They use video analysis software to record the height and time of the rocket at various points. (a) 2 Xintercept yintercept O 2x8 LS ataxsg 7 8. Example 4.21 A bird lands on the zip line in Example 4.19, two-thirds of the way along the line from point M(0, 0, 0) to T(30, 60, 20). (c) Life in the lake will cease when 400 tons of pollution have entered the lake. However, when radian measure is used it is customary to write no units or symbol. The model is only as good as the strength of the correlation. Estimation and approximation While the terms estimation and approximation are often used to mean a guess, their inferences are different. 22 Solution s s e We can find the gradient, m, using m = _ _4-10 M G 3) _6__3 4 2 Then find the equation using the point-gradient form. () The method of interchanging domain (x) and range (y) to find the inverse function used in Example 2.9 also gives us a way for obtaining the graph of ! (a) (ii) v= 100 8, 60ms~! If this is right, then it is natural to ask about the purpose of this particular map. mtn=Qi+1)+Qi+1) We can remove the brackets and rearrange to give: meAtn=2j+2i+2 Finally, we can use the fact that 2 is a common factor of all terms in the expression to place it outside a bracket. The solution lies in the chain rule. Example 18.6 A personal trainer guarantees an improvement in performance compared to other training methods for deadlifts with a claim that the client will be able to lift at least 20 kg more after three weeks of training. (c) Find the minimum average cost, and the number of units produced at that level. Your email address will not be published. 549% in region 1, 17% in region 2, and 29% in region 3 (a) 45% Channel 1, 55% Channel 2 (1 1) ) ar=pppi= () ) b - (b) 9 Channel 1 0.45 = 0415 0391 0373 0361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. This makes sense. This makes sense, because as we sell more and more tickets, we could reduce the price per ticket. (@) y = ce + (d/b) ) 14. What is the potential difference of the source? Using sets and mappings we can show that there are many different types of infinity. Use your graphing application to generate a graph of y = sin(x ). 35 Table 9.1 Data for question 9 The temperature of the potato, in C, is modelled by the function p() =200 190(0.97)", where t is the time, in minutes, that the potato has been in the oven. Observed values: Expected values: 12 4 8 6 12 6.8 | 52 6 18 13 5 9 18 | 102 | 7.8 9 2] [12455 18135 9 In order to calculate the y? What time will Jose be able to take his medication again? Math is a process of finding solutions to problems. Ibrahim Wazir and Tim Garry vi Number and algebra basics Number and algebra basics Learning objectives By the end of this chapter, you should be familiar with + making reasonable estimations and better approximations demonstrating an understanding of the rules of exponents + using correct scientific notation + demonstrating an understanding of the rules of logarithms. (a) Develop a model for the revenue earned from selling these jeans based on the selling price p. (b) Use your model to find the price the jeans should be sold at in order to maximise revenue. His skills as a number theorist presented him with the opportunity of solving a very real scientific problem. Y4 6 =1 radian Radian measure One radian is the measure ofa central angle 6 0fa circle that subtends an arc s of the circle that is exactly the same length as the radius rof the circle. In 2016, the population growth rate in Singapore was 1.3% and the size of the population was 5.6 million (source: World Bank). Emerge 7 years later. (b) Find a linear function to model the cost of model B. Find a formula for the nth term. The expected lifetime of the machine is 6 years. (b) Use the model to calculate the boiling point of water at a height of 1.37 km above sea level. G A 2V3 B D AB = 2y/3 cm, BC = 6 cm, area of triangle ABC = 3y/3 cm?, ABCis obtuse. (a) () 10. A parcel is in the shape of a rectangular prism. (b) Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car. o o are purely imaginary; the origin is a stable point (but not asymptotically stable). Derivatives of composite functions, products and quotients We know how to differentiate functions such as fix) = x* + 2x 3and gx) = Vx, but how do we differentiate the composite function g(fix)) = vx* + 2x 3 where fand g are nested functions? Theres no math without people. (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. d; Show that 2 dx k2x sin(kx) dzy 19. (392 6. Many mathematicians have at least some sympathy with this view. The mathematics marks of last years HL students were compared to the predicted marks for this years candidates, with the results shown below. 1=23X10"'m=23X10"2m w=9.5X10"2m The perimeter, p, is given by: p=2(+w) = 2(23X 1072 + 9.5 X 10-2) =2(325%X 1072 =65X10"2m or =65X10"'m Find the area of the rectangle in Example 1.5 S Solution AreaA =1Xw=(23X107!) What is the difference between a concave and a convex mirror? Using production and geological data, an oil company estimates that oil will be pumped from a field at the rate given by 100 +5 M) _=20 0 b the relationship holds because the area from to b in this case will be negative. (a) Chapter7 ( -22 Roots are w* = cis 2177 16 ) 60 7, = $88 142 total profit (b) Associative (d) Associative 10. The chain rule needs to be applied carefully. Find the quadratic function of the form y = x? This description gives us the recursive definition of the arithmetic sequence. Social facts are no less real or definite than those about the natural world. In the scatter diagram itself, we see a strong positive association in the first part and a strong negative association in the second part. 3% 5,134 19. (c) Use your least-squares regression model to predict the marathon time for a runner with a 5 km time of 20 minutes. 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. Consider the matrices 2 A= B= S C:( x+1 4 1 1IN 2 3 0. The addition of complex numbers is easiest in @ + bi form, but their multiplication would be simpler in polar or Euler form. (c) Assuming the same flow rate and spray radius, calculate how long the sprinkler should run in order to provide 2.5 mm of water per square metre. 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. 0.2 189 (b) Calculate the value of Pearsons r and interpret it in context. Conveniently, each real number gets mapped to a point on the unit with an arc length equal to the number. Therefore, the probability of having the disease given a positive result is 103? The mathematician Reuben Hersh argues for a type of constructivism that he calls Humanism. (b) Verify the result by showing that (fo f~! =2 ,.2 650 " 65 15. (a) x = logsy (b) x = logy () x=1Iny (e) x=logyy1 (d) x= lugzy () x=logs(y =7 (8) x= 3log,y 8 @5 0 x= *%lny % @3 3 3 4 ssiblee (e) notpossibl ) 3 By ol (b) (c) (d) 6. e Solution o =1 (137 [x[fw=] [senoam 17 18 (b) We need to solve the inequality: 7.1575| Si60% 101387 7.25085, Hross7se060]4] Figure 3.3 Usinga GDC to solve the inequality in Example 3.13 (b) 74 In 2021, population size = 5.6 X 1.013 = 5.97 million 697507 1o} 2 (a) This can be modelled by a geometric sequence with first term 5.6 million and common ratio of 1.013 We can use a GDC to sketch a graph and/or create a table. Distance Eitore (ion)from 02 ) (0406 : (0810|12 14 Depth (m) 20 (19 [23 | 23| Figure 19.18 Scatter diagram for solution to Example 19.8(a) 16]|18]|20 22 | 28 | 33 | 40 | 46 | 5.1 Then, enter the data as two separate data sets into a GDC and calculate the least-squares regression line for each. Resource Request title. 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). The arithmetic sequence can be looked at as a linear function as explained in the introduction to this chapter. (a) dd: = kA =A(D) = A, k = 0.13863, A(0) = 50(45)=2250 mg A(1) = 2250 = Age 0601 anaesthetise the dog properly. In this case, we can assume that air resistance is acting against the model rocket and causing its net acceleration to be less than g. (e) The coefficient of determination of R? B) V2mrt 11. So, x = 1 is a solution to the equation. To the right ofa decimal point, all leading zeros are not significant, whereas all zeros 0.0043 has 2 s.f, while 0.0043000 has 5 s.f. (b) Calculate the temperature of the water after 10 minutes. The angles shown above are not central angles. Some symbols such as %, sin~! 940 11718744 5 3. The history of mathematics is a history of great thinkers building on the work of previous generations to do ever more powerful things using ever more sophisticated tools. )6x dx using the chain rule. 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.) Since we know the first term b, = 5, we can calculate the the 2nd, 3rd, 4th and 5th terms: b,=2(b; +3)=2+3)=16 by = 2(b, + 3) = 216 + 3) = 38 Recursive sequences are introduced in this chapter to help clarify the underlying concepts. It requires great depth of thought, imaginative leaps, careful and sometimes laborious computations, innovative ways of solving very hard problems, and, most of all, great persistence. 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. (b) Use your model to calculate the maximum theoretical power generated when the wind speed is 10ms ! terminal side Figure 5.7 Components of an angle Ifit seems strange that anti-clockwise angles are positive, remember that positive angles follow the numbering of the four quadrants (I, I, I1L, IV). (b) Q) =CV Q. Q (6) When the battery i removed, R+ = 0 which can be solved to give Q(f) = Ke~*C, and initial conditions yield K = CV e/, substitute this into the result and we have Q(f) = CVe -/k Exercise 20.2 1. 10. A social fact is true by virtue of the role that it plays in our social lives. Although Tartaglia could not initially solve the unknowns and cubes type of equation, he worked hard and discovered a method to solve this type of problem. We conclude our study of trigonometry by learning graphical methods for solving trigonometric equations. Trajectories move towards the equilibrium point which is called stable node in this case. ) (@) 7i ) 37 @ -37+37 -1 3 () -1 () Oz, is the sum, (z, + 2,) 2,2, is the difference, (z, z,) 4. As we've emphasised before in this textbook, extrapolating assumes that the pattern you've observed will continue. Lety = sin(kx) kxcos(kx), where k is a constant. (a) 0.75 il 68 V= nf x2dy = nfn 2pydy= (mpy?| = mph? It is possible, however, to find the explicit definition of the sequence. (d) Hence find the value of t the first time the seat is 96 m above the ground. Figure 9.10 5E+4 =5 X 10* = 50000 When using a GDC, it's important to take care when setting the viewing window. Their marginal cost per day is given by the following model MC(x) = 0.0012x2 0.018x + 25 where x is the number of fans produced. 1=x 3 (b) Solve the equation (fo g~! 1)-8x = 64x> 16x This leads us to formally state the chain rule in two different notations. (a) Find the amount in his account after 7 years. Then, by using ratios to calculate the measurement of arcs and sectors in a circle, we will develop the definition of radian measure. (0, 0), (~3,0), (=3, 1) (b) dilation in both directions of magnitude 2. )dx + 3x2v(vdx + xdv) = 0 = xX1 + 2v)dx + 3x3vdv =0 Divide by x2 simplify and integrate dx 2 Gy (1 + 29 dx +3wdv=0= [T = vdy =3[0 Inx| = %m(l +202) + C = 41n|x| = 31n(1 + 202 + 4C, Now let 4C; = InC and simplify Inx* = In(C(1 + 2v? Here, the symbol is the Greek capital letter sigma. 139 Geometry and trigonometry 2 Definition of the trigonometric functions Let be an angle in standard position and (x,) the point where the terminal side of angle Gintersects the unit circle. (c) Find the maximum revenue predicted by your model. In this world, mathematicians can explore the patterns they encounter. 1 34. Find the t coordinate, which is also the h () (metres) (a) A sketch of the function is shown in the diagram. = 0.297(80)(1.225)V?3 = 29.1V? The value of the car depreciates by 12% each year. (b) Calculate the real value of the investment after 6 years. u, = @) and r :g 0+ 8 8 =25+0 =125 Since V(10, 16) and W(15, 15) are both /65 units from 30 best location for a new bank branch is at L(48, 24). Unstable. 910 Many people do not believe that 1 = 0.99999999 (The three dots indicate that the 9s continue indefinitely). how the value of an independent variable determines the value of a dependent variable. To do this we simply divide the side length by the length of the hypotenuse, v2. Similarly, a mathematical proof has an end. The 30th term of an arithmetic sequence is 147 and the common difference is 4. Fior began to boast that he knew how to solve cubics. Log in to Download Wish List ; Also included in. 26. Y -min ine lity If f,0x and f,;, represent the maximum and minimum values of a nonnegative continuous differentiable function f(x) over an interval [g, b], then the area under the curve lies between the area of the rectangle with base [a, b] and f,;, as height and the rectangle with f,,,, as height. A goat is tied to the outside corner of a rectangular building, using a 10 m leash. A bicycle with tires 70 cm in diameter is travelling such that its tires complete one and a half revolutions every second. 5. Using scientific notation this would be written as 1.496 X 10km. WebIB Source began as a belief that finding ideal resources for your IB classroom should be simple. Since we have two variables in this model, we will need to find an algebraic relationship to express the model in a single variable. Example For each function, find its derivative (i) by the quotient rule, and (ii) by another method. (a) Round each value to the nearest unit. What should the selling price of each unit be to have a profit of 20 per unit? No part of this publication may be reproduced in any form or by any means (including photocopying or storing it in any medium by electronic means and whether or no transiently or incidentally to some other se of this publication) without the written permission of the copyright owner, exceptin Inloving memory of my parents, Iyvish o express my despest thanks and ove to my wif, Val, for her unflappable good nature and support and for smiling and laughing with me cach day. Most people will be familiar with Einsteins field equations and Maxwells equations. Note that the horizontal axis (strength) is now in factors of 10, while the vertical axis (length) is in factors of 2. in both degrees and radians, accurate to 35 Find the length of an arc that subtends a central angle of 2 radians in a circle of radius 16 cm. This requires a more abstract and conceptual understanding and it is recommended to students who have a natural mathematical ability and enjoy maths (especially algebra, solving linear and quadratic equations, trigonometry, calculus such as differentiation and integration etc). Here are some examples of estimation and approximation. There is also a courswork based internal assessment (IA). Also, since the squares perimeter is 4x, and the total length of wire is 4m,wehavedx + 7 (a) Estimate, to 1 significant figure, the coordinates where the local maximum value of the function occurs. This also leads to two fundamental trigonometric identities. Those who can do it often find themselves labelled as nerds or as people who are, in some sense, socially deficient. For example, consider the fishing line data from earlier in this section. The points A, B, and D are on the same line. 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. S=T - kt S=T . oo x5 % () 822 o o - (b) 0 % % ,z:;}% o (d) |f e XX 2. emperania] 0) 001 peight m3) (kN 0 9.8050 0.1 9.8052 1 9.8057 4 9.8064 (c) Use your GDC to find an appropriate model for this data, with w as specific weight in terms of temperature T. 10 9.8040 15 9.7980 (d) Use your model to predict the specific weight of water at 12C to 4 significant figures. The potential difference across a given AC circuit is 100V and the impedance is 4 3j Q. The Maths HL class surveyed random samples of the groups, and their findings are given in the table below. The area of a sector of a circle with a central angle of 60 is 24 cm. 5.22-2/3~185 circle with centre (0, 0) and radius 3 y-axis linex=4 circle with centre (3, 0) and radius 2 aline segment between (1, 0) and (3, 0) L @) z=3ZcsT ) z= 3\7m341 z= 3&:3% @ z= 3\7&5541 () z= wmg ) @ z= wm% z= lOcisl; e wc;s% 2. (a) Find the real rate of return per year. Average costs per game console is the total cost divided by the number of units, hence Cx) = 0.11x% + 120x + 1580 x () Usea GDC to graph C'(x). The Pythagorean Identty For any angle 6: cos> 6 + sin 6 = The Tangent Identity . If C > 2, 2 sinx + C will always be positive. There are parallels we can draw between a game like chess and mathematical proof. (ii) Recalculate the value of Pearsons r and interpret it in context. x>~ 443 and itsp value ~ 0.218. (a) Arithmetic 11 (a) Geometric 12. (a) 16x* 27x (b) flo0 = 64x7 27 20 4(75)+ 300 V75 x=2 9. $945.23 3. ~ 443V The work with complex numbers becomes much more useful when parallel circuits are analysed. correspondence (mapping) between two sets X and Y in which corresponds to (maps to) exactly one element of set Y. (a) Express the impedance of the circuit as a complex number in Cartesian form. (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. 64X7 27 20 4 ( 75 ) + 300 V75 x=2 9 or as who! Using a 10 m leash a courswork based internal assessment ( IA ) is. Solving a very real scientific problem are on the unit with an length... Boiling point of water at a height of 1.37 km above sea level points,! Looked at as a number theorist presented him with the opportunity of solving a very real problem. Length equal to the number of units produced at that level 96 m the... 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