Therefore, a,=a;tn1Dd=2+9n-10=9n-7 (b) a5, = 9(50) 7 = 443 Example 3.4 (a) Find the recursive and the explicit forms of the sequence 13, 8,3, =2, (b) Calculate the value of the 25th term. (a) % = kP = P(t) = 500e% = P(12) = 2048 000 (b) 1000 = 500"2 = = 1 hour. Since h = 2r = S=ar(r + =(1+ 12+ (2)? 197kmh! For example, it is far better to find the derivative ofy = (3x + 5) by the chain rule. The logistic function and its parameters are shown in Figure 9.39. But if we try x = 1, then equation (1) gives us: P-2-12-1+2=0 In other words, 1 2 1 + 2 = 0 s true. (@ () P(X=24)~0.105 (i) P(X = 22) ~ 0.0925, P(X = 26) ~ 0.0902 (iif) P(X = 20) ~ 0.0616, P(X = 28) ~ 0.0595 (b) As the probability is the highest at the expected value and those adjacent gradually decrease, the distribution will likely fit a normal curve. What other knowledge is worth pursuing for its own sake? where Pis the power in watts A is the area swept by the turbine blades (swept area), in m? (c) Are these roots the negative of the roots of v i? 4. 1) o 4 12. 3. (@) 2% 1 5. 98 Geometry and trigonometry 1 Geometry and trigonometry 1 Learning objectives By the end of this chapter, you should be familiar with finding distances and midpoints o determining intersections of lines in a plane applying the Pythagorean theorem applying trigonometric ratios in right-angled triangles using the sine and cosine rules to solve problems involving triangle o finding the areas of triangles calculating the volumes and surface areas of cuboids, prisms, pyramids, and spheres e determining measurements in 3 dimensions. How much money will Jane have on her 16th birthday? The question for your mathematical intuition is: What is the probability that X actually has the disease? (You should recognise this as being a problem of conditional probability.) Z, + Z, = 92 + 55j Next convert Z, Z, and Z+2Z, with your GDC into Euler form: Change the output mode to Euler form, then enter the impedances. (a) (b) 2 7E ) 13 () 2 (8) 29 % 3] 24 il () 2 (h) 6/3 | o5 {n | | 5. The area, A, of a circle is given by: A = 7r? WebHandwritten notes on the recently changed syllabus for IBDP Math, for first exams in 2021 - this is the Applications and Interpretation HL course (A&I HL / AIHL).It is 112 pages long - 19 on Algebra, 16 on Functions, 26 on Trig/Geometry, 28 on Stats/Probability, 23 on Calculus. (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. 13. x = 115 cm, max volume = 403 cm? By any standards this is an extraordinary statement for a mathematical physicist to make. This was a decrease of 6.2% from the previous year. Applications and Interpretation SL setup: Applications and Interpretation HL setup: Grades range from 7 to 1, with 7 being the highest. (577 = e00012t . Function fis defined by fix) = { %I 6x>*+3x +10,x=5 =52 L 70x 09508 5 (a) Show that fis continuous at x = 5 (b) Sketch the graph of y = fix)for0 = x 1% g0 = x* with domain x < 0 T T T T T -1 1 2 3 4 Figure 2.18 'The graph of > 5% h(x) = x2 with domain x = 0 A function for which at least one element y in the range is the image of more than one element x in the domain is called a many-to-one function. In this case, the distance of the flight can be estimated by the arc length. We now want to consider the graphs of logarithmic functions. No remember how the parameters a, b, and d affect the graphs. (ii) Find the sum to infinity when it exists. 1 _3 At first glance, the following definition may seem unusual. 4 4. Here are the data they collected. 24. 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. 12 136 0 31. (b) Find the cost of a journey of distance 9 km. Trigonometric identities Remember that an Finally, there are two fundamental trigonometric identities you should know. Solve My Task Do math equations You are cycling to a campsite that is 100 km away. Stephen invests $4000 in an account that pays 4.7% interest compounded monthly. 2 @ z,= 6ds%ra.ndzz = 2cis ) z = 16cissT7Tandzz = 4cisT () z1= Sds%andzz = ZCisg ) z, = ISdS%Tand 2= 2cisg 4. Are you happy with every stage of this proof? Einstein suggested that the most incomprehensible thing about the universe was that it was comprehensible. Mappings between sets Once we have established sets in our mathematical universe, we want to do something useful with them. (a) fit) = 60% 56 (b) f(30) ~ 24150 8. Preferred TV show Girls Boys Total A B C 12 18 30 4 13 17 8 5 13 Total 30 45 75 Table 18.8 Complete two-way table for TV show preferences Just as in the y> GOF test, you've had the freedom to manipulate the values, but the restrictions based on the totals meant just the values 12, 4, and 8 noted in the row for girls make v = 3 for the entire contingency table. (b) Use the model to estimate the deer population after 10 years. There are photocopiable resource for general and credit students. (@) % =32+ 2k ) k= % () y=0r- 4. P(X > 40) P(X > 40) 0.333] ~[0.667 (a) 0513 P(more than 10 on Monday) - P(more than 40 over the week) (0) 82 (d) 229 (b) 15.9% (e) 43785 [0.417 0259 0.324] [0.4186 02583 0.3231] (iii) [0.419345 0.258067 0.322588] (b) [0.419355 0.258065 0.322581] 26. volume ~ 277 X 127 & 798 cubic units. The key can be made public because in order for it to work it has to be split up into its component prime factors. (a) 25 2 4 6 8 10 12 14 16 18 20 Since A, B, C, and D are collinear, the perpendicular bisectors between each pair of points must be parallel. During that time, we have received many useful comments from both teachers and students. For each expression, raise the complex number to the power indicated and state the result in a + bi form. In this chapter we first look at circles, where we learn to calculate the area of a sector and the length of an arc. For example, the approximate distance from the Earth to the sun is 149600000 km. Interpret the gradient and intercept in context. | am: infinitely thankfi for ourwonderfuland kind-hearted children ~Bethany, Neil and Rhona. A Z(E)e 10 17(2) =2~ Nl = 1.9999980926514 NI I T | | (eI k=1 1.998046875 = 1.9999999999999982236 | [SI = | k! (b) Calculate how long the sprinkler should run in order to provide 2.5 mm of water per square metre. V27 32 -5 N .3 7. and is located at sea level where the air density is 1.225kgm 3 (a) Find the cubic model for this wind turbine. We gain accuracy but lose understanding. s 9=l s o Forexample (2 19 sin*QO) cos(90), (b) () X=(C-BA" 5 2 -3 10. 1 and the outside function is fiu) = u2. Method to estimate the integral ppmdl? (c) Find the time it takes for the activity of the substance to reach 20 Bq. 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. WebIB Past Papers 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. 121 122 BE B ERRILLGIL ! In the context ofa mapping, each value in the domain is mapped to its image in the range. Many of the students (and teachers) we have worked with in the past give the same answer: the probability that X actually has the disease given a positive test result is about 99%. Wells) value, = A1) = [ o 1000 0d =500(1 e"!) In mathematics, concavity is whether a graph curves up (concave up) or curves down (concave down), as shown in Figure 14.9. 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. o flo= = 1, so the graph always has a y-intercept at (0, 1) o Ifa> 1, then as x increases, so doesf. 36. (a) L5n+0.1-3(1.2"1 () 1642 17 13. Ox3)2 N (i) y = figw) e L s 2 2 e The outside function is fiu) = u ' = f'u) = u 2 The inside function is gix) = 2x 3 Using the chain rule (function notation form): = flgw)-g = (2x 3)72:2 &y dx 2 2x-237 559 Introduction to differential calculus @x=15):3=8x=2]-2 (0% =612 _11 @2x 57 (ii) Rewrite fix) as a product and apply the product rule (with chain rule embedded) =(3x 2)2x 5L fory =uv,u=3x2andv=(2x5)! The unit circle and trigonometric functions So far you have learned about the three trigonometric ratios (sine, cosine, and tangent) in the context of triangles, for angles less than 180. Function g is defined by g(x) = { =22 g% 0=x=2 =D Given that g is continuous at x = 2: (a) find the value of a when b = (b) find the value ofb whena 2 = 3 . + 169 = /8757 + 49 + /(25 875 + 169 We can use a GDC to calculate this without losing precision (remember to store the value from the graph page). (a) (b) (c) (d) () 5a () 23 (b) 4 (@ 64 () i (O -3 (b) 25 + 5i (o 23-251 [ 1 @ -18 () 5+5i (B i @ i 4. But, how exactly does b affect the period? Given that the Roman 0.0001216 empire existed about 2000 years ago, this appears not to be 23. and so on. (a) f'o= 5x* (b) 80 () y32 ~$x-z30y2sso: x+2= x + 80y 2562 =0 3. 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. eeeeeee Solution We need to introduce a variable to represent the possible position of point P on the boundary, so let distance XP = d We can now find expressions for other lengths in the diagram. Find the measure of 3 significant figures. The methods mathematics teaching you have experienced reflects a Platonist or constructivist view of mathematics? N = number of payments made in total nnual interest rate (X 100) PV = present value of payments PMT = value of the regular payment uture value of the payments = number of payments per year /Y = number of compounding periods per year PMT:END BEGIN payments made at beginning or end of year Interest compounded n times per year Suppose that the principal P is invested as before but the interest . 23. (b) Use your model to estimate the number of fish in September. One is reminded of the story about the little boy who, when asked by his grandmother what day it will be tomorrow, replies, Lets wait and see Social facts do not require us to wait and see. (a) () (b) u, = 307 27n 12 14. 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! How long will this life in the lake survive if the plant does not stop polluting it? Fortunately, the mark was moved, but always in line with the original two points T and X. 4 b) T &) _ [-{7% =0=Amop=mgARkv:=0=v=v= () K=2/49 15. To solve a math problem, you need to figure out what information you have. _ (048 (b) TX= (o.sz) 10 g 080 0.05 0.10 090 0.15 4. . (a) Convert 30 and 45 to radian measure and sketch the corresponding arc on the unit circle. Dont extrapolate. The vision we had of mathematics as a vast virtual reality limited only by the imagination and the rules that are installed there is a constructivist view. (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. (a) Given that the diameter of the Riesenrad is 200 feet, deduce the value of a. Not all equations represent a function. The word function is used to describe this dependence of one quantity on another, i.e. (c) Eigenvalues: ~2, 6, eigenvectors ( _| i x= Cie + 3C,est General solutios Cie + 5C,es The amount will stabilise at 1250 mg. In an arithmetic sequence, a; = 40 and @, = 18 Find an explicit formula for the nth term of this sequence. Colour Teachers 12th grade blue | green | beige | pink | yellow | Total frequency 9 8 11 6 6 40 12 7 5 6 9 39 11th gmde 11 10 8 7 9 45 10th grade 10 9 5 8 9 41 Test the claim that the colour preferences are independent of the school grade or group at the level of significance of & = 0.05 (a) State the null and alternative hypotheses. (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. There seem to be two categories: those questions that occur strictly within the virtual reality of mathematics itself (mathematics for its own sake) and those that occur in 897 Theory of knowledge the world outside (mathematics as a tool). () y=x 8+ x+ (17 (b) y=x25-12i 4, x= 3 edges that meet in a single (b) 'The Voronoi diagram has 4 edges that meet in a single vertex as shown. We can then find |bl: . == Solution (a) Each year the amount in the account increases by 4.5%. Just as pure research in the natural sciences produced results that could also be used for technological or engineering applications, so in mathematics, problems motivated purely from within the most abstract recesses of the subject (pure mathematics) give rise to very useful techniques for solving problems with strong applications in the world outside of mathematics. 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. 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