ib mathematics: applications and interpretation pdf

(d) Atwhat time is the velocity of the tennis ball equal to zero? Thus, the amount of the gift should be 1 Present value = | o 20000 095 dt ~ 134726 19. They feel that there really is a mathematical world out there and that they are trying to discover truths about it, much like natural science discovers truth about the physical world. : 4 3 This gives C =3 2 + 0.5d,08 We can write these two results as one piecewise function: = { 3 + 0.5d, 0=d=38 7+03d8, d>8 B 2 b b sariri :. The unit circle and trigonometric functions So far you have learned about the three trigonometric ratios (sine, cosine, and tangent) in the context of triangles, for angles less than 180. Here are some examples of estimation and approximation. 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. Developing trigonometric models Now that we have an understanding of the parameters of trigonometric models, we can model some real-life periodic phenomena. 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.) The interest is calculated at the end of each year and added to the amount outstanding. For example, to convert 225 to radians, use the 5( ) = 57=7 5457 == 5(7) fact that 225 = 545).eSince 450= T7, then 225 o = 5450 Example 5.6 (a) Convert the following radian measures to degrees. This will be more powerful for modelling real-life phenomena. Find the indicated roots. All courses, with the exception of Math, Environmental Systems and Societies and ab initio language, are taught at the HL level. v oy bl General solution { = =1y(~1),( (-17)) Eigenvalues: v towards the equilibrium. Numberof spoonfuls | 1 | 2 [ 3 [ 4 [ 5 [ 6 Mass remaining (g) 280 | 255 | 235 | 200 | 175 | 145 (a) Find an estimate of the common difference, by using an average of the differences. This is an important question that motivates much of what follows. The relationship of nature to geometry was explored by the Scottish biologist DArcy Wentworth Thompson in his magnificent book of 1917, On Growth and Form. It is important to note that the 3% interest is paid annually and is added to the account, so that in the following year it will also earn interest, and so on. (a) 2memhr! According to data published by our local airport, Approximately 2 million passengers used the airport in December. For example, finding the 100th term in the sequences in Example 3.8 would be laborious, using the recursive formula. dt Population growth rate in cases where the birth and death rates are not variable i proportional to the size of the population. 64 If we repeat the steps above using the general terms, it becomes easier to develop the formula: Time (years) | Amount in the account ($) 0 1 A, =P A,=P+Pr A, +Pr=P+2Pr t A,=P+Prt Table3.2 Developing a formula Note that since we are counting from 0 to , we have + 1 terms, and hence using the arithmetic sequence formula, a,=a,+m1Dd= A=A+ OAr=P+ Prt Example 3.7 $3500 is invested in an account paying simple interest at a rate of 4.2% per year. (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. (d) On Earth, at sea level, acceleration due to gravity is the constant =9.81ms 2 Compare to the value of a and interpret the difference. @ fion =23 b) finy =257 ( foo =& x=0 (d)f(x):x}rz,x;ez @ foo=4x4x>0 ) foo=Vi=5,x>5 (g) fix) =ax+ b,a#0 (h) floy = x>+ 2x,x< 1 5 = 0 foo =555 x=a (a) Write down the least possible value of a. 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 ! The number of cells in the culture at 12:00 on Monday is 47. Let A be the amount per kg, When there is no battery, the capacity will go down till t it is completely exhausted. The rounded values in part (a) of Example 1.1 produced differences of (i) 256.4 256 =0.4 (iii) 63.5 64 = 0.5 (i) 1.49 1=10.49 (iv) 700.9 701 = 0.1 What are the percentage errors in rounding if the original values are assumed to be precise measurements? Express exactly. . o o are purely imaginary; the origin is a stable point (but not asymptotically stable). L T 4x 4 = Consider the relationship hx) = Vx 4,x=a (a) Find the least value of @ which makes 4 a function. 10. /3-8 15.972. IB Applications and Interpretations SL and HL Resources. The midpoint of [AB] is ( i 2* = 622) =(1,2) Therefore,y 2=(x L)ory=x+1 The perpendicular biseeiok Al Fyment [AB] is the normal to [AB] at its midpoint. - @) P = To (b) 70 tigers (0) 7.37yr. The zero after the decimal point is also significant. He makes his first deposit at the end of the first year after taking out the loan. For part (b), to find the minimum time Figure 14.17 GDC screen for the solution to Example 14.19 Again, we could use aGDC to analyse the function directly. These seem to be rather modest beginnings on which to build the complexities of modern mathematics. A series AC circuit has a resistor of R = 12 (), a reactance across an inductor of X; = 12 Q) and a reactance across a capacitor of X. The Analysis and Approaches course is suitable for future mathematicians, engineers, scientists and economists. T & Figure 5.6 Angles are not only in triangles - in this section we study angles of rotation, which can be used to describe this snowboarders 3607 jump 129 Geometry and trigonometry 2 Angle Remember than an angle in a plane is made by rotating a ray about its endpoint, called the vertex of the angle. How can politics enter mathematics? As you will see in the next section, this method of regression minimises the sum of the squared residuals, S, What can go wrong? by 8 m. For example, ( 2 3 2 7 ) and ( 2 xAa X) areequal onlyifx =3and x> 4 =75 7 which can only be true ifx = 3 Adding and subtracting matrices We can add two matrices A and B only if they are the same size. The lifeguard will leave point L, run along the waters edge (the x-axis) to point C, then swim to the swimmer at point S, as shown below. n X p matrix.% The product ABis: an 7 X p A= [a;)isanm X n matrixand B = [b;] isan matrix AB = [;] where aikbiy = ayby+ apby+ + ajyby foreachi=1,2,,mandj=1,2,,n For the product of two matrices to be defined, the number of columns in the first matrix must be the same as the number of rows in the second matrix. c 12 0 2 d 6 4 ) 3. 4, R A farmer, F, uses a tractor to travel to his house, H. The tractor can travel at a speed of 25 km per hour on roads but only at 15 km per hour over fields. 2. It would appear that the o sum to infinity, Z(Zl-) = (1\k-1 , equals 2. 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. 4. ()6 =6 5 2X2 2X%X2 2X2 A4 50 =2 3 & 213 3X3 2 3 3 7 10 - 7 9 7 4 5 7 7 3X3 7 16 | _ |7 5 1 0 0 010 0 0 1 7 3X3 When a matrix of order 2 is multiplied by the matrix ( . (a) C =45a=10cm,andB = 30 (b) A =110%a = 8cm,andB = 40 (c) c=12cm,B = 36 and a = 18cm (d) a=12cm, b =6cm,and c = 8cm 18. Determine the impedance in a + bi form of a series AC circuit if it contains a resistor, inductor, and capacitor with R =40, X; =7Q, and X, =3Q 199 Complex numbers 24. Finally, use the GDCs Zero tool to find the x value where T'= 0 The GDC tells us that T'= 0 when x = 8.75 Furthermore, we can see from the graph of the derivative that T" is negative to the left ofx = 8.75 and positive to the right of 8.75, so we conclude that x = 8.75 is a relative minimum for T. Therefore, to minimise the length of the rope, it should be anchored 8.75m from the base of the shorter post. WebAuthor: Jane Forrest Publisher: ISBN: 9780198426981 Category : Languages : en Pages : 672 Download Book. The domain is set X (independent variable) and the range is set Y (dependent variable). Sarah is spending her holiday camping. Mathematics can challenge our intuitions and can push our cognitive resources as individual knowers. Find the distance between each pair of points. Conclusion We have seen that mathematics is really one of the crowning achievements of human civilisation. 4. Given that the safe level of DDT is 40 units, for how long will the area be unsafe? Cut out and discard corners. Refer to Figure 5.21. (a) Find an equation to model the total cost C in terms of the distance travelled d. (b) Find the total cost for a journey of distance 12 km. = (AB')* + (z, z)* =@ x)+ () (g 2) =>AB=[tc,x)+(,~y) + (& 2) Given two points in space, A(x,, y,, z,) and B(x,, y,, z,), the distance AB is found by taking the square root of the sum of the squares of the differences in each dimension. Analysis and approahces is a mainly based on applications and gives a student the 'know how' and the ability to apply maths to real world techniques. Consider the function y = tanx 8 sinx. Ebook. (b) Use your model to find how much pollution entered the lake in the first three years of the plants operation. Formalism is connected with rote, the traditional method, which is still common in many parts of the world. This unit takes students on the journey in to Calculus by building on work they have done in modelling. WebFollows the new 2019 IB Guide for Mathematics: applications and interpretation Higher Level Available in the series Mathematics for the IB Diploma: Analysis and approaches SL Student Book ISBN: 9781510462359 Student eTextbook ISBN: 9781510461895 Whiteboard eTextbook ISBN: 9781510461901 Mathematics for the IB Diploma: Analysis and (a) 69 () u,=11n 19 () ty =ty + Nandu,= -8 19. He does not invest any further amount or withdraw any money from the account. Points (2) and (3) show us that the actual content of the model depends on something outside mathematics namely some well-established results in physics. Mass (g) 10 Extension (cm) | 24 | 20 48 | 30 70 | 40 50 60 70 80 95 | 11.8 | 142 | 164 | 189 It is believed that the data can be modelled by an arithmetic sequence, according to Hooke's law. 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. V=27 fz xfx)dx; we use Trap. (a) A(1,3,5), M(2,1,3) b)IC2:7, =9); B(8.5,=7) (c) F(7,1,3V6),8(4,3,V6) | Solution (@ AM=/2-12+(3-1?2+B-52=3 (b) CP=J8-122+(B-72+(2+2) =2/5 (c) S=y4-72+(B-1>+(/6-3/6)2=8 Example 4.19 A zip line is to be constructed across a rectangular, sloped piece of land from a point up a tree (T) in one corner to a mound (M) at the diagonally opposite corner on the ground. We will figures. The GDC shows that the bestfit model for the first part of the data is d = 0.400h + 1.90, and the best- fit model for the second part of the data is d = 2.94h 0.747 Now, we combine these two into a piecewise model: q= {0A400h +190,0sh o g S SIS Year Figure 19.34 Zooming in on the bottom left corner of Figure 19.33 Graphs such as those in Figure 19.33 and Figure 19.34 are called log-lin graphs, since the y-axis is logarithmic and the x-axis is linear. B) V2mrt 11. A large lizard known as a Gila Monster is about 16 cm long at birth. Calculus. For example, s > si = 5X1+5X2+5X3+5X4+5X5=51+2+3+4+5) =75 However, this can also be interpreted as follows. If it's an objective fact that there are nine apples on the table, thats just as objective as the fact that the apples are red, or that theyre ripe or anything else about them; that's a fact. For example, 6 = 30 degrees, or 6 = 30. This is the thinking that has inspired architects since the times of ancient Egypt and generations of painters, sculptors, musicians, and writers. Use the rules of exponents to write each of the following in the form 3, wheren Z. 1. 18371 7. Maybe they were right. There is some evidence of a nonlinear relationship (but maybe only because we are looking for it!). 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. The pizza is cut into 8 equal slices as shown in the diagram. When something is proved, we can say that it is true. Trigonometry Trigonometry is the study of angles. Also, note that if b is negative, then the graph is reflected across the y-axis. Free Exam Papers Ib Math Studies Pdf Pdf is available in our digital library an online access to it is set as public so you can download it instantly. a) A Assuminging thatthat the the p percentage tage d decrease remains i constant tant atat 6.2% 6.2% per year, estimate the UK total greenhouse gas emissions in 2030. for your model. If we use (1)X, then the first term will be negative, hence we can use (1)**! For example, the display for the model in Example 9.5 shows how this default view can be very misleading. The Fibonacci sequence is related to the golden number . 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. (b) Find the difference in height between low tide and high tide. Express each as a complex number in the form a + bi (a) V=16 (b) 25 + V=25 (d) 3V2+ /=18 (e) 23 V=12 (g) Vi* (h) (=3 + (c) 5+v25 ) 7 /=9)i 197 Complex numbers . (a) A random sample of size n = 18 is taken from a population assumed to be normal, to test a claim that the average hours of sleep IB students get per week night is w = 5.5. Geometry and trigonometry go well together as they provide visual models for analyses. 2. Don't try to predict x from y using the model (especially if the correlation is not strong). ~ 443V The work with complex numbers becomes much more useful when parallel circuits are analysed. If not, state why not. One way to think about math problems is to consider them as puzzles. Webyou could enjoy now is Mathematics For The Ib Diploma Analysis And Appro Pdf below. (a) Vertical asymptote occurs when denominator equals +1= 3 =15=415 0 + signoff Therefore, ; (i) local maximum is atx = 29) + P(X= 12) X P(Y>28) + .. + P(X = 40) X P(Y > 0) 3 Px ). Mass (g) 10 Extension (cm) | 4.4 | 20 30 40 50 60 70 80 87 | 132 | 174 | 21.7 | 26.1 | 30.3 | 34.7 It is believed that the data can be modelled by an arithmetic sequence, according to Hooke's law. Geometry and This textbook comprehensively covers all of the material in the syllabus for the twoyear Mathematics: Applications and Interpretation Higher Level course of the International Baccalaureate (IB) Diploma Programme (DP). e e e e ec0e 4 e e e e o000 o000 e . The formula V = IR becomes V = IZ with AC circuits. In this case we are able to express this function in explicit form. In general, the number of degrees of freedom, given m groups and n outcomes is v = (m 1)(n 1) Calculating the chi-squared statistic We will now use the algorithm to calculate a y? By the Triangle Inequality, the sum of the lengths of any two sides exceeds the length of the third. The trick now, as you know; is to factor out (x 1) from equation (1) to give: @ G-DF-x2)=0 We can now try to find values ofx that make the second bracket in (2) equal to 0. Also, at the end of each chapter there is a set of practice questions, which are designed to expose you to questions that are more exam-like. Scribd is the world's largest social reading and publishing site. (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. 2. You need to examine the scatter diagram and make sure a linear model is appropriate. Alternatively, the equation can be written as fix) gix) = 0 This equation can be solved graphically on a GDC by sketching the graph y = fix) g, then finding the zeros. WebMathematics Applications SL YEAR 1 Mathematics Analysis HL YEAR 1 12 AP Calculus AB Mathematics Applications HL YEAR 2 Mathematics Applications SL YEAR 2 Mathematics Analysis HL YEAR 2 11 Mathematical Studies SL YEAR 1 Mathematics Applications HL YEAR 1 Mathematics Applications SL YEAR 1 Recommendation 1 4l Sequences and series Example 3.10 can also be solved using the built-in financial package on a graphing calculator. cA by factoringc out of entry in A. ) = (cos\*goi sin(90) 55% = @ happens again is 30%. s ) 5 7. So this expression means the sum of all the terms a; where i takes the values from 1 to k. We can u also write _a; to mean the sum of the terms @, where i takes the values from =m m to n. In such a sum, m is called the lower limit and the upper limit. Consequently, having four TV shows produces 3 degrees of freedom. 3. This makes sense. X )2-i 5. It is very helpful to be able to find an explicit expression for the partial sum of an arithmetic series. (c) Interpret the values of v, and s, in context. We hope could proceed by the result is even: 1+3=4even 5+ 7=12even 13-F9:=22/ven 131 + 257 = 388 even You can see that this method will not serve as a proof because we would have to check every possible pair of odd numbers and, since this set is infinite, we would never finish. Table 1.1 Significant figures rules and examples Indicate the number of significant figures in each value. Hence, the bird is at B(30 - %, 60 - %, 20 118 %) = B(ZO, 40, %) i A cuboid measuring 10 cm x 12 cm x 20.cm is cut into a rectangular pyramid, keeping the face that is 12 cm x 20 cm as its base and 10 cm as its height. A research vessel is mapping the floor of the sea in a certain area. 143 Geometry and trigonometry 2 Solution Since cos? + sin?f = 1 cos?f + el N2 are 1. 13. Unstable. (0,0), (0, 3), (1, 3) dilation in y-direction of magnitude 3. Given g(x) = 3x 7, find g7'(5) 3. 900 3 26. WebMathematics: applications and interpretation formula booklet . It is center. ~ 95.49297 The degree measure of angle AOB is 95.5 (3 s.f.) (Answers will vary) (a) height (b) swimming competition times (c) distance between cities (d) photo file size () conversion of 0.5 inches to cm (f) grocery purchase (g) cold day in Singapore (h) current in amps 4. Therefore, the inverse function is g ~'(x) = *\/ Figure 2.21b GDC screen for solution to Example 2.10 (c) (ii) 40 (ii) See GDC screen on the left. 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. It is an upper (upper triangular) or alternatively when i < j (lower triangular). (a) Geometric 8. Not all equations represent a function. The matrix 10 01 ), the product is the . The data in Table 7.3 can be represented by a triangular matrix. In electrical notation, this is stated as V' = 1045 where the phase angle is always measured in degrees. Coordinate geometry in a plane lengths of segments B(x, y,) Alx,y) Figure 4.1 Aand B Given the Cartesian coordinates of two points A(x,, y,) and B(x,, y,), the length AB of the segment [AB] is the hypotenuse of a right-angled triangle whose side lengths are the differences in the abscissae (x-measurements) and ordinates Coordinate points 4B= G =X T G P (y-measurements). &= @, provided a0 3. () 6. So, a5 =3 +49 X 5 =248 = S5 = 25(3 + 248) = 6275 metric series A geometric series is the sum of the terms in a geometric sequence. The constructivist is a victim of the success of mathematics in fields such as the natural sciences. An even number is 2 times a whole number. (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. Any sum in the series Sy will be called a partial sum and is given by: Sc=a,ta,++ta_,+a For convenience, this partial sum is written using the sigma notation: Sigma notation is a concise and convenient way to represent long sums. Give your answer in the form fix) = ax? Theoretically, of course! Mathematics uses symbols to describe these amazing structures in the basic language of sets and the mappings between them. Because the value of a can cause the graph to be reflected across the x-axis, when a < 0 cosine will begin at a minimum and then increase, and sine will begin on the principal axis and decrease at first. They also identify common errors and pitfalls. It is precisely the simplification (literally what makes it false) that makes it useful. It is not a contradiction to say that the moon is composed of green cheese. When we enter the data into a GDC, we can use it to estimate the logistic model for this data as shown in the screenshot. Last updated: My lov foryou ll s immeasurabe. The area, A, of each site is given by the function A(x) = x(200 x) where x is the width of the site in metres and 20 =x =180 (a) Site S has a width of 20 m. Write down the area of S. (b) Site T has the same area as site S, but a different width. Could we allow ourselves to be guided to truth in mathematics because of the beauty of the equations? A lobster trap is made in the shape of half a cylinder. and interpret it in context. Note that the ratio of consecutive terms equals a fixed constant.

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