Here are some more examples of sequences: 16, 12, 18, 24, 30 3,9,27,3% 1. (a) 0 (b) 'The future value of the income stream at 12% compounded continuously is 5 FV = 01255[ o 5000600 0121t x 37 545 Y 1 We will use Trap. Mathematics and the knower English poet John Keats said, Beauty is truth, truth beauty - that is all/ Ye know on earth, and all ye need to know. In this section we will see how mathematics impinges on our personal thinking about the world. The measurements are given in Table 19.13. 4 b) T &) _ [-{7% =0=Amop=mgARkv:=0=v=v= () K=2/49 15. 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. 10. (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. (a) Use these values to find a linear function that converts Celsius to Fahrenheit. As the value of x increases, the value of y also increases and there is no limit on how largey can become. 5 Figure 3.1 Triangular pattern of dots The pattern can also be described, for example, in function notation: f) =1f2)=3, f(3) = 6, etc., where the domain is Z* The first unit represents one dot, the second represents three dots, etc. 68 125 shown in Table 19.16. 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(b) Normal body temperature is 37C. The imaginary value was stored in C and can be reused. . o Ifa =1, then f(x) = 1, which is a constant, so this is not an exponential function. 2. Given that sin 6 = %, and g < 0 < , find the values of cos and tan 6 without finding the size of angle 6. (a) Generate a scatter diagram, find the value of , and describe the relationship between width and length for these leaves. In this form a, b and d are usually written as integers. 55,667 1(3)= V94 V(2516 As before, we find that the rope should be anchored 8.75 m from the post, and that the minimum length of the rope is 32m. (i) 256.4 (i) 1.49 (iii) 63.5 (v) 700.9 (b) Round each value to the nearest one-hundredth. From building bridges to controlling strategy in football, mathematics lies at the heart of the modern world. 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.) During that time, we have received many useful comments from both teachers and students. We can use our knowledge of the function to choose more appropriate settings. One of the more surprising aspects of mathematics is the twoway link to the arts and beauty. Therefore d = 112 (c) By graphing the model, we obtain the display shown. (a) Assuming that the reflector is at the topmost position at time t = 0, develop a model for the height of k, in cm, of the reflector above the ground at any time in seconds. Use the rules of exponents to write each of the following in the form 27, wheren Z. 3 T El 4m 1 27 2 T Table 9.12 Values ofb and Make sure to set b to at least the following four values: b = 1, %, 2= You should see that b determines the wavelength of the graph. However, our data indicates that the height of the rocket at time t = 0 is zero, so there must be error present in the data collection or in the model. They are coincident lines. ~ 95.49297 The degree measure of angle AOB is 95.5 (3 s.f.) By symmetry, we know the coordinates of the points on the unit circle from our discussion at the beginning of this chapter. (e) Yes.d = 7,usy = 341 2. Z~5.15 + 0.355]Q) 13i (@ {m. B EEEESES S S S _.) 126 | Solution (a) First we must calculate the arc measure. We say that fis an increasing function. Function fis defined by fix) = { 2 Ry 3 ax b x =3 Given that fis continuous at x = 3: (a) find the value of a when b = 8 (b) find the value ofb whena = 4 . 45. All we have to do now is to add two of these general odd numbers together. Can you think of an example ofa model that does not represent the world well but is nonetheless useful? (c) Jashanti does not want to wait too long and wants to buy the car two months after she started saving. 1. We will not spend too much time on these matters here, but there are a few aspects to highlight. e v ww . 2560 4 ==10 _1-kInx (b) [y> + 2xy ) e*=1+Inx? (b) Indicate the number of degrees of freedom, v. (c) Test the claim at the level of significance of a = 0.05 6. ib mathematics analysis and approaches sl inertialearning web ib mathematics analysis and approaches sl study notes ib practice questions with detailed answers and much more is waiting for you question bank syllabus The IB Math Analysis and Approaches course is designed for students who enjoy the abstract nature of mathematics and have a strong interest in exploring the (c) For the reflector to start at the rightmost position, the model must have an initial value equal to the location of the principal axis. Therefore, the area of one slice is (lZO) = 15cm? Since the rows represent the sales of the different types of camera, create a row matrix of the different prices and perform the multiplication. The origin is an asymptotically stable focus, also known as a spiral sink. Subtract the exponents. )(0.55) (o.z X =TX, (0.8 03)10\ _ (0.600 _ruy (0.400) (o.z 0.7) (1) ) X, = T"% long term will stabilise around 60% donation and 40% no donations. Hence, this method of measuring arcs is given the name radian and leads to the following definition. 1.0 0.444 (d) Hence, find the non-linear model I = ;e ~* where I, and k are constants to be determined. At the level of significance of& = 0.05, test the statistic given by the app. 1 and the outside function is fiu) = u2. A language act would be a registry officer saying I pronounce you married The use of language in a performative manner creates social facts. (577 = e00012t . Complex numbers are used in fluid dynamics, control theory, quantum mechanics, and Fourier transforms, for example. 5. Conceptually challenging but very rewarding, these topics are a great insight in to the potential of mathematics for solving applied 19. (a) W' ? 7. A statement such as odd + odd = even has such power that we can say that it is certain. A family of isosceles triangles is built upon the line segment [PQ] as the base where the coordinates of the endpoints are P(3, 6) and Q(5, 10). In this case, we can construct a piecewise linear model by partitioning the data set into two or more subsets. Call this point A(xl, Yo z). (f) Use your model to predict the best fuel efficiency (minimum fuel consumption) and the speed at which this occurs. in context. Be careful not to confuse this calculator notation with the reciprocal A The reciprocal, should one be needed, is known as a cosecant and sin expressed as csc A = %. (a) Neither 5. @ y=3x1 (b) y=5x6 y=x+7 y=2x+3 (c) x=2y=11 (d)y=3x+2 2% =/ =16 @i=6%1 (e)y:x+13 y=x-7 x+y=1 3x2y =12 5. (iv) 1344m 16. that at t = 1, production is greatest. (a) 9x 8 (b) 27x 26 (c) 81x 80 d) 3% 3"+1 (e) 10 9. (a) 2308 (e) 570.03 (b) 2300 (f) 0.0003 () 1000 (g) 0.00320 2. - 20dx 16.5 Evaluate (a) fej"dx (b) fsinlxcosxdx ( [2sinGx 5)dx () (@ [Joxds [emtndx Fa)=2 671 Solution (a) Letu = 3x, then du = 3dx tdy petde= Ly f ax dx = 3fe3x( 3dx) = fedu 1 s = 3e3+c du = cosxdx, and hence (b) Let u = sinx = fsmzxcosxdx f 2clu*:; 3+c*sm3x+c (c) Letu =3x 5,thendu= 3dx f25m(3x 5)dx = 2-fsm(3x = *%cosu (e *gcos(Sx = o)k 3 (d) Let u = mx + n, then du = mx fe"*"dxL 5)3dx = 3fsmudu femetnmdy= L m [en du*leu+[ lemx+n+c () F(x) = [x(xdr= [xide=2% Ry =21i+c=2+ Therefore F(x)= % '+ Examples 16.4 and 16.5 make it clear that Table 16.3 is limited in scope, because we cannot use the integrals directly to evaluate composite integrals. are not significant, whereas zeros to the right of an explicit decimal are significant. 0.0346 (b) 6mm a(t) =500 24,0 < < 250 225 minutes 935cm? (b) 2.04cm 8. However, it is a sort of idealisation that we accept because the model provides an approximation to the behaviour of the stone (although not such a good one for deeper wells) and more importantly it gives us understanding of the system. (a) R=62 2.3 6.3 10. (c) Calculate the number of seconds the bucket is underwater during one rotation. (b) Find the value of b. s = 4550 1. Please get in touch via the contact form if interested. This fact can be used to convert between degree measure and radian measure and vice versa. @) a=7b=2 10. Morning exams must start after 7 AM and finish by 1 PM local time (the usual start time is 8:30 AM or 9:30 AM). 5. 12. How tall would the pyramid have to be for the prism and pyramid to have the same volume? Scribd is the world's largest social reading and publishing site. (a) Eigenvectors: ("6) and ( ): Images of vctors along the x- and y-axes are multiples of the original vectors. 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