graph 0 Since each period is 4 seconds, we add i(4) = 1 second to t: 8t (d) A point on the outer edge of the wheel will have a similar model, but the amplitude of the model is now equal to the radius of the wheel. The atmospheric concentration of carbon dioxide (CO,) in parts per million (ppm) in the month of June at the Mauna Loa observatory in Hawaii, USA, is given in the table. 15. The first one is derived from the Pythagorean Theorem. 3. (a) The image s 3x + 2y = 18 (b) image is 2x + 3y = 18 1 o1 0 _(-1 0 X =TX (o.z _ oy, (0.8 0.7)(0.7) (0.55) 03)(045) _ (0.525 (0.475) . Precipitation 5s o= eoe && (b) The pattern is not sinusoidal because there appear to be unequal local maxima. The owner of a small company buys a company car for one of his employees. Note that in each case, the location of the horizontal asymptotes is given by the value of in the model fx)=k(1 Figure 9.15 Graph of wombat population +py+c For the wombat example (Figure 9.15), = 0 so the asymptote is at y = 0. 17. (e) 2 The bases are the same. Find the combined impedance. The diagram shows the graph of /, for0=h=72 The point A(6.25, 0.6) represents the first low tide and B(12.5, 1.5) represents the next high tide. It is increasing for ] oo, 0[, and decreasing for 10, oo Therefore, f does not have an inverse f ! Cubic models are also widely used in computer graphics, in everything from modelling the curves of the letters in this textbook to smoothing computer-generated effects and animation. Then dx = du, and f% dx = f}u(*du) = j% du = Inu + , u > 0, butu = x, therefore whenx + 2,and so du = 3x*dx. (a) d =433 () () 304 1L (a) d= 27 (i) 0.330% () 307g @ (i) An underestimate. The difference in voting preferences between three areas was found through random sampling by students in the Geography class and is summarised below. (a) 1+2i (b) x=3+i () x=5%2i (b) 2 +3i () x=4+3i (f) x=4+4i (c) 35 2i 7. V27 32 -5 N .3 7. (a) 27.047 (b) 800.008 (c) 3.14159 (d) 83.47 (d) 0.0009 Give an example that would justify the use of measurements given to the following levels of precision. (a) Find an estimate of the common difference, by using an average of the differences. The domain of f " s equal to the range of f; and the range of ~ is equal to the domain off Consider the function fix) = Vx +3,x= -3 (a) Determine the inverse function f~! ()1>7+(1)679 0 5 10 300 w? -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. info@dokumen.pub @@ M=(2 +21001 Y +2350 L 350-0 @) m=150 3% (e) y 150 = 3.5(x 0) = = zero; therefore, x> = 0 = x = 0. Using Pythagoras theorem, AP = VAX? Example 6.16 Three AC sources are placed in series, with T v, = 4cos(mt 2 7 ), and V, = s(cos ot potential differences V, = cos wt, TT2 ) Find the components of the total, V.=V, + V, + V;, Solution In electrical notation, V; = 10, V, = 4/45 and V, = 690 = V=14 2/ +2/21) + 6i = (1+2/2) + (6 + 2/2)i = |V =1 +2/2) + (6 + 2/27 ~ 9.62 and arctan (M 1+2v2, ) ~ 66.6 .V =9.62/66.6 or 9.62e%5" To convert your answer to a + bi form, use your GDC. [Remember VI = |1 - |ZI] 6. Ifyou use a graph to answer a question on an IB mathematics exam, you mustinclude a clear and well-labelled sketch in your working. Solution Since the sprinkler sprays an arc of 150, the watered area is % of the total Qo circle area. (b) Find the missing side(s) and angle(s). 1=x 3 (b) Solve the equation (fo g~! (c) Find the maximum revenue predicted by your model. ) = (cos\*goi sin(90) 55% = @ happens again is 30%. (f) Find the time when the two substances have equal activity. Plotl Plot2 Plot3 nMin=1 ~.U(n)E2(u(n1)+3 ) U(5) U(20) ] U(nMin)E5H Find the first five terms of each infinite sequence. L original matrix. The domain of f! That is kS T and T(0) = T, AT _ 37 ) where & where k > 0 and T, < , i.e,, the initial temperature is less than the temperature kis a constant and of the surrounding medium. What is the significance of the two diagonals of the parallelogram? Example 20.11 The rate of decay of a substance y at any time is directly proportional to the amount of y and also directly proportional to the amount of another substance x. (a) log;x = 2 (b) log,(x 3) =5 () log,3=2 (d) log; (x> +2x + 1) =0 " Number and algebra basics Optional Which is larger, or 77 The numerical values of and 7r were calculated in Exercise 1.3, Now, consider an analysis using the graph off(x) = 1% 2.11%Y 2 with its maximum at (2.72, 0.368) |22.71 -3.94; Note that the maximum is found at x = e (which can also be found using calculus later), and 7ris further to its right, below the maximum. Evaluate: @) [t35rdt c G smvt 2V ) ogdo [ 91 tan 2t cos?2t dt @ ffv?Jrz) Kz @) [VT+cosOsingdy 1 dr (k) o (m) 242 dt xard fxz+6x+7 () [3e/xTdx G [AT=tar [()] e e "' fex1+e = 5. Hersh called his theory of mathematics humanism because its saying that mathematics is something human. 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. (b) Find the minimum cost of building the pipeline. If the units are not comparable, they need to be converted to a common unit. are inverses by showing that: Af ') = xand f(fin) = x L = r faxt2)=a(ter2)-s=xt8-8=x f(4xs>:i(4x8>+2:x2+2:x This confirms thaty = 4x 8 andy = ix + 2 are inverses of each other. (Can you think of an exception?) Show that an expression for the minimum value of Q is 5000 + 1.028 5000 +o+ 1.0282 5000 1.028" (ii) Hence or otherwise, find the minimum value of Q that would permit David to withdraw annual amounts of $5000 indefinitely. (a) Sketch a graph of the function. Horizontal asymptote For the coffee example (not shown), 0 = 20 so the asymptote is aty = 20. et s apties of radioactive decay. The higher level courses each have 3 papers and the standard level courses each have 2 papers. Certainly, these problems were of interest from the 7th century in what is now the Middle East - the home of algebra. Convert this into Celsius. For students taking their exams in 2021 there is a big change to the IB syllabus there will now be 4 Determine the equation of the line perpendicular to y = 24 + 4 that 3 5 passes through the point (4, 5) 9. Topic : SL 1.2 Arithmetic sequences and series. Now since we know that the temperature T is less than the surrounding temperature, then In|s | = (s The general solution then is In(ST=ktc = S il Ree = T=eche The initial condition implies T= ek = T;=8el= =e=c=8 T =c=-In(S T Therefore, substituting this value in the general solution In(SD'=kt c=kt +1n(S Tp) =In(S ! Bivariate analysis 9. (e) Eigenvalues: . Infinity is not something that the human mind can fathom in its entirety. An experiment is carried out in which the number 7 of bacteria in a liquid is given by the formula n = 650e, where t is the time in minutes after the start of the experiment and k is a constant. Replace fix) with y. Looking for a quick and easy way to get detailed step-by-step answers? The interest is compounded half-yearly. For example, most cars have an instrument called a tachometer that indicates the number of revolutions per minute (rpm) at which the engine 130 is operating. 325 Modelling real-life phenomena Graphical interpretation It's worth looking at the graphs of the last few models. John buys a car for $12,000. In this case, the law of acceleration due to gravity. Who would have guessed that recent work in prime number theory would give rise to a system of encoding data that is used by banks all over the world? 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. Given that a straight line has gradient, m, and passes through the point (x,, y,), we can first write the equation using gradient-intercept form as y=mxtc substituting the given point for (x, ) gives n=mx +c rearranging for gives c=ymx which can be substituted into the original equation to give y=mx+y, mx, rearranging and factorising gives Vs S Find the equation of the straight line passing through the points (1, 4) and (=3, 10). The mathematics is only a tool, albeit an important one. Stable system with the origin asanode. 9.89 grams. (@) P(6,~1), Q(4,5) (b) R(7, 2),5(1,8) (o) T(1, 4), U(8,0) 19 Geometry and trigonometry 1 . Using more steps/smaller step size gives better approximation. 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: ! If the difference in the buildings heights is 16 m, how far apart are they? 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. (x) = 1 3x solvefory solvedfory replace y with g ~'(x) 4 Functions gis a linear function with domain x R and range y R, therefore, for g ! A language act would be a registry officer saying I pronounce you married The use of language in a performative manner creates social facts. 998 AR R NN N N[N x=Ce Cet General solution: { y=2Ce%+ Gt Trajectories approach equilibrium and then move away. 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? Thats certainly what makes a lot of people hate mathematics () There are various kinds of Platonists. Mathematics uses symbols to describe these amazing structures in the basic language of sets and the mappings between them. (a) (b) 2 7E ) 13 () 2 (8) 29 % 3] 24 il () 2 (h) 6/3 | o5 {n | | 5. on the same set of axes. Your bank has a digital device that scans the waiting line of customers and suggests the approximate waiting time in line is 6 minutes. Applications and Interpetation covers a wider range of topics, however Analysis and Approaches covers a narrower range and in greater depth. DOWNLOAD FILE Each interval has a length Ax = b=n bi:i: n n Since Ax = ) == 18 PR The average (mean value) of an integrable function f{x) over an interval [a, b] is given by b av(f) b-a=51 [ foods 682 n ni=1 Ax i the average value of f(x) is given by ba . Find the measure of 3 significant figures. (b) Explain why f does not have an inverse function. Person X takes the test and it shows positive. If no units are given, assume the angle is in radian measure. (e) Since the maximum acceleration force increases by a factor of m? 25. With radian measure, calculating the area of a sector is a bit . 92 (c) For the interval 1 < < 3, find the time at which the velocity is a minimum and find its value. Tartaglia did not take Fiors money though; the honour of winning was enough. 1. The sprinkler is now adjusted to spray in a 100 arc. e Solution Start with an accurate diagram. So, you cannot say, for example, Jeosdx = feosi? At her daughter Janes birth, Charlotte put $500 into a savings account. (a) 2m 4. They are motivated by the internal beauty and elegance of their subject. 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. Paper 1. 2014 370 2015 432 2016 488 (h) The number of years it takes for global wind power production to double is 7. Examples of many-to-one functions that we have already encountered are y = x%, x R andy = As Figure 2.19 illustrates fory = |, a horizontal line exists that intersects a many-to-one function at more than one point. Year 1960 | 1970 | 1980 | 1990 | 2000 | 2010 | 2017 | 2018 O, concentration | ), | 350 | 341 | 356 | 372 | 392 | 409 | 411 (ppm) (a) Generate a scatter diagram for these data and describe it. (@) 2% 1 5. 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