled by the function A(t) = 12%4 (a) Find the number of bacteria in colonyA after four hours. (d) 5: All digits between the leading non-zero digit and the decimal point are significant. Calculus. They feel that there really is a mathematical world out there and that they are trying to discover truths about it, much like natural science discovers truth about the physical world. Use your graphing application to generate a graph of y = a sinx. Ifthe original values in questions 1 and 2 were precise measurements, find the percentage errors in their rounded values, to two decimal places. Einstein suggested that the most incomprehensible thing about the universe was that it was comprehensible. In the circuit in question 7, R, = R, =8, X; =6 Qand X, = 6 Q What is the impedance? Geometry and This textbook comprehensively covers all of the material in the syllabus for the twoyear Mathematics: Applications and Interpretation Higher Level course of the International Baccalaureate (IB) Diploma Programme (DP). To the nearest metre, how much of a head start should the second runner have in order for them to arrive at the opposite corner at the same time? (a) State the null and alternative hypotheses. Use the model to calculate the height above sea level of Mt. (a) 3.09 1 - 20 15. @ 55 XL By the Net s Change Theorem, the increase in velocity is (d) 49ms' (e) 12465 ) 73.08ms"! For example, (0 28 13. (a) GAUSS IS A GREAT MATHEMATICIAN (b) Answers vary Exercise 7.3 Exercise 7.4 Lm=2orm=3 2. a=7b=2 () (-1,2,-1) 3m=2 4. 2x+6 We can show this on a GDC. 2 seconds. 3920 cm? 1060 W (c) (c) () () (c) (c) 133m 5.7 seconds 1L1s 625 X 10m 235m 148ms! Afternoon exams must start after 12 PM and finish by 6 PM (the usual start time is 12 PM or 1 PM). Maybe they were right. Hersh thinks that this is not really the same nine. All DP mathematics courses require students to appreciate the use of technology in mathematics and become proficient with graphic display calculators. Note that the ratio of consecutive terms equals a fixed constant. Figure 4.2 Same line (coincident lines) 101 Geometry and trigonometry 1 Parallel lines The first function is flx) = 2x 3 in blue. The domain of f! (iv) Write down an appropriate domain for your model in context. (a) The points T(1, 3), X(4, 6), G(6, 4), and C(8, 8) are coordinates on a treasure map. 14. It is center. (b) 3,3.1,3.3 (e) 0.64 0475 (@1 14 @ de) _=202000 ) s =519+ 54 0.0549x(x + 95)dx + 2 (b) 233310 85 ~ 27449 (o) 134 barrels Z 95 ~233310m L V=mf 15 5 w5 womf -Saer 35),17() 0 + 2(0.087 + 0.226 + 0.405 + 0.633) + 1) L=1-2X0.3702 o 1.0041s 204 (b)b) =~ %= 166 13 () =6 h 166 &5 5 (b) 2725m () 10s (b) 10625 10. (a) Find a function for the amount, @ m/, remaining in the bag at time minutes. dt Population growth rate in cases where the birth and death rates are not variable i proportional to the size of the population. In section 5.1 we used degrees to measure angles. Purpose: mathematical models Unlike pure mathematics, which is about the solution of exclusively mathematical puzzles, applied mathematics is about solving real-world problems. Time (years) Amount in the account ($) 0 2500 1 2500 + 2500 X 0.03 = 2500(1 + 0.03) 2 3 2500(1 + 0.03) + (2500(1 + 0.03)) X 0.03 = 2500(1 + 0.03)(1 + 0.03) = 2500(1 + 0.03) 2500(1 + 0.037 + (2500(1 + 0.03)2) X 0.03= 2500(1 + 0.032(1 + 0.03) = 2500(1 + 0.03) 4 2500(1 + 0.03)* + (2500(1 + 0.03)%) X 0.03 = 2500(1 + 0.03)(1 + 0.03) = 2500(1 + 0.03)* Table 3.3 Compound interest This appears to be a geometric sequence with 5 terms. () 44i (c) 88/3i 936 Z = 1500w |V =15V [V] = 60V 2% 636 128/Q Chapter 6 Practice questions 2. 21 Functions Linear and piecewise functions Linear Figure 2.4 Graph of the line y=21 ctio You will probably be familiar with linear functions from your previous studies in Mathematics. 6. The number of cells, C, in a culture is given by the equation C = p X 2% + g where t is the time in hours measured from 12:00 on Monday and p and g are constants. Some conventions represent the number % by the decimal 0.3, others by 0,3. 4. (f) Eigenvalues: L A Vb v v ke . (b) Explain why fdoes not have an inverse function. View Bundle. Solution (a) The common ratio is 2 and a recursive formula is u, = 2u,_, u; = 1 (b) The common ratio is % and a recursive formula is u, = %u,, u; = 600 (c) The common ratio is 4 and a recursive formula is u, = 4u,_,, u; = 3 In practice, the recursive formula is not useful to obtain the terms of a sequence. (a) $3809.01 0. WebAuthor: Jane Forrest Publisher: ISBN: 9780198426981 Category : Languages : en Pages : 672 Download Book. (b) Coincident: the second is a scalar multiple of the first. on the same set of axes. The owner of a small company buys a company car for one of his employees. Alternatively, the equation can be written as fix) gix) = 0 This equation can be solved graphically on a GDC by sketching the graph y = fix) g, then finding the zeros. (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. Cubic models are also widely used in computer graphics, in everything from modelling the curves of the letters in this textbook to smoothing computer-generated effects and animation. This PDF download accompanies the IB Mathematics: Applications and Interpretation SL study guide, providing worked answers to problems presented throughout the book. Therefore, the probability of having the disease given a positive result is 103? In order to account for the rate of inflation, the nominal rate is adjusted to take inflation into account. 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. (b) 94 years (b) $1650 15. Ifa o central angle is greater than 180, then the subtended arc is referred to as a major arc. The height of the ball from ground level can be modelled by the function h(t) = 4.9 + 11t + 50 where h(#) is the height of the ball in metres and t is the time in seconds after the ball was thrown. So, we simply need to turn the wheel by adding a quarter-period to t. 0 TT> 2 4 6 Figure 9.28 The sine model above produces the same h=17 cos(l;(t i ])) + 30 as shown in the graph. (a) a=4 ) (i) d=-5 6. One of the more surprising aspects of mathematics is the twoway link to the arts and beauty. For example a=(35 YJanan=(2 ) tenan= (] 5)(2 = 2 5 & 5 2/ & BA:( 208 2 5 6\(3 )( 6 5 2 ):( 36 24 20 32 )AB:BA 20 232 Multiplying by an identity matrix is also commutative. (e) The company plans to produce 400 units per day. Now consider the 9999 900 people who don't have the disease. (b) Write down an equation for the volume of the lobster trap in terms of r,land . A geometric sequence has a fifth term of 6 and a seventh term of 8.64 Find the possible values of the second term. 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. (c) 1:Only the 8 is significant. Solution Using the cosine rule, AB = y20? The partially completed matrix below shows the number of direct routes between these cities. '(a) Quantity produced = f(m + S)dt = 1001In(t + 1) + 5t + , with initial production at 0, c=0 (b) approximately 404 thousand barrels 13. It is also called a center. On a coordinate plane, an angle having its vertex at the origin and its initial side lying on the positive x-axis is said to be in standard position (Figure 5.8). (a) $5057.31 e a3 u,= 2(8) (b) 59.2% ) 12 () 178 44. (x) (ii) State the domain and range of f ! 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. 998 AR R NN N N[N x=Ce Cet General solution: { y=2Ce%+ Gt Trajectories approach equilibrium and then move away. For this expression, we will rely on a GDC to find the numerical derivative. (a) 125250 15. u,= landd =5 16. WebMathematics - Applications and Interpretation SL - Hodder 2019 - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. However, once an organism dies, it stops taking in carbon and the carbon-14in its body begins to decay. It is immensely powerful because it covers an infinite number of situations. WebMathematics Applications and Interpretation for the IB Diploma Higher Level 9780435193447. He found great satisfaction in working with patterns and sequences of numbers. 7. The position of an orbit around the attractor depends on the initial numbers of prey and predator. The IB diploma is earned when students pass exams in six different subject areas and successfully complete the Extended Essay, Theory of Knowledge course and Creativity, Action and Service requirements. (a) ) + 1L,xeR 4 x+9 130 8x 18 1728 10( | 5 (@ Doy =_ =57+ () y=x Ao (b) 20 ( -1 (b (b) =2 y intercept (3,0 Exercise 2.5 C] 1. First, since the graph must pass through (0, 0), it must be true that 0=a(0?2+b0 +c = So c=0 y=ax?+ bx Next, since the graph must also pass through (251.5, 118) and (503, 0), we can generate a system of equations by substituting each coordinate pair into the model: 18 = (25157 +b(2515) 0 =a503? A geometric sequence has a second term of 2, a seventh term of 243 1nd 19 683 an nth term of 3107 512 Find the value of n. 38. This means that they cannot be explained in terms of more simple ideas. (a) V(4, 3), W(2,5) (b) K(3, 1), L(1,7) () T(6,7), G(2, 8) 2. Find the value of d. (c) Calculate the number of minutes until a person riding the Riesenrad reaches the top of the wheel. That is out by a whopping 99 000%. We therefore need to revise some of the derivative formulae. This book is designed to complete the course in conjunction with the Mathematics: Core Topics SL textbook. Comment on the reasonableness of this prediction. One way to take this into account is to assume that the death rate is directly proportional to the present population, and that the birth rate remains constant. (e) Find the maximum speed of the tennis ball and the time at which this occurs. Copyright 2023 DOKUMEN.PUB. (iii) Explain the meaning of the percentage difference in context. This is best done application such as GeoGebra that allows you to observe each parameter has on the shape of the graph. $500 is invested in an account paying simple interest at a rate of 3.2% per year. Moreover, this is a very public beauty because it can be dissected and discussed. 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. In this chapter we first look at circles, where we learn to calculate the area of a sector and the length of an arc. T & Figure 5.6 Angles are not only in triangles - in this section we study angles of rotation, which can be used to describe this snowboarders 3607 jump 129 Geometry and trigonometry 2 Angle Remember than an angle in a plane is made by rotating a ray about its endpoint, called the vertex of the angle. (31 9.91.812 10. Piecewise functions are often used to model situations that display behaviour that fits with one or more type of function. What problems does it solve or what questions does it answer? What does it tell us? Determine the number of significant figures in each of the following. @ 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. But the story doesn't stop here. Find the amount of Thai Baht he receives. On any wall, floor, or ceiling, a diagonal drawn is the hypotenuse of a rightangled triangle. It follows that: When f7(x) > 0 for all xin some interval of the domain of f, the graph of fis concave up in the interval. The domain is set X (independent variable) and the range is set Y (dependent variable). For example, the display for the model in Example 9.5 shows how this default view can be very misleading. represent the name of the function. We can also see from the table that the product of |b| and the period has a constant value of 277. The activity of a second radioactive substance can be modelled by the function b(t) = %, t = 0, where t is the time in days. It may or may not be the expected value of 3.5. 3 23. The wheel rotates clockwise and takes 4's to complete one revolution. (a) Convert 30 and 45 to radian measure and sketch the corresponding arc on the unit circle. 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. This implies that the ratios expected from the girls and from the boys should match the ratios for the entire group. (a) (a) (a) (a) (a) (a) S=075d 9.8 1.62 4.00 X 10" 7.664 0.613 () x=16 (b) x=10 (b) x=4 (b) x (b) x=10 (b) (d) (b) (d) sometimes never never never =100 () x=3 (b) (b) (b) (b) (b) (b) 5.25m 39.2 40.5m 7560ms ! 9. adjacent The same logic applies for the cosine, so that cos (45) = = hypotenuse which implies that the value of the cosine ratio is always equal to the . (b) Verify the result by showing that (fo f~! Find: (a) the slant length [ (b) the radius r (c) the arclength of the sector (d) the angle 6 in radians. = OO Mark 4 fold lines, equidistant from the edges of the paper. Although the test is quite good, it is not perfect it is only 99% accurate. TheInternational Baccalaureate(IB) offers a continuum of international education through four challenging, high-quality educational programmes to students aged 3 to 19. Students have the option to choose between two different courses: IB Math Analysis and Approaches (AA) and IB Math Applications and Interpretation (AI). The word function is used to describe this dependence of one quantity on another, i.e. In this case, the law of acceleration due to gravity. What does this give us? 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. More worrying is that an expression such as the smallest real number larger than 1 doesnt actually mean anything either. Either way, the mathematics is the same and these do not really count as different mathematical cultures. The potential differences across a resistor (V; = 6 V), an inductor (V; = 11.5V), and capacitor (V. = 3.5V) are individually measured in a series circuit. The problem occurs when we make a subconscious switch to nine as an abstract noun in the sort of problems we deal with in Mathematics class. The least number of significant figures in any measurement determines the number of significant figures in the answer. We say that fis a decreasing function. Now, instead of merely keeping a tally, after rolling the dice 6 times, find the mean of the outcomes. In mathematics and science, we refer to this as the period of the function. (b) Estimate, to 1 significant figure, the coordinates where the local minimum value of the function occurs. On this view, mathematics is just like the glass bead game. Since the semi-circle formed by the equator measures 180, the arc measure must be 180 47.5 25 = 107.5. The function g is defined by gix) = 12 =2 (a) Show that g'x) = m g zzi Z for x = bwherebc R (b) Hence, find the smallest value of b for which the inverse function g ! The graph works exactly the same; just interpolate the values from the nearby axis labels. + 7x+ 3 24706 Ge2+76 e @ 22 (h) + 2 cost + (n) e (m)lnS 25 +e 0 & : ,3 . *kdt 841 Integral calculus 2 Integrate both sides to get 5dP = Jkat = In|P| =kt + where is an arbitrary constant. y dy = 3x2dx then integrate both sides f 1+ 4y? Before repainting the interior walls, the school administration sought some colour preferences from the teachers and students. That makes sense because the centre of the wheel is 30 cm above the ground. Our course selection guidance contains subject briefs for both standard and higher level, including information about core requirements, aims and assessment. WebMaths: Applications and Interpretation Standard Level Print and eBook: 9780435193454: 64.00: Maths: Applications and Interpretation Standard Level eBook: 9780435193416: 58.00: Maths: Applications and Interpretation Higher Level Print and eBook: 9780435193447: 70.00: Maths: Applications and Interpretation Higher Level eBook: Download Mathematics applications and interpretation IB Past papers 2021 PDF and use it for your revision. (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? (b) Find the least-squares regression line for marathon time (M) based on 5km time (7). To find the dimensions of the box, we need to go back to our expressions for the length, height, and width of the box and we can use these to obtain the missing dimensions: height = x = 4.06 cm length = 30 2x = 21.9cm width =21 2x = 129cm (d) The volume of the open box is given in GDC's output, since the volume is the y-coordinate in our GDC; V = 1140 cm?. Instead of dividing a full revolution into an arbitrary number of equal divisions (e.g. Sarah is spending her holiday camping. This description gives us the recursive definition of the arithmetic sequence. 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 ,. There is another species of cicada that has the same cycle and no fewer than 12 types that have a cycle of 17 years. 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). phase shift = c = =T3 What does do? = (2x 572(6x + 4) + 32x 5)] we would apply either the 30y Taking out a factor of 2x 572 order to differentiate hoo = = 2)[= w520l = (6x + 4)2x 572+ is not equivalent to Fx @2x 5713 o =(2x = 5)"6x + 4 +6x 15] f= =11 s As Example 13.20 demonstrates, before differentiating a quotient it is worthwhile to consider if performing some algebra may allow other more efficient differentiation techniques to be used. y=sin(x) +2 principal axisaty =2 Use your graphing application to generate a graph of y = sin(x) + d. Experiment with different values of d to see how the graph changes. Give answers in terms of 7. Find a formula for the total tin consumption within years of 2013 and estimate when the known world reserves of 156 million tons will be exhausted. 10. So, we can say that x = 0 is not a solution to the equation. (a) How much bigger is the volume of the sphere than the volume of the cone? S0 0 =< p = 200 is a reasonable domain. @ =) & 2y, Example 4.2 Find the coordinates of the midpoint of the line segment joining each pair of points in Example 4.1 (a) B(8, 1), N(4,4) (b) C(12,6), P(3,2) (c) E(3,5),R(3V3,3) Solution o (54 ) o (27255~ (2323 o Coordinate geometry in a plane lines and intersections Given two lines in a plane, three possibilities exist: o The lines have the same gradient and the same y-intercept; hence, they are the same line (coincident lines). Figure 16.14 Approximate total area under f(x) We partition this interval into n subintervals of equal length in a fashion similar to the previous discussion. 10 05 0 (b) Y(4) =25 @1 1 (b) IV 11. (a) 20 M) 20 x1, fi=3, h=7oh=18 =2l =31 2. 37. (c) Find the value of k. (d) The number of bacteria in colony A first exceeds the number of bacteria in colony B after n hours, where n Z. You can improve your educational performance by studying regularly and practicing good study habits. (@) (-1+15 (b) (-1 1i* ( (=v3+it (@ (=3 5. Forgetting to apply derivative of inside function necessary, or by applying it improperly, is a common source of errors The chain rule in words: in calculus computations. Recall that the linear function fix) = mx + has a gradient (slope) given by m, and a y-intercept at (0, c). 20 9.7890 The specific weight of water varies with temperature according to the data given in Table 19.11. Than 12 types that have a cycle of 17 years seventh term of 8.64 Find the mean of the formulae! Is invested in an account paying simple interest at a rate of 3.2 % per year lobster... 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Arc is referred to as a major arc arithmetic sequence expression, refer. Mean anything either DP mathematics courses require students to appreciate the use of technology in mathematics and,. Write down an equation for the rate of 3.2 % per year DP mathematics courses require to... Level, including information about Core requirements, aims and assessment they emerge and a! And sequences of numbers the specific weight of water varies with temperature according to the equation way, the measure... Tally, after rolling the dice 6 times, Find the numerical derivative problems throughout. The smallest real number larger than 1 doesnt actually mean anything either in example 9.5 How. The most incomprehensible thing about the universe was that it was comprehensible explained in terms of r land. Lines, equidistant from the boys should match the ratios for the rate of %... In each of the following have a cycle of 17 years an arbitrary number of equal (... 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