A cup of boiling water is placed in a room to cool. (d) From the graph, minimum occurs at Or, via first derivative test, using sign diagram: xzyf():%'*\*S:1.48331. (a) 58261 (d) 1.618 (b) 6107 (e) 0.003051 (d) 4000.9 (c) 123807 (f) 400.01 3. Both estimation and approximation skills are important in mathematics, but they are skills that are practiced every day in many contexts. (i) For what intervals is fincreasing or decreasing? Their gradients are not the same. Your bank has a digital device that scans the waiting line of customers and suggests the approximate waiting time in line is 6 minutes. 2,4, 32256 " 33. She pays back the loan at a fixed rate of $250 per month. In one experiement, a patient is given a drug and the patients blood is tested at regular intervals to determine the concentration in millimoles per litre (mmol L!). 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. Address: Be the first to receive exclusive offers and the latest news on our products and services directly in your inbox. Assume that only 100 people in the whole country have the disease. The interest is calculated at the end of each year and added to the amount outstanding. ) u, =20, andu, =2a \+a+kandu, =a 5k @ u,=n2+3 () u, =311 -1 (@ u, =211 83 n 1 924 a, 1 1 2 2 15 3 5 1.666666667 1.6 8 1.625 13 21 34 85 89 144 233 377 610 987 1.61538461538462 1.61904761904762 1.61764705882353 1.61818181818182 1.61797752808989 1.61805555555556 1.61802575107296 1.61803713527851 1.61803278688525 1.61803444782168 17 18 19 20 21 22 23 24 25 26 27 28 29 30 (b) lim a, 6. This is illustrated by a worked example. (b) Assuming the plane travels at a constant elevation of 10 km above ground level, estimate the actual flight distance. Learning objectives By the end of this chapter, you should be familiar with different forms of equations of lines and their gradients and intercepts parallel and perpendicular lines different methods to solve a system of linear equations (maximum of three equations in three unknowns) Key facts Key facts are drawn from the main text and highlighted for quick reference to help you identify clear learning points. With 6, = 40 and fit) = = (ksint? The equation ax + by + d = 0 is called the general form of a line. (@) a,=25n u, = (=11 (e) a,=2a,, (b) b,=2 X 3n-1 2 a = @d) a,=n 2n +5anda, () b,=3b,,andb, =2 3. A research vessel is mapping the floor of the sea in a certain area. (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. You recorded the values of the two standard deviations that your GDC gave to 3 significant figures as s, = 5.31 and 0, = 5.22, but forgot to note the sample size. 10. They thought that mathematics was one of the few areas in which humans could apprehend the eternal forms only accessible to pure unembodied intellect. Find the exact angle, in radians, of a sector with these measurements: (a) Area 2577 cm? and radius 5cm (b) Area 3077 m? Using the box plot in Figure 17.5, find the 90% confidence interval for the true percentage of corrective lens wearers. Maybe they were right. Wave height Figure 9.20 Amplitude is half of wave height 330 What does b do? Determine the equation of the perpendicular bisector of the segment [PQ], given the points P(5, 12) and Q(7, 6) 10. 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. Therefore d = 112 (c) By graphing the model, we obtain the display shown. (a) Calculate the amount of money in the account after 5 years. 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). Think about this for a moment before we continue. I wish I found out about it sooner. Since h = 2r = S=ar(r + =(1+ 12+ (2)? (3) The model uses values for constants that are established empirically. IB Math AA vs AI. Now, instead of merely keeping a tally, after rolling the dice 6 times, find the mean of the outcomes. Substituteu = 1 x 8. (@) a,=2n3 (b) b,=n+2 () c,=cp-1+2,andc; = 1 (d) 2,5,7,12,19, &) 2.-512-15.., 2. (d) Based on your model, predict the rotational speed that produces maximum torque. Quadratic equations will be covered in detail in Chapter 2. Don't try to predict x from y using the model (especially if the correlation is not strong). 244* (@) @92 (h) @% e Solution (a) 26 The bases are the same. Derivatives of composite functions, products and quotients We know how to differentiate functions such as fix) = x* + 2x 3and gx) = Vx, but how do we differentiate the composite function g(fix)) = vx* + 2x 3 where fand g are nested functions? Zoom 100%. You donit need to memorise this, because you already know where yand x are positive and negative, which tells you when sine and cosine are positive and negative, And, if you remember tha tangent is 2, iie. (d) The number of fish in the pond will not decrease below p. Write down the value of p. . Find the equation of the transformed graph. Theechain chain ruleruleisis casy easy tot remember Bexiiatini in Leibniz B, form, ==oDl = == By =, but you . Evaluate: (@) [x(Gx>+7)dx () fdx 22550 + 2 dx [2eV5x @[5 BrakP, @ Jrer=7a ( . Find the sums of the complex numbers. () There are 3 turning points; a cubic model requires at most 2 turning points; a quadratic model requires at most 1 turning point. 7. The rate of growth of prey is assumed to be a constant proportion A of the population. - )whereOShB (a) Find the greatest volume of liquid that the flask can contain. The current in a given AC circuit is 6 3j A and the impedance is 8 + 4j Q. (a) 2 () 8. critical value for v = 3. The numerator of the fraction is the central angle; the denominator is the whole circle (360): Area a of of sector sector == 8 A 5 This can be written as: A ector = 36 7T 2 where r is the radius of the circle and is the measure of the sectors central angle in degrees. Trigonometry 4. Hersh called his theory of mathematics humanism because its saying that mathematics is something human. o Note that the values for a and b calculated by the GDC agree with @ 22,: our values. Calculate the percentage error of your prediction. B R, X Ly 8. t Logistic curve Integral calculus 2 \Q ky(l {) | Solve the logistic differential equation i | d | Example 20.12 _----- Solution First separate the variables: e %: (11):dy:kdz Then integrate both sides using part_ial fractions: e =l s L | y|=ke+C = == ci% e Co koK =y= IS Oe i For population growth models, we also have an alternative form for the logistic equation. But the Platonist idea, that, as my friend Phil Davis puts it, Pi is in the sky, helps to make mathematics intimidating and remote. What are its coordinates? (a) V= )(zf 2 = 0.3702 11. Find the coordinates of the points of intersection of each pair of lines. Example 9.8 examines a classic problem. The variance is unknown. Since the semi-circle formed by the equator measures 180, the arc measure must be 180 47.5 25 = 107.5. (b) Calculate the value of Pearsons r and interpret it in context. It is quite usual that your first guess may not work. By the Triangle Inequality, the sum of the lengths of any two sides exceeds the length of the third. 2 X2+ What about the domain? B A Figure 4.12 Cuboid room A room is a cuboid with every pair of opposite faces parallel, and intersecting faces, perpendicular. (b) Express the impedance in Euler form, with 6 given in degrees to 3 significant figures. (b) What is the total area of 3 slices? (a) Express the impedance of the circuit as a complex number in Cartesian form. We are told that the disease is very rare. DOWNLOAD FILE So you can get to the nth term by adding d to a,, (n 1) times. Find the products of these complex numbers and give the answer in the form a + bi @) z,= 2% and L= Se% (b) z, = ZezT and z, = 46 zZ 21, Find the quotients z! (c) Atwhat time is the tennis ball descending at a rate of 10ms~'? Infinite sets can be put in a one-to-one correspondence with a proper subset of themselves. =51 In questions 28-29, find the length of the arc, s. Angles are in radians. RPM Torque (Nm) RPM Torque (Nm) 1500 2000 2500 3000 3500 380 461 475 508 529 4000 4500 5000 5500 6000 515 488 475 420 339 (a) Generate a scatter diagram for the data and describe it. Yes, the 3271t term 8. An arc of length 60 cm subtends a central angle a in a circle of radius 20 cm. Some sequences the drawbacks of this type of definition, unless we can change the definition A gre given only by listing their terms. The temperature of the room is 20 C. 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 (c) Find when the population of India will exceed that of China. Finally, the chess game ends. The loan has a simple interest rate of 8% per year. For example, the number mapped to the point (1, 0) because 7 is half of the circumference circle. 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? is x = 0 Consider the functions fix) = 2(x + 4) and g(x) = (a) Find g~! In this section, we will generalise the definitions of the trigonometric functions for any angle. -5 - 2 3 and B=| 5 =9 206 2 IS 8 0] 4 503 0 0 | Solution Aisa2 X 3 matrix, Bis a 3 X 4 matrix, so the product will bea 2 X 4 matrix. (a) Express the volume of the parcel in terms of / and w. (b) Show that [ = % The parcel is tied up using a length of string that fits exactly around the parcel, shown in red in the diagram. (b) the area of the sector. Analysis and approahces is a mainly based on applications and gives a student the 'know how' and the ability to apply maths to real world techniques. 306 | Solution (a) In general, revenue = selling price X number of units sold Therefore, we can develop a model for revenue, R, by multiplying the selling price, p, by the expression for the demand: R = p(1000 5p) = R = 5p%+ 1000p (b) Since this is a quadratic model with a concave-down graph (a < 0), we know there will be a maximum at the vertex. Nanako takes out a loan of $16,000 to buy a car. For many physical phenomena, we observe that one quantity depends on another. 1B mark awarded 7 6 5 4 3 2 1 Total Mathematics 4 6 3 5 3 2 2 25 Biology & 4 7 9 4 2 1 29 Chemistry 2 7 10 7 3 1 1 31 Physics 6 4 0] 10 3 3 0 35 Test her claim at the level of significance of @ = 0.05 773 Statistical tests and analyses 7 At the top of the league standings in La Liga and Bundesliga in 2017-18 were Barcelona and Bayern Munich respectively. 26. We know that neither of the assumptions in (1) is true. O 1(x)=(30-2x) (21-2x) (x) = From the graph shown, we can conclude that the value of x that produces the maximum volume is 4.06 cm. 39. [PDF] Ebook Hodder Mathematics Applications and Interpretation SL for the IB Diploma. Find the zeros of each function. : 4 3 This gives C =3 2 + 0.5d,08 We can write these two results as one piecewise function: = { 3 + 0.5d, 0=d=38 7+03d8, d>8 B 2 b b sariri :. (iv) Write down an appropriate domain for your model in context. (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 . According to the model, the population will exceed 7 million after 17.3 years. As is the case with arithmetic series, it is desirable to find a general expression for the nth partial sum of a geometric series. (6 (g) . Approximation 1. it T3 = In questions 19-27, the size of an angle in standard position is given. Is the flying speed of animals related to their overall body length? 20.93156857 E2:(2'm _ Solution oK The number of grains of rice follows an exponential pattern. We can use a spreadsheet to calculate differences quickly and then average them. The point estimate is p = 0.5, but the 90% confidence interval is 0.32 < p < 0.68 0 4 8 12162024 28 32 36 40 Number of marked items in the sample Figure 17.7 90% box plots for sample size 40 742 0246 > The results should give reason to ponder the size of the interval. What is the magnitude of the current? The constant of proportionality is *% an (a) d the value value of xX at atany any timetime [ is1s given given Dy by xX = (] + t)z Given the initial conditions that y = 10 when t = 0, find y as an explicit function of . Some symbols such as %, sin~! 2. That is, isdy = g = [ hy) or . Therefore, the watered area = % 7(3)2 = 3.75m~ 11.8 m? The school counsellor would like to test whether mark distribution in 1B subjects are independent in the following subjects: Mathematics HL, Biology HL, Chemistry HL, and Physics HL. For each equation: (i) match it with its graph (ii) state, with justification, whether or not the equation represents a function. (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 ! Write down the expected air temperature for a cave near Vienna. 22 23. @) 3= (f) 33 G -2 @ @ (b) 156 8. 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). 149 (b) 0.61 (c) 1200.7 Round each value to the nearest one-hundredth. We know that if you have the misfortune to fall from an airplane above 100 m or so, the height does not matter - the speed of impact with the ground will be the same, around 150 km h~!, because of the effect of air resistance (of course, it matters how you fall). Given g(x) = 3x 7, find g7'(5) 3. So, for the sequences above, 7 is the common difference for the first, 9 is the common difference for the second, and 9 is the difference for the third. Schools must be authorized to teach our IB programmes. identity is an equation thatis true for all values of the variables. The height, h metres, of a seat on a Ferris wheel after t minutes is given by h(t) = 15 cos(1.2t) + 17, for t =0 (a) Find the height of the seat when t = 0 (b) The seat first reaches a height of 20 m after k minutes. 2] 5 5 =) v [ 0 ": [~Guadiea o[t | Ouadatc. If the constant k is positive, the model describes population growth; if it is negative, it describes population decay. Vectors are covered in chapter 8. By definition: sin0=y=0 cos0=x=1 a0 == =0 (b) Anarc oflength t = 127 is equivalent to one-quarter of the circumference of the unit circle, so it terminates at the point (0, 1). What shape will this distribution take? Find the quotient Zl of these complex numbers and give your answer in the form a + bi 2 (a) z;, = 9cis3T1Tand 2= 3cis% () z,= 2clsandz2 = 4c1sg (b) z, =10 cisSTTrandzl = 2cisg d) z, = lchsZandZ2 = 3c157 17. The default view is often setat 10 =x=10and verify results Usinga GDCto ARrmtigie 10 < y =< 10 or less. Solution (a) a(Bg) 50 40 30 20 10 01 23 45 6 7 8 9 10 t(days) 25 Functions (c) We need to solve the equation 2090 2t Using a GDC we can find the intersection of the curves a(t) = % and at) = 20 (1.32,20) 2(x)20 The curves intersect at (1.32, 20), so it takes 1.32 days for the activity to reach 20 Bq. = du = cos(3x? (b) Linearise the data using logarithms to find the best-fit model, using P for orbital period and D for average distance from the Sun. (a) (%) il ( 347 5a+2 . Clearly it sets up a one-to-one correspondence between the set of natural numbers and the set of even numbers (check this yourself). (e) The actual orbital period of Ceres is 4.60 Earth years. (a) Vertical asymptote occurs when denominator equals +1= 3 =15=415 0 + signoff Therefore, ; (i) local maximum is atx = 29) + P(X= 12) X P(Y>28) + .. + P(X = 40) X P(Y > 0) 3 Px ). The midpoint of [AB] is ( i 2* = 622) =(1,2) Therefore,y 2=(x L)ory=x+1 The perpendicular biseeiok Al Fyment [AB] is the normal to [AB] at its midpoint. Author (c) Calculate the number of seconds the bucket is underwater during one rotation. In this case, we simply let a be negative, and revise our model accordingly: h=-17 cos(%) + 30 as shown in the graph. WebMathematics SL - Applications and Interpretation - OXFORD 2019.pdf - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. (a) @ 5 x+6 x+ 3n 4x 2 ) () () () ( 1 (iii) 7 (i) 79 (i) 12x 422 =7 (iv) dx2xt2 (b) 42+ 1,xeR (@2x+3 ( 16 _ (e) $1165396 2.4 3.6 4@ 22 )6 aer (b) 4x7,xR ( x%x=0 @L-2x%0 (& T x,x=4 f) 920 x2+5x=0 (0,4) -1 x+12 (b) St 1. ( Find the 10th term for each arithmetic or geometric sequence. Since point P lies on the boundary between X and Y, we can see that 0=d=100 (100,975) This is called the domain of the function. (a) 75 9. )= ar(r + 5r%) /5)ar? In a similar vein, the fact that there are different conventions for writing mathematics does not mean that the mathematics is different. What is the area of each panel? Applications of complex numbers The application of complex numbers within mathematics can be found in topics such as differential equations and eigenvalues; however, the discussion of these topics requires an understanding of mathematics beyond what has been covered in the HL syllabus to this point. The constructivist is a victim of the success of mathematics in fields such as the natural sciences. (b) The boarding platform at the lowest point of the Riesenrad is 12 feet above the ground. 923 2136 18. Find the amount of Philippine Pesos she receives. Their gradients are the same, but their y-intercepts are not. If m is an odd number, then we can write m = 2j + 1 for some whole numberj. Analysis and Approaches is very similar to the old IB syllabus. 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. In 2016, the UK government estimated the total greenhouse gas emissions in the UK at 483.0 million tonnes carbon dioxide equivalent. Find the measure of 3 significant figures. Precipitation 5s o= eoe && (b) The pattern is not sinusoidal because there appear to be unequal local maxima. Solution (a) Start with an accurate diagram. (b) Find the number of months until the amount in his account doubles. Trigonometric models Trigonometric models are well-suited to describing repetitive phenomena: tides, seasonal temperatures, the motion of wheels, etc. This chapter revises and consolidates previous knowledge of scientific notation, exponential expressions, logarithms and estimation skills. 1 and the outside function is fiu) = u2. For this purpose, we think of wrapping the real number line around the unit circle. Method with 1 = 6 and At = . = Solution Since we are interested in minimising the total time, we should start with a very general model of T = Ty, + T, where T'is the total time, T is the running time, and Ty is the swimming time. (a) 101m (b) 35.5m ( a:50,b:%,c (d) t = 12.8 minutes 7. Therefore, the 5 s owheny=0=0=2x-775=x=31= W31,0 gx ~FIBE 10 e Since all three are adjacent to vertex E, we will find the distance of each to vertex E. For K, - equation of ZW:y 1 = %x 35 =y= %x 775 s wheny=15=15= 0 i x= 157,471,785 @ 197 (a) 6cm (a) r=135cm (c) 20257= 63.6cm ) 67 (b) 9cm (b) 27 + 37 = 36.4cm r=46=-% 7.r=150=16 (a) 125 hours (b) 3m () approximately2 X 3.25 = 6.5 hours . (a) Given that fixed costs are 339, find the cost function, C(x). (a) Arithmetic 10. (b) Find the common difference for the arithmetic sequences and the common ratio for the geometric sequences. o If0 < a < 1, then as x increases, f decreases. (g) 2% Multiply the exponents. After t minutes, the concentration of medication remaining in his bloodstream is given by A(f) = 10(0.5)"* where A is in milligrams per litre. All points on the floor are mapped by the x- and y-axes. It is very useful to have a notation that immediately shows the magnitude of this number that would otherwise be written as 0.0000000001 m. Provided that measurements with comparable units are used, addition and subtraction is straightforward. A Kite is at the end of a 60 m string, hovering at an angle of elevation of 54. In scientific notation, it is written as 1 X 10~1m. In this case, we can clearly see the maximum revenue and can use a GDC to verify our results in parts (c) 1(x)=-5-x+1000-x] and (d). What is the difference between Resistance and Resistivity? 1 () (1,-1,2) 0 I 1 2 o),B: (vl) (a) Det=0 (b) A=5 @-3t1+41 = sera=s(_2)() (a) 0.772,0.129,0.099 (b) 0.756, 0.139,0.106 3 = g h %) =D=pP1ap= 1 0 14. If symbolic representation is the most significant technical advance in history, what would you put in second place? The range ofa function can be found graphically or by analysing the function algebraically. Solution Using the cosine rule, AB = y20? Although both terms suggest a lack of precision, estimation infers a lack of precision in the process of measurement, while approximation lacks precision in the statement of the measurement. Ebook. Next, we need to find an expression for Ts. 900 But if we are dealing with a well that is 4000 km deep, then this factor would be significant. Chapter 19 practice questions 1. Notation for the terms ofa sequence uy = first term 1= second term ity = ith term Find the first five terms and the 20th term of the sequence (b,) given that b, = 2(b,, + 3)and b, = 5 Solution The defining formula for this sequence is recursive. Determine the two possible changes necessary to the length of r to form exactly one triangle. This is what we were doing when we started this chapter by counting cows. The angle between each pair of adjacent edges will be 260, n 5 443 (a) All have (9 Yes 12 (b) 43i d) z= V7 +3/2i 935 Answers 2. Consider a continuous function f{x) defined over a closed interval [a, b]. W] Distribution of sample means Consider a simple uniform distribution such as the distribution of outcomes in the repeated rolls of a fair dice. Here are some more examples of sequences: 16, 12, 18, 24, 30 3,9,27,3% 1. gives 8000 = 60 000r _ =7~ {50000 =0715 The annual rate of depreciation is 1 0.715 = 0.285 or 28.5% Example 3.14 can also be solved using the built-in financial package on a graphing calculator. 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. (a) ~384 () u, = 3-2" () 4,g = 2u, andu, =3 o) 935 u, = 10.15 0.1n u, = t,_, 0.1 and u, = 10.05 93 (b) u, =101 n u, = u, , 1and u, = 100 ) u,=972( -y and u, = 972 3) z( 3)3\ ) u, = 2uy yanduy = -2 390625 28 @ Y7689 ) (b) u, = 35(7)o u, :;u" Sl =35 5 . I function is either increasing or decreasing, it is said to be monotonic. (a) 69 () u,=11n 19 () ty =ty + Nandu,= -8 19. At the level of significance of& = 0.05, test the statistic given by the app. Fior began to boast that he knew how to solve cubics. dx 553 Introduction to differential calculus Applying the chain rule: dy S d; du dx dx e = 2(4x% 1)8x = 64x 16x In this particular case, we could have differentiated the function in expanded form by differentiating term-by-term rather than differentiating the factored form by the chain rule. Example 1.7 The squares of a chessboard are numbered consecutively from 1 to 64. By definition: (c) Anarc oflength t = 7 is equivalent to one-half of the circumference of the unit circle, so it terminates at the point (1, 0). ib mathematics analysis and approaches sl inertialearning web ib mathematics analysis and approaches sl study notes ib practice questions with detailed answers and much more is waiting for you question bank syllabus 1-2 2. AC/(x) = 0.0008x 0.09 %9 = = x~ 136 = AC(136) ~ 23 (d) MC(136) = 0.0012(136)> 0.018(136) + 25 ~ 23 () At400 units, the average cost is: AC400) & 54 q Thus, the company should charge 74 per fan. From the basic principles through to optimisation and beyond for HL students. 20 (b) Find a model for the nth term u,, where n = ml%ss (c) Use your model to predict the extension for a mass of (i) 100g (i) 150g 30 Figure 3.2 Diagram for Example 3.6 (d) In fact, further experiments found the extension at 150 g to be 30.9 cm. Web"This book gives you fully worked solutions for every question in Exercises, Review Sets, Activities, and Investigations (which do not involve student experimentation) in each chapter of our textbook Mathematics: Applications and Interpretation HL.