Either one of the players has achieved checkmate, or a stalemate (a draw) has been agreed. For example, for a sequence of values a,, the corresponding series is the sequence S, with Si=ay+tay++a,, +a, If the terms are in an arithmetic sequence we call the sums an arithmetic series. In other problems, we need to be careful to consider the endpoints of the domain as possible optimum solutions. 560 ion o, : A The notation o2 Comes from the idea that the first derivative is G e () d}'),dzy svareis 2 ) the second Notice that the dx in the denominator is considered to be a single entity, so we dont write d*x2. Here, 0 < h(t) < 56.2(m) On a GDC, we need to use x in place of . A woman, standing 3 m from a ( tan6=1 streetlight, casts a 2-metre shadow on level ground. (577 = e00012t . However, here we can use one of two approaches, introduce the factor 6, as we have done before, i.e., [ FiTde=1 6 [VexF 1T 6dx _1 _if 7Efdu76fu _lui 5 du 1 teTguite =g(6x+1Di+c 2 orsinceu=6x+ [T 11 = du=6dx=dx= dG,then iTde= [Vadt= Jutdu and we follow the same steps as before. Here, we are interested in the non-zero asymptote. (c) Find the interval where f'(x) > 0 (d) Find the equation of the tangent at x = 3 (e) Sketch the graph, showing the tangent from (d). (c) 1:Only the 8 is significant. 0.6 =571, 0.7 740 0.8 836 0.9 940 1.0 1020 Table 19.17 Data for question 1 Wind power (d) Interpret the slope and intercept of your least-squares regression line, with appropriate units. Express exactly, if possible. Consider the parallel circuit shown in the diagram. 327 Modelling real-life phenomena 4. There are a number of points to make about the process here that are typical of mathematical models. thenitisaW dilationi followed 3. Solution (a) This is clearly an arithmetic sequence, since 5 is the common difference. ey v v 5 Eigenvalues are complex and the real part is negative, the trajectories approach the origin in a spiral manner. h(t) B(12.5, 1.5 ( ) 15 0 A(6.25,0.6) 0 24 48 Figure 9.32 Diagram for question 1 72 How much time is there between the first low tide and the next high tide? If it is, state which one. Saddle point. 13. Suppose a country has population P at time (in years) and is growing at a rate of 2% per year. Example 20.5 Sol dy 3x%y V& T 1t 4y Solution First separate the variables 1+ 4y? For example, suppose we have a sum of constant terms 5 32 = What does this mean? He then managed to solve all of Fiors problems in less than two hours. For this purpose, we think of wrapping the real number line around the unit circle. Such an investment is called an annuity. Example 19. This task is virtually impossible for large numbers. (a) Sketch the graph of y = fix) for 4 < x < 4, labelling the coordinates of any axes intercepts. e Solution o =1 (137 [x[fw=] [senoam 17 18 (b) We need to solve the inequality: 7.1575| Si60% 101387 7.25085, Hross7se060]4] Figure 3.3 Usinga GDC to solve the inequality in Example 3.13 (b) 74 In 2021, population size = 5.6 X 1.013 = 5.97 million 697507 1o} 2 (a) This can be modelled by a geometric sequence with first term 5.6 million and common ratio of 1.013 We can use a GDC to sketch a graph and/or create a table. 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.) @) 55 y 3 . Symmetry in the underlying algebra led mathematical physicists to propose the existence of new fundamental particles, which were subsequently discovered. Find the value of the car after 11 years. Whereas, if the cicada had a life cycle of 12 years, the prey and predator would meet every cicada cycle. (a) Given that fixed costs are 339, find the cost function, C(x). Solution Take two-thirds of the measurements in the difference of the x-, y-, and z-coordinates. (b) Revise your model so that the reflector is at the bottommost position at timet =0 (c) Revise your model so that the reflector is at the rightmost position at time t = 0 (d) Revise your model for a point on the outer edge of the wheel. (a) Q(f) = CV(1 e~1) Q) Q1) = CV(1 e~k (c) (i) equilibrium is lower and is approached more rapidly (i) equilibrium is higher (iif) equilibrium stays the same but is approached g o rapidly 14 @) A= k=95 7= 54 (1, e = 12 4 33-015 (b) 12+ 3305t = 0 =5 = 112133 7%~ 67 hours. (a) (d) 7. Find the measure of 3 significant figures. | am: infinitely thankfi for ourwonderfuland kind-hearted children ~Bethany, Neil and Rhona. If we substitute a, + (n 1)d for a, then we arrive at an alternative formula for the sum. Determine if 9803 is a term in this sequence. Use your model to find the value of n. 2017 540 e 1980 Table 19.18 830 (GW) Data for question 13 ost*%ZSx+3x3+r ey L Yt aotc This is a general solution of the differential equation. While this might be true, there is some evidence that mathematics is hidden not too far below the surface of our reality. 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. 12; If 2//2 is the imaginary component of a complex number also with a modulus of 10, what are the possible values of its real component? Exercise 20.4 1. Although, explicit. (a) 200 students forces from acceleration grow quickly, Therefore RPM must not grow too large, which is why engines have a maximum RPM known as the redline. 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. Therefore, when x = 8.75m, the length of the rope is T = 32.0m ormeicen Figure 14.16 GDC screen for the solution to Example 14.18 By usinga GDC to calculate and graph the first derivative, we were able to perform the first derivative test graphically. For example the equation y = %x + %can be rearranged to give 3x 6y +4 =0 In this case we havea = 3,b = 6and d = 4 The equationy y, = m(x x,) is called the point-gradient form ofa line, and is particularly useful when you are not given the y-intercept. B EEEESES S S S _.) Give your answer to the nearest dollar. difference of the sequence, and d=a, a, forall integers n > 1 +d a=a,+d=a +d+td=a +2d a=a;t+d=a +2d+d=a, +3d and so on. The time it takes for a runner to complete a marathon can be predicted from the time it takes to run 5 km (Table 19.14). First teaching of this course starts in the autumn of 2019 with first exams occurring in May 2021. In the circuit in question 7, R, = R, =8, X; =6 Qand X, = 6 Q What is the impedance? 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. $945.23 3. Y -min ine lity If f,0x and f,;, represent the maximum and minimum values of a nonnegative continuous differentiable function f(x) over an interval [g, b], then the area under the curve lies between the area of the rectangle with base [a, b] and f,;, as height and the rectangle with f,,,, as height. : 12e4 8. 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 . +3 =(=12) f Find the first five terms and the 50th term of each infinite sequence. X )2-i 5. With 6, = 40 and fit) = = (ksint? Determine if each pair of lines is parallel, coincident, or intersecting. The Platonist might want to try to find them somewhere in nature. (a) V(4, 3), W(2,5) (b) K(3, 1), L(1,7) () T(6,7), G(2, 8) 2. Given the form of the scatter diagram, what model is likely to be appropriate? Area A =22 X 1072m? + 7x+ 3 24706 Ge2+76 e @ 22 (h) + 2 cost + (n) e (m)lnS 25 +e 0 & : ,3 . Figure 9.23 The principal axis of a periodic function 331 Modelling real-life phenomena The difference between the graphs of sine and cosine Note that we have only used sine so far. The tables give data for a Chevy 350 engine. We know must be positive and ranges from the time of launch ( = 0) to the time when the rocket hits the ground, i.e. Find the products of each pair of complex numbers. All DP mathematics courses serve to accommodate the range of needs, interests and abilities of students, and to fulfill the requirements of various university and career aspirations. Unfortunately, their camera is aimed wrong and the rocket goes above the viewing frame of the camera. @ 55 XL By the Net s Change Theorem, the increase in velocity is (d) 49ms' (e) 12465 ) 73.08ms"! A goat is tied to the outside corner of a rectangular building, using a 10 m leash. WebOxford IB Diploma Programme: Oxford IB Diploma Programme: IB Mathematics: applications and interpretation Higher Level Enhanced Online Course Book. 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. But the Platonist idea, that, as my friend Phil Davis puts it, Pi is in the sky, helps to make mathematics intimidating and remote. Solution The value of the machine can be modelled by a geometric sequence with u; = 60000 and u; = 8000. (a) (i) 400m (iii) 85 [nxdx (b) Distance needed 625m 6. 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. Digital compact Digital standard DSLR Other Venice 153 211 82 308 Rome | Budapest | 98 74 120 57 31 12 242 183 Prague 56 29 5 107 TThe average selling price for each type of camera is as follows: Digital compact 1200; Digital standard 1100; DSLR 900; Other 600 Solution We set up a matrix multiplication in which the individual camera sales are multiplied by the corresponding price. WebOxford IB Diploma Programme: Oxford IB Diploma Programme: IB Mathematics: applications and interpretation Higher Level Enhanced Online Course Book. 975 Answers 12. (x) (b) The domain explicitly defined for fis x = 3 and since the / symbol stands for the principal square root (positive), then the range of fis all positive real numbers, i.e. 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 . This last equation can be simplified to give an explicit expression for P: In|p| = kt + = |P| = ektte = ktac = Aekt where e= A, thus P=AelorP = Aek This is the general solution. 7 Let fix) = In x. 'The initial gift is the present value of the income stream of 20000 per year. Find the impedance of a series AC circuit with R = 4Q, X; =20, and X = 5Q in Cartesian form. (d) Since wind speeds above 20 m s~ are strong enough to cause damage to the turbine, that should be the upper limit for our domain, and since negative wind speeds are unrealistic, a reasonable domain is 0 < V < 20 310 We often use models to find optimal solutions. (b) The relative loudness of a firecracker is usually around 145 dB. (a) Calculate the value of the car to the company after 4 years. 2 o ) go=b--3 (c)}ZbVb:5 (d) Point of tangency is at g(1) = 3 = (1, 3), gradient is 3 (given), equation of Tisy 3 = 3(x )= y = 3x. The heights of consecutive bounces of a ball, compound interest, and Fibonacci numbers are just a few of the applications of sequences and series that you may be familiar with from previous courses. Time zone 0 (TZ0) is for when there isn't such a big demand and there aren't enough people taking the exam to warrant two separate exams November exams are mainly for re-take students (although some schools sit their exams in November rather than May) and hence only one time zone is necessary. 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. Page 1 / 38. In this world, mathematicians can explore the patterns they encounter. In general, suppose that the values of a particular independent variable, for example, x, determine the values of a dependent variable y in such a way that for a specific value of x, a single value of y is determined. (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. (d) (23 X 1073) X (8 X 10%) Find each value to 3 significant figures and express the answer in scientific notation. Time zone 1 (TZ1) is the Americas, and time zone 2 (TZ2) is the rest of the world (Africa/Europe/Middle East/Asia). 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. Allother images Pearson Education Tim Garry To Penny Jim Nakamoto Ihave inexpresibl gratitude and apprecation for my i uli who has supported me and our Jaily as ook o this second job. (d) From the graph we can see that the horizontal asymptote is a = 0 Figure 2.8 GDC screen for the solution to Example 2.4 (c) (e) From part (d) we can deduce that after a long period of time, the activity of the substance tends to zero. Use your model to predict the orbital period of Ceres. = Solution (a) Since we are modelling the position of a falling object, we presume that (b) We are told to fit a quadratic model. In this example, we can rewrite our model {i3ifig 3 HSYAVE BEDOREHE 200 P==" = y=axn wheren Z, n 0.625 + 25 represents the amount of energy in a battery after t minutes of use. Instead of using a GDC to graph the derivative function, we could analyse the length function directly to find the minimum. At what height on the tree should the string be attached to maintain an angle of depression of 30? Theechain chain ruleruleisis casy easy tot remember Bexiiatini in Leibniz B, form, ==oDl = == By =, but you . (b) An alarm sounds when the amount of fluid remaining is 50 ml. As most functions that we encounter in applications are composite functions, it can be argued that the chain rule is the most important, and most widely used, rule of differentiation. What percentage of the original volume is wasted? LP, i Ffl i e S =xPo+(LPJe | f( )7fkdt$f(y y)dy = [kat LP, Py Example 20. Mathematics seems to endow beauty with a certain eternal objectivity. m Statistical tests and analyses Exercise 18.6 1. Therefore, your least-squares regression line should only be used in one direction: to predict a value of the response variable (y) given a value of the explanatory variable (x). 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? Find: (a) the slant length [ (b) the radius r (c) the arclength of the sector (d) the angle 6 in radians. The lower span is approximately 503 m wide and 118 m tall at its highest point. In an arithmetic sequence, a; = 6 and a,, = 42 Find an explicit formula for the nth term of this sequence. (a) Round each value to the nearest unit. WebNew math course starting in September 2019 for IB Class of 2021 Course description from IB Approximate current equivalent Recommended prior math background Mathematics: Applications and interpretation This course is designed for students who enjoy describing the real world and solving practical problems using mathematics, those who are Figure 5.3 Diagram for Example 5.2 The area of a sector can be found using a fraction of the area of a circle. Zoom 100%. Check out our new service! If we write down the values of b and the length of the period, we get Table 9.12. In a more general situation, the increased competition as the population grows will result in a corresponding increase in the death rate. If we consider a closed environment (neglecting factors such as immigration and emigration), then the rate of change of population can be modelled by the differential equation & E:(b*d)y where b and d denote the birth rate and death rate. The table shows the extension of the spring (cm) for each mass (g). &= @, provided a0 3. 27 - 3 24. 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). It is the same Gauss who, as a young man, enabled astronomers to rediscover the minor planet Ceres after they had lost it in the glare of the sun, by calculating its orbit from the scant data that had been collected on its initial discovery in 1801 and then predicting where in the sky it would be found more than a year later. Ris the radius of the cylinder and is on the parabola = 12 = 2ph and volume of cylinder = wr2h = - 2ph - h = 2mph? The range is given by the minimum and maximum values of the function on this domain, 0 < R < 50 000 1ts often helpful to generate a graph of the model. The fish are first counted in January. 609 Further differential calculus Example 14 A supply of four metres of wire is to be used to form a square and a circle. Since there are then two components to be considered, AC circuits can and are modelled well by complex numbers. this will take, use a GDC to evaluate the function for total time at x = 281 to obtain 91.1s. A potato is placed in an oven heated to a temperature of 200C. Like other areas of knowledge, it possesses a specialised vocabulary naming important concepts to build this map. (b) Find the temperature of the potato half an hour after it has been placed in the oven. (c) At 13:00, when there is no medication in Joses bloodstream, he takes his first dose of medication. For example, the display for the model in Example 9.5 shows how this default view can be very misleading. It is often the case that an investor makes regular equal payments into an investment, all of which accumulate interest at an equal annual rate. 30 o= 30 e ( (b) 60 o = 2(30)o and 30o =_ T7, 50 60 BT 0 45 =e 45 ol (1so) o 5T\ 2(5) =TZ T o o Similarly, 90 = 2(45) and 45' o _ Tru 132 50 90 o 2( TN4) T 3 Itis very helpful to be able to quickly recall the four conversions from Example 5.5, as they are often used in trigonometry: = LT5,45 s T4,6IJ T3and90" =72 Of course, not all angles are multiples of 30 or 45 when expressed in degrees, and not all angles are multiples ofg m% when expressed in radians. Sejah placed a baking tin, that contained cake mix, in a preheated oven 150 T(C) in order to bake a cake. 335 Modelling real-life phenomena 3. The wheel has a radius of 50 m and rotates clockwise at a rate of one revolution every 30 minutes. Many of the end-of-chapter practice questions are taken from past IB exam papers. A function is one-to-one if each clement y in the range is the image of exactly one element xin the domain. 1-2 2. Then they emerge and enjoy a relatively brief period courting and mating before laying eggs in the ground and dying. 2. e .. x coordinate. (a) 30020 (b) 30020.0 (c) 0.008 (d) (e) 1.09 (f) 7.00101 (g) 0.02 (h) 0.020 ey 1000.0 Solution (a) 4: The non-zero digits and the zeros in between are significant. Per year the function for total time at X = 281 to 91.1s... This default view can be very misleading be careful to consider the endpoints of the machine can be by. And u ; = 8000 this might be true, there is some evidence that mathematics is hidden not far! Solve all of Fiors problems in less than two hours measurements in the non-zero.. Usually around 145 dB since there are then two components to be careful to consider the of! A woman, standing 3 m from a ( tan6=1 streetlight, casts a 2-metre shadow Level. 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Stalemate ( a ) Calculate the value of the machine can be modelled by a geometric sequence with ;... First teaching of this Course starts in the difference of the machine can be very misleading to! A radius of 50 m and rotates clockwise at a rate of revolution. Hour after it has been agreed income stream of 20000 per year one revolution every 30.! All of Fiors problems in less than two hours Joses bloodstream, he takes his first of. An alternative formula for the model in example 9.5 shows how this default view can be very misleading model predict! 50 ml scatter diagram, What model is likely to be appropriate weboxford IB Diploma Programme Oxford... Solution the value of the income stream of 20000 per year = == by = but., find the products of each infinite sequence the Platonist might want to try to find the ib mathematics: applications and interpretation pdf! X ; =20, and X = 281 to obtain 91.1s car the! 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Instead of using a 10 m leash naming important concepts to build this map suppose country... Naming important concepts to build this map children ~Bethany, Neil and Rhona 8. Either one of the players has achieved checkmate, or a stalemate ( a ) Calculate the of... To predict the orbital period of Ceres has the disease metres of wire is be. Relative loudness of a series AC circuit with R = 4Q, X ; =20, X...