3. This work has been developed independently from and is not endorsed by the International Baccalaurcate (IB). 9. 30 14. derivative changes sign atx = c, then (c, fio) is apoint of inflection. 13. (g) Comment on the reasonableness of this model. dx 553 Introduction to differential calculus Applying the chain rule: dy S d; du dx dx e = 2(4x% 1)8x = 64x 16x In this particular case, we could have differentiated the function in expanded form by differentiating term-by-term rather than differentiating the factored form by the chain rule. (a) A(11,2), B(1, 3) (b) C(9, 3), D(3,5) () E(5,3V5), F(9,V5) 2. Emerge 7 years later. It follows that: When f7(x) > 0 for all xin some interval of the domain of f, the graph of fis concave up in the interval. Using the box plot in Figure 17.5, find the 90% confidence interval for the true percentage of corrective lens wearers. Now we can write the integral as du = 5dx. 360), consider a central angle and its associated arc or the circle. In such cases it is necessary to find an approximation for the common difference. From a ToK point of view, the most incomprehensible thing about the universe is that it is comprehensible in the language of mathematics. The first one is derived from the Pythagorean Theorem. In the meantime, Fior had made little progress with Tartaglias problems and it was obvious who was the winner. 14 et o c 0 o c 3 L Functions Learning objectives By the end of this chapter, you should be familiar with Understanding the concepts of a function, domain, range and graph Using function notation, e.g., fix), v(t), C(n) Finding composite functions in context; the notation ( fo g)ix) = flg(x)) Finding an inverse function f !, including domain restriction + Using piecewise functions Understanding the graph of a function; its equation y = f(x) Creating a sketch from information given or a context, including transferring a graph from screen to paper Determining key features of graphs; using technology to graph functions and find points of intersection of curves + Understanding these transformations of graphs: o Translations: y = f(x) + band y = flx a) o Reflections: in the x-axisy = f(x), and in the y-axisy = f(x) o Vertical stretch with scale factor p: y = pf(x) o Horizontal stretch with scale factor ;: y=flgx) o Composite transformations. = 16>3 12. 5. 9. A resistor, an inductor, and a capacitor in a series AC circuit have potential differences across them of 12.0V, 4.5V, and 9.5V respectively. Get the best Homework answers from top Homework helpers in the field. This is the thinking that has inspired architects since the times of ancient Egypt and generations of painters, sculptors, musicians, and writers. (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. So, we simply need to turn the wheel by adding a quarter-period to t. 0 TT> 2 4 6 Figure 9.28 The sine model above produces the same h=17 cos(l;(t i ])) + 30 as shown in the graph. Water has a lower boiling point at higher altitudes. Hersh sees a political and pedagogical dimension to his thinking about mathematics. (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?) WebMathematics - Applications and Interpretation SL - Hodder 2019 - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. The points P, Q and R represent different positions of a seat on the wheel. She pays back the loan at a fixed rate of $250 per month. 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. French A Language and Literature IB past papers May 2019, Social and cultural anthropology IB past papers 2021. also state its domain. when RPM increases by a factor of m, the (d) p(x) = 2x* + 12x* 15x () p'oo = 6x* + 24x 15 (f) poo increases for 0.775 < x < 3.22, decreasing elsewhere (g) Profit of 9350 THB for producing 322 baseball caps. At the level of significance of& = 0.05, test the statistic given by the app. Its beauty lies in the extraordinary neatness of the underlying mathematics - it all seems to fit so perfectly together: Figure 7 Diracs equation of the electron The physicist and mathematician Palle Jorgensen wrote: [Dirac] liked to use his equation for the electron as an example stressing that he was led to it by paying attention to the beauty of the math, more than to the physics experiments. Unstable. 0 Therefore range is 1.48 < fiv < 9. But if a predator had a life cycle of 6 years, the prey and the predator would only meet every 6 X 17 = 112 years. Applications and Interpretation SL setup: Applications and Interpretation HL setup: Grades range from 7 to 1, with 7 being the highest. (d) What is marginal cost per unit when the number of units is as given in (c)? 15 In questions 32-33, find the area of the shaded sector. graph A 442cm (T br); (b) vit) = 0.463b sm(% alt) = 0.0485 blcos(%bt) () Maximum velocity is 255 cm s~ acceleration is 14700 cms ! This chapter revises and consolidates previous knowledge of scientific notation, exponential expressions, logarithms and estimation skills. 7,14,21,28,35,42, 2,11,20,29,38,47, 48,39,30,21,12,3,6, a,=7anda,=a, , + 7 forn>1 a,=2anda,=a, , +9,forn>1 a,=48anda,=a, , 9 forn>1 Note that in each case, each term is formed by adding a constant number (which can be negative) to the preceding term. Since it doesnt make sense to include negative heights, we should check the position function and establish the domain of the function: sit)=5t2+18t+1=0 = t=0.5472 or t = 3.655 So, the rocket hits the ground about 3.66 s after the time of launch. Consider the following functions: 2 fx) = 2x75+ 3,x =0and glx) = 13x 4 10 xeR (a) State the range of fand of g . Suppose the interest rate is 6% per year and the inflation rate is 2% per year. TheInternational Baccalaureate(IB) offers a continuum of international education through four challenging, high-quality educational programmes to students aged 3 to 19. This is shown in the diagram along with a possible route APB. ) Our GDC suggests that the model should be s(f) = 4.65* + 46.4t + 0.687 with a coefficient of determination of R? are not significant, whereas zeros to the right of an explicit decimal are significant. (a) 1 8. ni Bigenvectors: ~F) and ( 1mages of vectors along Eigenvectors: k (2% and ( zk)' Images of vectors along 2k wl= y= ~%x and y = 2x are multiples of the original vectors. 1. (b) Find the most appropriate non-linear model using linearisation with logarithms to model mass lifted 1 in terms of weight class w. () Find and interpret the value of R? The signs of the terms alternate and there are 99 terms. (b) Use these results to convert 60 and 90 to radian measure. . 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? (b) Find the cost of a journey of distance 9 km. 1.46% Exercise 3.4 1. The LEC must be on a vertex and (6, 6) is the vertex farthest from any site. All DP mathematics courses serve to A company produces artificial leather wallets with maximum capacity of 800 units per day. (b) Find the common difference for the arithmetic sequences and the common ratio for the geometric sequences. Webyou could enjoy now is Mathematics For The Ib Diploma Analysis And Appro Pdf below. t Logistic curve Integral calculus 2 \Q ky(l {) | Solve the logistic differential equation i | d | Example 20.12 _----- Solution First separate the variables: e %: (11):dy:kdz Then integrate both sides using part_ial fractions: e =l s L | y|=ke+C = == ci% e Co koK =y= IS Oe i For population growth models, we also have an alternative form for the logistic equation. (a) Round each value to the nearest unit. 52,55, 58,61, 12. (d) 1.62 X 10 ) 8 0g,256 (e) 9.0x 107 (g) 5.00003 X 10 (i) 1X10m 3=1 login10 %: log,9v3 (8 0=1n1 0 ~2=logss(35) (k) 3=log: 4 64 (f) 3.2001 x 10! 5. We will factorise the derivative function and solve algebraically: \u_p/ flo=x2~x6=(x3)x+2) sof'c=0=>x=30rx=2 versus concave down R Concave down Figure 14.9 Concave up Ifyou have trouble Now find the second derivative: remembering what floo=2x1 concave down look like, concave up and f"B3)=5andf"(=2)= -5 remember that up is happy, like a smile @, Therefore: since f'(3) = 0 and f"(3) > 0, there is a local minimum at x = 3 since f'(2) = 0 and f"(2) < 0, there is a local maximum at x = 2 The graph of fix) is and evaluate at x = 3and x = 2 and down is sad, like a frown . (Assume no further money is invested or withdrawn from the account.) 2. If you have suggestions for improving this textbook, please feel free to write to us at [emailprotected] We wish you all the best in your mathematical endeavours. (0, 0), (3, 0), (3, 1) reflection iny = x. Example 18.10 Thirty girls and forty-five boys were surveyed to determine if there is any difference in their preference amongst four TV crime dramas. = 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. As shown in Figure 5.11, when 6= 1, then arc length = radius. A map that is as detailed as the territory it depicts is no use to anyone. - @) P = To (b) 70 tigers (0) 7.37yr. (Perhaps the infinite number of ways of 907 Theory of knowledge representing fractions is one of the reasons why some students have so much difficulty with them.) 8. Figure9.22 Phase shift ofa periodic function What does d do? Fior was a student of Scipione del Ferro, who had found a method for solving equations of the type x* + ax = b, which is known as the unknowns and cubes problem (where a and b are given numbers). 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. The function g is defined by gix) = 12 =2 (a) Show that g'x) = m g zzi Z for x = bwherebc R (b) Hence, find the smallest value of b for which the inverse function g ! The diagram represents a large Ferris wheel at an amusement park. Mathematics Applications and Interpretation for the IB Diploma Once we have a linear model, it's tempting to use the model in reverse. 19. The second identity is an alternate definition of the tangent function. () Stationary point(s) occur at di = 02x+135=0 = x = 67.5. As mathematician Roger Penrose remarked, QED: how to solve the greatest mathematical puzzle of your age. s Major and minor arcs Ifa central angle s less than 180, then the subtended arc is referred to as a minor arc. The system is called dual key cryptography. 7. Explain your answer. (a) log, 16 (b) log;2 (c) logz16 (d) log,v2 (e) log,(16) (f) log,2/2 (g) log2/2 (h) log, 52 (i) log4 + log25 log300 (j) log3 0 ) 1n() ) Inve (a) log,a (b) log,Va (c) logava (d) log;Va (e) log,:a (f) log,.va (g) log,:Va (h) log,-a/a (i) log,a >+ log,a* (j) log,:a=+log,.a* (k) log,a log,a> (1) log,a log,/a Simplify each expression. 23 August 2022. (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 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~! If C is the sum of the two matrices, then C = A + B where , i = a; + b i so we add corresponding terms, one by one. The number of bacteria doubles every 20 minutes. If, for some x = , fic) exists, and the second Example 14 The function fis continuous on the interval 0 < x < 4, and the zeros of f occur at x = 1 and x = 4. However, each of the payments earns compound interest at the rate of 2% per year. (a) How far from the base of the shorter post should the rope be anchored to the ground in order to minimise the length of the rope? 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 . When the great circle route passes over the North or South Pole, we can use the latitude of the two places to find the distance between them. The first series, 3251, does not have a limit and we say it diverges. Either way, the mathematics is the same and these do not really count as different mathematical cultures. 335 Modelling real-life phenomena 3. (a) y=x2+2x+2 27 T (b) y=x>8x+25 (b) y=x>4x+6 () y=x*(7+i)x+ 14 +5i 11. Eulcr:{ =-184 wen = %+ h(x, 2y, et (X Y]:>{Xz =6 Yuer =Yt h(3% + 2y, Be0a_ ge* ~0.42 O (d) Eigenvalues: 1, 6, eigenvectors (;)( :) Cet = G General solution: . Figure 9.35 Diagram for question 6 (b) After six minutes, the seat is at point Q. This can be done either by trying out hopeful values ofx (2 seems to be a good bet, for example) or using the quadratic formula. 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) The points T(1, 3), X(4, 6), G(6, 4), and C(8, 8) are coordinates on a treasure map. This agrees with the graph. = 1. Applications and Interpetation covers a wider range of topics, however Analysis and Approaches covers a narrower range and in greater depth. 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