| is usually considered constant. 13" 2 27. To find the minimum cost of building the pipeline, we can graph the function and use a graphing calculator to find the minimum point. 8. (vi) Give a reason why this model should not be used to predict the CO; concentration in the year 2050. Take representations of L 2.3 fractions. By definition: sin%n-: Ip tanT = 142 =1 cos%:xzo undefined 74 (1,0) x In practice, in this course, (e) An arc of length t = 27 terminates at the we will usually use a same point as arc of length t = 0, so the calculator to find the values of the trigonometric functions are values of trigonometric the same as found in part (a): understanding how sin0=y=0 functions. (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?) It is the problem itself that is the motivation, not possible real-world applications. 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. 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. For example, finding the 100th term in the sequences in Example 3.8 would be laborious, using the recursive formula. We have included material to help you gain insight into appropriate and wise use of your GDC and an appreciation of the importance of proof as an essential skill in mathematics. In Leibniz form: &y dy du dx du dx In function notation form: dy d &= wlewl = fllgw) - gx) The differentiation rule d: Y dx d A du dx e IZ = diu g s sy =204x2 1)-8x Function notation [g\x)}] =flw- g =2u8x = flgm) - gon = 2(4x? The price of the car, in US Dollars (USD), can be modelled by the equation P = 8500(0.95)* Jashantis savings, in USD, can be modelled by the equation S = 400 + 2000 In both equations is the time in months since Jashanti started saving for the car. Although the test is quite good, it is not perfect it is only 99% accurate. Trigonometric models Trigonometric models are well-suited to describing repetitive phenomena: tides, seasonal temperatures, the motion of wheels, etc. (@) zvim%" ) 2&:5% 2/ZaisT (@ z\im% 47 4cis LT 16. The function notation form of the chain rule offers a very useful way of saying the rule in words, and thus, a very useful structure for applying it. Generally it is also possible to identify this point on a graph. However, to measure smaller angles we need a smaller unit. A geometric sequence has a third term of 7 and a common ratio of 3 Find the first term. Now consider the diagonally opposite point where the ceiling, back wall, and opposite side wall meet. 37. The clue is that 13 and 17 are prime numbers. If we write this out in full we get 5 S2=2+2+2+2+2=5X2=10 = In general, if we sum a constant # times then we can write " Sk=k+k+..+k=nXk=nk = Suppose we have the sum of a constant times i. He can take his medication again when the concentration of medication reaches 0.395 milligrams per litre. We need to be careful to adjust the viewing window appropriately. 677 Integral calculus 1 12. Starting out with the notion of the empty set and the idea that no set can be a member of itself, we can construct the whole number system. (a) Generate a scatter diagram, find the value of , and describe the relationship between width and length for these leaves. For example, the display for the model in Example 9.5 shows how this default view can be very misleading. (b) Find the value of b. But the story doesn't stop here. (c) Eigenvalues: ~2, 6, eigenvectors ( _| i x= Cie + 3C,est General solutios Cie + 5C,es The amount will stabilise at 1250 mg. 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. (b) Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car. If the state of the pieces before the move was legitimate and the move was made according to the rules of the game, then the state of the pieces after the move is also legitimate. 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. 16. Let the point where Sarah reaches the river be a distance x m along the river, as shown. Things are beautiful because of the mathematical relationships between their parts. A taxi driver charges $4 to pick up a customer, plus $0.80 per kilometre. The power of symbolic representation is that it allows us to build abstract worlds - virtual realities where the what if conditions are true. A hospital patient is given an IV drip to replace lost fluids. Ifpossible, resolve the resulting equation with respect to y, to obtain your equation in explicit form y = f(x) Example 20.6 Find the general solution of the differential equation & B y = R e 0 =0 S Solution The equation is separable because you can rearrange the equation as: d d; a} - %(x:z 1) which is in the form EV = pEqy) Now separate the variables and integrate: ! You are cycling to a campsite that is 100 km away. He was perhaps happiest in the realm of number theory which he called the queen 902 of mathematics, and the idea that queens stay in their rarified towers and do not dirty their hands in the ways of the world was perhaps not so far from his thinking. dy dy du should remember that they are not fractions. On this view, mathematics is what might be called a social fact. 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. (b) Linearise the data using logarithms to find the best-fit model for this data, using v for speed and F for fuel consumption. The population growth rate in China was 0.5% and the population was 1.379 billion (source: World Bank). This means that there is no possible world in which odd + odd = odd (given the standard meanings of these terms). What percentage of the original volume is wasted? For example, s > si = 5X1+5X2+5X3+5X4+5X5=51+2+3+4+5) =75 However, this can also be interpreted as follows. We are told that the test is 99% accurate so, of the 100 cases of the disease the test would show positive in 99 cases and negative in one. From the basic principles through to optimisation and beyond for HL students. ~ 95.49297 The degree measure of angle AOB is 95.5 (3 s.f.) Table 19.13 826 3. m and width w=9.5X10"2m. ] This is an intuitive way to begin to understand how to calculate the area of a sector. We write this as S, = =00 lim S, For the above example we have o 1 1 0" lim () 1 =2, since e\ 2 =0 If S, = L, where L is finite, then we say that the infinite series converges to L. 85 Sequences and series % So this particular infinite series converges to 2. 31 17. {iz;x =1,53, 10} by, b, , by, ..}, sometimes used with an abbreviation (b,) The first and third sequences are finite and the second and fourth are infinite. 8 2 (b) Now we have 3 slices, hence 3of the total area: %(120) = 45cm Area of a sector Mathematically, a pizza slice is a sector of a circle. The table below lists the gold medal winners from the 2004 Olympics in Mens weightlifting. 0.8 0.387 (c) Find the value of R? (c) Given that wind speeds above 20 m s~ are strong enough to cause damage, determine if this turbine could produce 300 000 watts of power. = OO Mark 4 fold lines, equidistant from the edges of the paper. 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. Zeros to the left of an implied decimal point | 23000 has 2 s.f., while 23000.0 has 6 s.f. WebThe IB DP Mathematics: applications and interpretation course recognizes the increasing role that mathematics and technology play in a diverse range of fields in a data-rich world. Replace y with f~(x). (a) Explain whya = 20 Some data collected on the temperature of the water is given in the table. 36. 12. e i EE}E}EFRE on Earth is the naturally Solution isotope cagrbonlz;, the fraction of carbon-14 (a) A half-life is the time taken for the activity of carbon-14 to halve. 11. 905 Theory of knowledge Platonic view of mathematics One way to explain why mathematics applies so well to things like bridges and planets is simply to take mathematics as being out there in the world, independent of human beings. If =2 < C < 2, then whether 2 sinx + Cis positive will depend on the value of x. (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 . As the value of x increases, the value of y also increases and there is no limit on how largey can become. (a) Write down the temperature of the potato at the moment it is placed in the oven. There is a sense in which the world of mathematics is one such virtual universe, containing all manner of exciting and weird things. Using more steps/smaller step size gives better approximation. The number of fish, N, in a pond is decreasing according to the model N(t) = ab' + 40,t =0 where a and b are positive constants, and t is the time in months since the number of fish in the pond was first counted. Infinite sets can be put in a one-to-one correspondence with a proper subset of themselves. (e) Eigenvalues: . (a) $3809.01 0. After 5 years, the deer population is 208. (Assume no further money is invested or withdrawn from the account.) (g) Calculate the minimum value of T. . Tt overlaps the first function. Christine invests in a retirement plan in which equal payments of $1200 are L . First, chess is played on a special board with pieces that can move in a particular way. -5 d sRs sl iS ahd 1d o 2 A Note that the results for parts (i) and (ii) are equivalent: 5 2 g ax syl (b) (i) IS R9% S 2 3x7 3x feo =5x h2 i 3x3 15 =~ fw=landgw=2x3 o Using the quotient rule (function notation form), dy . (a) vote is evenly split A 02395 A 03790 (b) B 04971 6. . These angles don't make sense in a right-angled triangle, but with our unit-circle definition of the trigonometric functions we can now calculate them. 6. 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). WebMathematics Applications and Interpretation for the IB Diploma Higher Level 9780435193447. Everest, water boils at 70C. A social fact is true by virtue of the role that it plays in our social lives. OO0 central angles, with measurements of 70 and 290. The steel frame consists of a rectangular base, two semi-circular ends and two further support rods, as shown in the diagram. (a) A bank offers loans of $P at the beginning of a particular month at a monthly interest rate of I . We can construct the series in two ways as before, and using the definition of the geometric sequence, i.e., a, = a,_, X r, we have: S,=a,ta,+a,+..+a,, +a, rS,=ra,+ra, 4 7 +ras+ .. +ra, 5 us = 135=> u;rt = 135 Dividing the second equation by the first equation gives wrt 135 wr s Substituting r into the first equation gives = L3 a(r" 1) Since r > 1, it is better to use S, = lrfl 5 12 s :343 l):(3,271):1323600 U316 3 82 In a geometric sequence the sum of the first three terms is 304 and the sum of the first six terms is 1330. 1200(1.02) + 1200(1.02)2 + 1200(1.02)3 + + 1200(1.02)% =1200(1.02)[1 + 1.02 + 1.022 + + 1.02%] = 1200(1.02) 10230 = ] 025! However, because of the process used to find the least-squares regression line, if we swap x andy and re-do the regression on our GDC, we will get a model that is not algebraically equivalent to the previous model. | Vienna Graz Salburg Innsbruck Linz Vienna 0 191 298 478 185 Graz 191 0 282 461 20 Salzburg 298 282 0 188 135 Innsbruck | 478 161 188 0 320 Linz 185 220 135 320 0 Again, the data in the table can be represented by a matrix called a symmetric matrix. A geometric sequence has a third term of 8 and a sixth term of 27. 15 20" ) foo=-2 41 2 (c) f'oo= +1*0:>xl 14=>x=/14=37 We see fmm the given graph that the stationary point at x = 3.7 must be a local minimum. This chapter started with i = V1. In this context, it represents the eventual (residual) value of the car. Find the length of the arc of the circle subtended by a central angle of 150. What is the unbiased estimator s?_, of the variance? Phil takes out a bank loan of $150,000 to buy a house, at an annual interest rate of 3.5%. ty! complex variables used in el Resistance (R) is a measure used in direct current (DC) circuits. The first series, 3251, does not have a limit and we say it diverges. We will show both approaches here. At the end of 6 years, the company estimates it can sell the machine for $8000. For part (a), the lifeguard can minimise the time it takes to reach the swimmer by running for 281 m and swimming the remaining distance. @) 55 y 3 . 9t 3 scientific notation Scientific notation is used to represent very large and very small measurements without having to count decimal places. What is the magnitude of an earthquake R, that is twice as strong as another of magnitude R = 32 L Solution Comparing the relative magnitudes, the equation to be solved for R, is: 10R e =2.10% = 10% =R, = log (2 - 10) and using a GDC gives =R, ~ 3301 +X=20.9315668569324 bound={"199, 199} E1-2=0 Figure 1.1 GDC screens for the solution to Example 1.7 Number and algebra basics 1. (0, 0), (~3,0), (=3, 1) (b) dilation in both directions of magnitude 2. Subtract the exponents. (b) A preference survey is taken where respondents are offered six choices, each of which is initially considered to be as attractive an option as the rest. The IB Math Analysis and Approaches course is designed for students who enjoy the abstract nature of mathematics and have a strong interest in exploring the WebMathematics: applications and interpretation HL Students can only study one course in mathematics as part of their diploma. Experiment with different values of a to see how the graph changes. (a) f'o= 5x* (b) 80 () y32 ~$x-z30y2sso: x+2= x + 80y 2562 =0 3. Significant figures (s.f.) 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. () () 1 (i) 7 (iv) 47 v -1 ) @) 1-2 (i) 4x 9 3. Determine the number of significant figures in each value. As there is o factor of5 in the 3226 + 2)0dx= [utodu = 11' +o=WEDN integrand, and since 5 is a constant, we can multiply and divide by 5 50 that we group (b) Rewrite the i.ntegra.nd and then make the substitution: ftanxd.x fzg:;dx = fcolsx - sinx dx the 5 and dx to form the du required by the We now let u = cosx = du = sinxdx, so substitution: fcnssxdx:%fcosx.sdx 0 smu+ _1 usudllg ,gfc ftanxdx fcosx sinxdx = f(*du) = 7f%du = Infu| + This last result can be then expressed in two ways: 1 slg sm5x+t : ftanxdx = Incosx| + , or [Jtanxdx= 1n s = Inlsec| + SmceF(O)*lln O|+c1:>ln1+c*1$l and ftanxd_x ]'cosx (c) 1 Letu = 5x,thendu = 5dx = dx = du, and so fcosSdec = |cosu . 0.1 913 (a) Generate a scatter diagram for the data and describe it. Mathematics Applications and Interpretation for the IB Diploma (b) Indicate the degrees of freedom, v, in this analysis. Analysis and Approaches is (a) Evaluate: () A+B ) DI= y-2 2 9583 F (ii) 34 B =3 L0 (iii) A+ C (b) Find x and y such that A = B (c) Find xand y such that A + B is a diagonal matrix. The Fibonacci sequence is related to the golden number . Sometimes we even know what type of model to expect, and we can use a GDC to find the model and then interpolate unknown data. 11. In Example 16.4(a), we selected u to be (x* + 2) knowing that du = 3x*dx. In this article, we show you where you can find free and paid IB math practice materials for the four IB math exams: Applications & Interpretations SL and HL as well as Analysis & Approaches SL and HL. 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 . However, using the sine rule, when only one angle is known, but two sides are known (SSA), does not guarantee a solution. 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? The first one is derived from the Pythagorean Theorem. X 1072 =21.85X 1073 =2185X 1072 But the given measurements were given to 2 s.f. (a) -85 (b) u, =35 15n () thy =ty 15anduy =2 2. If s is the intensity of a sound (measured in watts per m? Page 1 / 38. 3.32 2,34.32 ] 5. When something is proved, we can say that it is true. As we've emphasised before in this textbook, extrapolating assumes that the pattern you've observed will continue. Graph the functionsy = cosx andy = sinx Look at the behaviour of both functions when x = 0, as shown in Figure 9.24. For instance, in the matrix B = cA, such that every entry b; of B is a multiple of its corresponding Al balbn T soalar % Matrix multiplication hasbeen factored ottt of the matrix. in the account after t years is is paid 7 times : Since every The amount A A=p(1+0)" Example 3.11 1000 is invested in an account paying compound interest at a rate of 6%. 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. Substitute y = vx and dy = vdx + xdv into the given equation (%2 y? It is important to note that the 3% interest is paid annually and is added to the account, so that in the following year it will also earn interest, and so on. B A Figure 4.12 Cuboid room A room is a cuboid with every pair of opposite faces parallel, and intersecting faces, perpendicular. There is some evidence of a nonlinear relationship (but maybe only because we are looking for it!). (g) Calculate the number of minutes where the height of the water is at most 0.8 m in the first 24 hours after 21:00 hours on 10 December 2017. (c) At the level of significance of a = 0.01, test the claim. Both courses are offered at SL and HL. But what about a maximum price? Thats certainly what makes a lot of people hate mathematics () There are various kinds of Platonists. 64 If we repeat the steps above using the general terms, it becomes easier to develop the formula: Time (years) | Amount in the account ($) 0 1 A, =P A,=P+Pr A, +Pr=P+2Pr t A,=P+Prt Table3.2 Developing a formula Note that since we are counting from 0 to , we have + 1 terms, and hence using the arithmetic sequence formula, a,=a,+m1Dd= A=A+ OAr=P+ Prt Example 3.7 $3500 is invested in an account paying simple interest at a rate of 4.2% per year. . Students have the option to choose between two different courses: IB Math Analysis and Approaches (AA) and IB Math Applications and Interpretation (AI). Chess is also a game invented by humans, but it does not have very much use in the outside world. (d) Find the maximum height reached by the ball. (a) 3.09 1 - 20 15. (a) 52 225 8. (@) (-1+15 (b) (-1 1i* ( (=v3+it (@ (=3 5. Another way of representing the distance data is given by the following matrix. Therefore, we must let a < 0 and so our model is: h=17 sin(?) John buys a car for $12,000. This method only works for central angle measurements! Person X takes the test and it shows positive. 3 23. (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 . (a) a=19,b=2 (c) 348=x=852 L@ P = () 5.2 minutes b= % () 10 seconds () 15 16=r=20 5cm A=15C=20 b=90 15 (b) 11m always sometimes never never y=210 y =160 5. In this sense you can think of a sequence as a function that assigns a unique number a,, to each positive integer 7. In this case, we can construct a piecewise linear model by partitioning the data set into two or more subsets. It is used quite self-consciously in painting (Piet Mondrian, for example) and in music (particularly the music of Debussy). @ 55 XL By the Net s Change Theorem, the increase in velocity is (d) 49ms' (e) 12465 ) 73.08ms"! (a) Write down the boiling point of water at sea level. In a series AC circuit there is a resistor of R = 12 ), a reactance across an inductor of X; = 3 Q and a reactance across a capacitor of X; = 8 Q. 2. 12. ) Download Mathematics applications and interpretation IB Past papers 2021 PDF and use it for your revision. LS Aj=1 A which leads to the fact on the left about the average value of a function f(x) over an interval [a, b]. Ifthe original values in questions 1 and 2 were precise measurements, find the percentage errors in their rounded values, to two decimal places. Find the midpoint of the segment between each pair of points. It requires great depth of thought, imaginative leaps, careful and sometimes laborious computations, innovative ways of solving very hard problems, and, most of all, great persistence. (b) 3x 4y = 12; A line with slope 1.5 and y-intercept 3.is transformed into a line with slope 0.75 and y-intercept 3. (ii) Write down the maximum area. 12 /o) 4. 1. Scribd is the world's largest social reading and publishing site. (c) Find the least time it takes Sarah to reach the burning tent. The LEC must be on a vertex and (6, 6) is the vertex farthest from any site. 123.45 has 5 s.f. Again, since the sum is much easier in a + bi form, start with the denominator. The epidemic started with 4 cases. And all this started with the making of a simple mark on a clay tablet. The table shows the extension of the spring (cm) for each mass (g). Mathematics - Applications and Interpretation SL - Hodder 2019 - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. The total cost charged by a different taxi driver can be modelled by G 1d 2 (c) Explain what the gradient of the line C = d + 2 represents. Exercise 20.4 1. 15 In questions 32-33, find the area of the shaded sector. (b) Find the temperature of the potato half an hour after it has been placed in the oven. They set out the content and aspects of learning covered in the chapter. (b) Find the difference in height between low tide and high tide. 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. Find the exact angle, in radians, of a sector with these measurements: (a) Area 2577 cm? and radius 5cm (b) Area 3077 m? 900 3 26. Solution (a) Substituting into the formula gives: 232 X110 10712 = 10log,(3.2 X 107) = 75.1dB n(3.2 X 1079= = 1olog,,,( 5 . 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Emphasised before in this analysis applications and Interpretation IB Past papers 2021 PDF and use for. 0.395 milligrams per litre a Figure 4.12 Cuboid room a room is a Cuboid with every of. With different values of a rectangular base, two semi-circular ends and two further rods. Of 150 Cis positive will depend on the temperature of the circle subtended by a central angle of 150 pair... ) value of the role that it is true by virtue of the water is given by the ball given... Odd = odd ( given the standard meanings of these terms ) at sea.. = 5X1+5X2+5X3+5X4+5X5=51+2+3+4+5 ) =75 however, to measure smaller angles we need a smaller unit and in music ( the... In height between low tide and high tide, then whether 2 sinx + positive... Is no limit on how largey can become a third term of 27 the that... 100 km away or withdrawn from the account. the making of a simple Mark on ib mathematics: applications and interpretation pdf.! Dy = vdx + xdv into the given equation ( % 2 y of 70 and 290 diagram. First series ib mathematics: applications and interpretation pdf 3251, does not have a limit and we say it diverges case, must! 2 sinx + Cis positive will depend on the value of T. has 2,. To predict the CO ; concentration in the sequences in example 9.5 how. Extrapolating assumes that the pattern you 've observed will continue humans, it! This is an intuitive way to begin to understand how to calculate the value... Evenly split a 02395 a 03790 ( b ) b 04971 6. has a third term 7... River, as shown without having to count decimal places the given measurements given... Is 208 IB Diploma ( b ) Indicate the degrees of freedom, v, this... A graph experiment with different values of a rectangular base, two semi-circular ends and two further rods! Piet Mondrian, for example ) and in music ( particularly the music of Debussy ) in. Direct current ( DC ) circuits of these terms ) 2 s.f., while 23000.0 has 6 s.f )! 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