What is the magnitude of the potential difference? 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. The constant, ;is called the common ratio of the sequence, and r = forall integers n > 1. (@) V=49 () V=9-y=T (b) V=18 ) v=12-y=27 2. (@) (-1+15 (b) (-1 1i* ( (=v3+it (@ (=3 5. (a) Chapter7 ( -22 Roots are w* = cis 2177 16 ) 60 7, = $88 142 total profit (b) Associative (d) Associative 10. =51 In questions 28-29, find the length of the arc, s. Angles are in radians. There are several other applications of geometric sequences, linked to exponential growth or decay. Because 277 is approximately 6.28 (to 3 significant figures), there are a little more than six radius lengths in one revolution, as shown in Figure 5.13. The rate at which predators eat prey is B, which is assumed to be a constant proportion of the product of predators and prey. If the world of mathematics is populated by some rather esoteric objects that are literally like nothing on Earth, then it is very important that these objects are precisely specified. A skydiver records the difference between her velocity and terminal velocity every 5s and obtains the data shown in the table. + 1000(100) = 50 000 euros (d) Clearly, our model doesnt make much sense if the price p is less than zero. Point B is 100 km to the east of pointA and 30 km directly south of the boundary. )6x dx using the chain rule. The interest is compounded half-yearly. (a) 74 12 | 10 8 6 4 g 0-+ 0 2 (b) x=15 ) 934 0 2 (11,2) 4 6 8 10 12 14 16 18 X 6 8 10 12 14 16 18 Chapter 6 () Exercise 6.1 1. Rectangle area = 5 X 37, Triangle area = 3 X 18 X 37 Total area of cross section = 727.62 Thus volume needed is 727.62 X 175 =127000 m* 9. 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. Schools must be authorized to teach our IB programmes. The difference in decimals is: 1000 000. Scribd is the world's largest social reading and publishing site. = 0= (7(6.22 21) = 0=r=0 2 + 2;3(5}( + 2150 () The minimum average manufacturing cost per device 34 %(Z'r =75 oo =12.5cm 8. x =2.64km Tdt The circumference of the base of the cone is equal to the arc length of the semicircle. In 99% of these cases the test does its job and records a negative result. That piece of paper acquires its status ultimately from a whole set of social agreements. 1 2 5 6 7 8\9 10% Figure 6.9 Sinusoidal curve All AC waveforms have three distinguishing features. Note that value 5E+4 for the y-coordinate of the vertex is the GDC's version of scientific notation (or standard form). Assume that the point is at the leftmost position at time t = 0 332 Solution (a) Since the radius of the wheel is 30 cm, and the reflector is 17 cm from the centre of the wheel, we know that the minimum height of the reflector is 30 17 = 13 cm. From a ToK perspective, mathematics, with its absolute and unchanging notion of necessary truth, makes a good contrast to the natural sciences with their reliance on observation of the external world, experimental method, and provisional nature of its results. 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. There is no competition between either prey or predators. v = (2x 5)! 1. log,(m-n) =log,m + log,n 2 logb(%) = log,m log,n 3. log,(m") = n-log,m 4. log,1=0 5. The temperature of the room is 20 C. (a) Convert 30 and 45 to radian measure and sketch the corresponding arc on the unit circle. Figure9.22 Phase shift ofa periodic function What does d do? 3 Modelling real-life phenomena 1 2x _ = The width and length of the paper get reduced by twice the length of the When we expand the right hand side of the corners we cut out. This is the thinking that has inspired architects since the times of ancient Egypt and generations of painters, sculptors, musicians, and writers. Book Description Featuring a wealth of digital content, this concept On the complex plane, connect the origin O to any two points a + bi and + di then construct a parallelogram with sides parallel to those segments. A geometric sequence has a third term of 8 and a sixth term of 27. + Fora 0) or minimurm (et < 0) o, use a sine model with a phase shif. A correspondence between two sets that does not satisfy the definition of a function is called a relationship or a relation. 60 251508 3.5 =270 4. Precipitation 5s o= eoe && (b) The pattern is not sinusoidal because there appear to be unequal local maxima. But this is precisely the definition of an even number that we started with. The Fibonacci sequence is related to the golden number . First published 2019 242322212019 IMP 10987654321 dedicate this work 0 the memory of my pareisand my brother, Saced, who passed avay during the caly stages of work on this editon. (nzv,%e 02 5 184 w1 Euler:{x" ! (a) x=log,y ( x=Iny Chapter 1 Exercise 1.1 1. If the linear correlation is weak to begin with, then you cant expect your model to make accurate predictions. (b) Use these results to convert 60 and 90 to radian measure. (@) :kP$whcnt:0,k7T Q(t) = CV(1 e~1k%) 0'07823 Q) = CVet-tyke .01081 (b) P(t) = 7.6 000811 = 60010816 = 18,64 billion 1n7.895 (c) 60 = 7.6e001081 = { = 0.01081 ~ 191 years, in the year 2208. dN _ NN ~1n0.65 ~ 3543 years. Table 19.11 Data for question 2 (e) Use your model to predict the maximum specific weight of water and the temperature at which this occurs. Find the angle 6 in both radians and degrees. Therefore, we know that volume equation, we can verify that this is a cubic length = 30 2x, width = 21 2x, height = x model. 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. At a level of significance of a = 0.05, determine whether the choice of crime drama is independent of the gender, given the observed data in Table 18.8. IB Math AA vs AI. Given h(x) = x? Thats certainly what makes a lot of people hate mathematics () There are various kinds of Platonists. Ay Ax In the graph of fix) = 2x 1 we can see that each time increases by 2 units. 14. 27 - 3 24. Michael Browner Features @) Geometric 2. The number of social media postings per week by students and business owners is presented below. + bx + c has real coefficients a, b, and c. Find the function if one of the zeros is known. Here are the data they collected. She picks her largest cooking pot and the river to fill it up, before to the burning tent to pour river she runs at 22 km h ', A :200m _.--7"Burning 7 L runs to Sarah running the water onto the fire. Platonists would certainly argue that mathematics is out there in the universe, with or without human beings. Let A be the amount per kg, When there is no battery, the capacity will go down till t it is completely exhausted. It is a thought-provoking chapter that will stimulate you to think more deeply and critically about the nature of knowledge in mathematics and the relationship between mathematics and other areas of knowledge. An IB education also fosters diversity, curiosity and a healthy appetite for learning. . Write down the equation of L. (c) To download a game to the smartphone, an electrical charge of 2.4 units is needed. (c) Calculate the value of Pearsons r and interpret it in context. 46. Read Download PDF Mathematics - Applications and Interpretation SL 2 Free, Update the latest version with high-quality. =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. (d) Determine a reasonable domain for this model. The commission believes that the environment cannot support more than 400 deer. (77&) =360 = 1= /360 = 6/10 cm 113 Geometry and trigonometry 1 4 Example 15 The distance from A to B cannot be measured due to a pond that is between them. It's really best for common maths which is needed in life. 3. Thus although not essential it is recommended that you read and study the chapters in numerical order. They thought that mathematics was one of the few areas in which humans could apprehend the eternal forms only accessible to pure unembodied intellect. identity is an equation thatis true for all values of the variables. Note that the function y is in terms of u, and the function u is in terms of x. Y 2by_. (a) 104 \{ (d) 20 (e) 1.55,5.21,8.79 (ii) () is not a function, it fails the vertical line test. Ax=2m = 2 area =~ E(O +2(62+72+68+56+5+48+4.8)+0) =80.8m 14.8x=5 1 > distance~ 5~ (0 + 2467 + 7.34 + 8.86 + 973 41022+ 1051 + 1067+ 1076+ 10.81)+ 10.81) = 4449m i T 15. A woman, standing 3 m from a ( tan6=1 streetlight, casts a 2-metre shadow on level ground. His skills as a number theorist presented him with the opportunity of solving a very real scientific problem. The intensity of radiation, I, from a certain source is measured at regular Data for question 3 4. This is something that students of ToK should think about carefully. Thus, the inverse of a many-toone function will not be a function. These arrangements are political. Their data are given in Table 19.12. If symbolic representation is the most significant technical advance in history, what would you put in second place? For the relationship at the start of this section, we can write: Population P is a function of time #: P(t) = P,(1.02)" As with the population P and the time , many mathematical relationships concern how the value of one variable determines the value of a second variable. September Table 9.10 question 8 Data for (a) Find the value of a and of b. Give answers in terms of 7. Consider the following functions: 2 fx) = 2x75+ 3,x =0and glx) = 13x 4 10 xeR (a) State the range of fand of g . On any wall, floor, or ceiling, a diagonal drawn is the hypotenuse of a rightangled triangle. To do this we simply divide the side length by the length of the hypotenuse, v2. So, x = 1 is a solution to the equation. So, we want to take two odd numbers, lets say m = 2j + land n = 2i + 1 where j and i are whole numbers. Approximate the diameter of a circle that has a circumference of 109.2 cm. (a) fiv) = 4 6x 5 113 _ 5t ST 1 13 ; B is the inverse ofA 1, @0 00 0 3 "1) 1+/5\ 2 :l+~, (1=/5 2 1 = 12'5)( 1 d) L= 5. ora=ts | 1+ 2 1 0 (o6 oe Y 5 M 2 6x+ 52 =50 O)L 1-V5 V5 a=1; (71 ' G il 94 0 1 11 B 3 5t -3 2 36 -0 1 4 X = (@ + DA+ abix = aora=ts (1)) 3 4 -6 0 0 -2 ) ( =2 "f) 21 01 5)( 2)(0 1 & 5 11 (a) 3 (0 3 3 _ (1+2v5 7t-g 10. m= 0\/-2 i G) M 3y, =A- A= b) flv) = %(7 mx? %17 mx+m oom=2|_, 19 1o1\/4 6 ok e 1 = 40r A=5; (51), 1) @ =1+ 100 = Tora=10; () () 10 94t ) () A = 9n + 20 e 2 =30+ 230 =10a=2 (%) 406 wim oo 1oo -316 3t ) () A= 6A+ 54 = Lora= (5 ),(;1); 36 2o h Loo5 1 -1 010 5i =3 2 23 4 e o o 2o 1) 8 0L 0010 ) 2 -22-3a=-1ora=3(_f)(1); () k=1 100 -2 -3 @fo 10 3 3 001 -2 4 71 % iV251 6 () k=0 100 3013 2 @fo 1052 4 -3 7t ) 5o 3l (h) (4,-2,1) fo 2 (d) 2> =20 - 34 =lorA=3 (i) (;) (8 (-2,4,3) 7@ Sk ) 2 =sx+6a=20ra=3 10 (& (~1-2-31 (f) Do A A -9 =23 (2), (7h); (b) (5,8, 2) 13 ( =+ 16 =30+ 2A=1lor (d) 1672 -16 184 -1\/5 0 13X (m)X2 = a2b2;A = abor A = abs (),(T}); (b 70 bzl s (m) A2=30 1A =50rA = ~2 (ii)( Gy B 209 939 Answers (0 AZms;). 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. % 9. = u(wdx 5ltb0) = Jau=u+e ) %{sin(u)} = cos(u)u'(x) = d(sin(u)) = cos(u)u'(xdx f cosudu = sinu + %(cos(u)) = sinw)u'(x) = d(cos(u)) = sin(uu'(x)dx fsinudu = cosu+c e Alm) = T A wwn= -1- = 4] = wwds | fwdu= 00+ ane - -1 tan = L) d ey (o) == = dtanw = w()de euy e(n) = dle W) == e ety () de inlul) == 29 Juce = dlinful) ul) == ioful) () ) ok du=tanu+ U gy = et+c Jerdu=et dx [i du=1nlu| d=tulud++ Table 16.4 Integration formulae Example 16.6 Evaluate: 672 [(sx + 2)8dx (@) [V6x+ 11 dx () [t VX' 8x+13 (d) [xsint(3x?) Likewise, you may seey,y,y"" o j, j, j for first, second, and third derivatives, respectively. Can the treasure still be found along the line defined by [GC]? (a) 2,-1%3 (b :(%Ar%i) V2i, v2 2i - & (b) 43es o 6. P(X=10) = 1 P(X = 9) ~0.0139 (a) 0.817 (b) 1 (c) 0.0162 (a) 0.0338 (b) 0.0245 (c) 0.783 (@ 3 (b) 0.101 (c) 0.000215 (a) 0.0025 (b) 0.9826 (c) 0.9999 (a) 0.1396 (b) 0.1912 (c) 0.9576 (a) 0.8187 (b) 0.5488 (a) 0.9877 (b) 0.999998 (c) 0.0000244 (a) 0.0334 (b) 646 (c) 98" P(X=7) =1~ P(X < 7)~0.212 (Note: In a continuous 2.00004 5. Example 3.2 shows how this is used. We applied logarithms to both variables to obtain the best linearisation, so it is appropriate to plot that data on a log-log graph (Figure 19.35). Assuming the data is correct, is it appropriate to remove the outliers? Experiment with different values of b to see how the graph changes. much? Students have the option to choose between two different courses: IB Math Analysis and Approaches (AA) and IB Math Applications and (0, 0), (6, 0), 3V21 =7\ (=7 =3V2T 5 10 )| 10 1 1 321 -7 7-3V21 10 10 1 13 + 3721 5 2 5 o 13-3Val|3va1| | 2 (p) A2 81,4 =9or A = 9 ;:) (s0) G ao 955 | 7+32T 10 3/20-7 10 N 1 3 I ) % - 7 1}/2 0O since (0 2\_(0 2)= (9 5)(& ). (ii) Giving a reason to support your answer, explain if you think this is an overestimate or an underestimate. (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. (b) Use your model to find how much pollution entered the lake in the first three years of the plants operation. Therefore, given L, we can determine a relationship between C and k. Since the curve must pass through (x, %) we can substitute and simplify as follows: fo = 1+ Ie"x - LZ: 1+ ek" 2=1+Ce* 1ew - The equation x = lnTC also gives us some insight into the location of the point of inflection. () 44i (c) 88/3i 936 Z = 1500w |V =15V [V] = 60V 2% 636 128/Q Chapter 6 Practice questions 2. Using the TVM solver, we make the following entries and solve for the interest rate, I N=6 1% = 285 PV= 60000 PMT =0 FV = 8000 Py=1 cYy=1 PMT: END Note that the negative interest rate corresponds to depreciation. The equation fix) = gix) can be solved graphically on a GDC by sketching the graphs y = fix) and = glx), then finding the points of intersection of the two graphs. Saddle point. = 0.996 indicates that the Recall that the x-coordinate of the vertex of a parabola with equation quadratic model accounts for 99.6% of the variation in the height, using time as the explanatory variable. Mathematics Applications and Interpretation for the. P Solution (a) To find the equation for f~*, start by switching the domain (x) and range () since the domain of fbecomes the range of f! The origin is an asymptotically stable focus, also known as a spiral sink. In other problems, we need to be careful to consider the endpoints of the domain as possible optimum solutions. It is the problem itself that is the motivation, not possible real-world applications. iy 4xland = e+ x Lan e*+ linto (b) y(1) ~ 0.32768 () y(1) ~ 0.348678 4401 (d) Actual value is 0.367 879 4412. (vi) Give a reason why this model should not be used to predict the CO; concentration in the year 2050. Mathematicians often practise their art as art for its own sake. Another way of defining a sequence is the recursive (or inductive) definition. for your model. 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. s ) 5 7. Figure 5.8 Angle in standard position Two angles in standard position that have the same terminal sides - regardless of the direction or number of rotations - are called coterminal angles (Figure 5.9). 192 The frequency for AC circuits is constant and is generally set at 50 Hz or 60 Hz, and can be taken out of consideration. This will be more powerful for modelling real-life phenomena. The tables give the temperature in a growing chamber at various times since midnight. Check out our new service! S=T - kt S=T . v oy bl General solution { = =1y(~1),( (-17)) Eigenvalues: v towards the equilibrium. 16. (a) logsx= 3 1o (b) logxz o (c) log;(x?2x5)=1 11 Given that 2 = 8 and 2" = 16, (a) write down the value of m and of n (b) hence or otherwise solve 82! s Major and minor arcs Ifa central angle s less than 180, then the subtended arc is referred to as a minor arc. When something is proved, we can say that it is true. But what about a maximum price? 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. is a composite function, so we need to use the chain dv The function haxy = x =1 looks similar to the rule le tofind to find flo= %(uv) = u% + vjz function g in Example 13.20 (a) (they are reciprocals). (b) Find the number of years until the amount in his account exceeds $5000. 8. 3, 2, eigenvectors ( ;)( 3 General solutior Particular solution: 1000 b P x=Cie + Cpe y= 3Cie + 2Cye? (b) 2 The bases are the same. 5. This is shown in the diagram along with a possible route APB, where P is x km east of X. 104 (b) It appears that over time, the X drawn on the ground has been redrawn and moved! (c) Listany assumptions you have made about the farmers journey. WebNew IBDP Mathematics: applications and interpretation courses- Syllabus Topic 1: Number and algebra SL content Topic : SL 1.1 Operations with numbers in the form a 10 k where 1 a < 10 and k is an integer. Then they emerge and enjoy a relatively brief period courting and mating before laying eggs in the ground and dying. Width Length 44 43 45 48 52 43 46 50 102 99 103 111 119 100 106 114 (mm) Table 19.15 (mm) Data for question 5 6. Evaluate: (@) [x(Gx>+7)dx () fdx 22550 + 2 dx [2eV5x @[5 BrakP, @ Jrer=7a ( . () () 1 (i) 7 (iv) 47 v -1 ) @) 1-2 (i) 4x 9 3. Water has a lower boiling point at higher altitudes. State each value in scientific notation, correct to 3 significant figures. 67 Sequences and series The value of an asset, such as a car, decreases over time due to wear and tear. Find the measure of angle AOB in degrees to 3 significant figures. For example, the number mapped to the point (1, 0) because 7 is half of the circumference circle. Trigonometry Trigonometry is the study of angles. However, these special angles often appear in problems and applications. Note that the number of terms is 5, as both the beginning and the end of the first year are counted. Determine the number of significant figures in each of the following. Either one of the players has achieved checkmate, or a stalemate (a draw) has been agreed. 549% in region 1, 17% in region 2, and 29% in region 3 (a) 45% Channel 1, 55% Channel 2 (1 1) ) ar=pppi= () ) b - (b) 9 Channel 1 0.45 = 0415 0391 0373 0361 Channel 2 0.55 | 0.585 0.609 0.627 0.639 () Channel1 z% Channel2 ~ 23 17. 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