The -coordinateof the vertexi therefore given by = 1- + The yintercept o the graph s (0, ) * The x-intercepts of the graph, also called the zeros of the function, can be found using the dac quadratic formula: x = b /B~ 2a Remember that a quadratic graph may have zero, one, or two x-intercepts. To do this, we can use a GDC to graph the model and look for a maximum value. The current in a given AC circuit is I = 2 + j A. The problem is with the limited sample size of 20. (c) Find the value of p. (d) Find the value of g. (e) Find the value of r. (f) There were two high tides on 12 December 2017. 2 @ z,= 6ds%ra.ndzz = 2cis ) z = 16cissT7Tandzz = 4cisT () z1= Sds%andzz = ZCisg ) z, = ISdS%Tand 2= 2cisg 4. 335 Modelling real-life phenomena 3. (d) The acceleration due to gravity of the model rocket is @ = 9.3 ms~2, This is less that the gravitational constant of g = 9.81 ms 2. Such a doll can be made from plastic blanks with the head being the same diameter as the body, then painted. Now, by using trigonometry, we have another possibility. Call this point B(x,, y,, z,). Lety = sin(kx) kxcos(kx), where k is a constant. The building measures 5 m? For the function fix) = x2Inx, x > 0, find f'x) 15. A goat is tied to the outside corner of a rectangular building, using a 10 m leash. 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 . (a) their nearest site, there are two largest empty circles, centred at each point, both with radius V65 ~ 8.06. (a) (ii) v= 100 8, 60ms~! @) 3= (f) 33 G -2 @ @ (b) 156 8. Then we compensated for the absence of 3. However, we can find a term in a recursive sequence B . @ Ll 31248192 " a 2a 8a> 32a7 W5 =g s () 4. a 5k,2a 4k,3a 3k, 4a 2k,5a k, Find an explicit formula that gives the nth term of each sequence. complex variables used in el Resistance (R) is a measure used in direct current (DC) circuits. (a) ) + 1L,xeR 4 x+9 130 8x 18 1728 10( | 5 (@ Doy =_ =57+ () y=x Ao (b) 20 ( -1 (b (b) =2 y intercept (3,0 Exercise 2.5 C] 1. It follows that the angle will be a central angle of a circle whose centre is at the origin as shown in Figure 5.12. 13. Horizontal asymptote For the coffee example (not shown), 0 = 20 so the asymptote is aty = 20. et s apties of radioactive decay. Express each rational expression in a + bi form. (a) 3-i ) -53 100 (b) 273 + 4i 9. It starts with a collection of statements in mathematical language called axioms. Find the tenth term. Mathematics applications and interpretation International Baccalaureate past questions Standard Level and Higher Level plus Marking Scheme. Determine the equation of the perpendicular bisector of the segment [PQ], given the points P(5, 12) and Q(7, 6) 10. The point-gradient form ofa line can be derived algebraically starting with the gradient-intercept form. In the 19th century, the German mathematician Georg Cantor showed by an ingenious argument that the number of numbers between 0 and 1 is a bigger type of infinity than aleph nought. Because symbols are abstract and not limited to representing things in the world, mathematicians can use their imaginations to create a virtual reality following its own rule system unhindered by what the world is really like, a counterfactual world. 607 Further differential calculus Example 14.19 When a lifeguard on a beach sees a swimmer in distress, she must make a decision about the fastest way to get to the swimmer: should she run along the beach until she is as close to the swimmer as possible, then swim, or jump right in the water and swim? Specifically the one from the oxford university press. 4. If we solve the equation for y, we get y=HV1x? (a) 2 Xintercept yintercept O 2x8 LS ataxsg 7 8. (b) 2 The bases are the same. Using scientific notation this would be written as 1.496 X 10km. My specal thanks ga a my wife, Lody, for standing besde me throughout witing his book. If the linear correlation is weak to begin with, then you cant expect your model to make accurate predictions. 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: . Significant figures (s.f.) . (b) The falling-object model is given as s(t) = %atz + vt + 5o where s(b) is the position at time , a is total acceleration due to gravity, v, is the initial velocity, and s is the initial position. Another beautiful example is the Lotka-Volterra model of prey-predator population dynamics in biology. 3. Simple interest is interest calculated only on the initial investment. ) u, =20, andu, =2a \+a+kandu, =a 5k @ u,=n2+3 () u, =311 -1 (@ u, =211 83 n 1 924 a, 1 1 2 2 15 3 5 1.666666667 1.6 8 1.625 13 21 34 85 89 144 233 377 610 987 1.61538461538462 1.61904761904762 1.61764705882353 1.61818181818182 1.61797752808989 1.61805555555556 1.61802575107296 1.61803713527851 1.61803278688525 1.61803444782168 17 18 19 20 21 22 23 24 25 26 27 28 29 30 (b) lim a, 6. 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. For instance, should a relation be expressed as y = a*, then x is the exponent of the base a which yields the quantity y, written simply as x = log, y When the base is 10, we typically do not write it. You will benefit by being able to quickly convert between degree measure and radian measure for these common angles. The point estimate is p = 0.5, but the 90% confidence interval is 0.32 < p < 0.68 0 4 8 12162024 28 32 36 40 Number of marked items in the sample Figure 17.7 90% box plots for sample size 40 742 0246 > The results should give reason to ponder the size of the interval. Unfortunately, their camera is aimed wrong and the rocket goes above the viewing frame of the camera. The problem occurs when we make a subconscious switch to nine as an abstract noun in the sort of problems we deal with in Mathematics class. 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. 31 17. On a certain area of a golf course, a lawn sprinkler is set to spray an arc of 150 with a maximum spray Figure 5.2 The parts ofa circle distance of 3 metres (Figure 5.3). It is usually not true that AB = BA. For example (2 3) 5 7 + (x y) a b, = (2+x 5+a 3+y) 7+, 205 6 7/ Matrix algebra We carry out subtraction in a similar way (2 3 1) (X y 8) (Z*X 3= J a b 2 5 7T-b 570 *7) 2 The operations of addition and subtraction of matrices obey all rules of algebraic addition and subtraction. (e) 2 The bases are the same. Consider the function y = tanx 8 sinx. The difference in voting preferences between three areas was found through random sampling by students in the Geography class and is summarised below. For the relationship at the start of this section, we can write: Population P is a function of time #: P(t) = P,(1.02)" As with the population P and the time , many mathematical relationships concern how the value of one variable determines the value of a second variable. 126 | Solution (a) First we must calculate the arc measure. Figure 9.18 Diagram for question 9 m Time | Temperature (minutes) [) 0 100 1 85 5 ? This will be more powerful for modelling real-life phenomena. (d) Find the total time needed for the water to reach a temperature of 35C. Distance from | Depth shore (km) | (m) 02 20 0.4 1.9 0.6 23 038 23 10 22 12 28 14 33 16 4.0 18 46 20 5.1 Table 19.4 804 Data for Example 19.8 Sometimes you have data that very clearly has two linear trends visible. The temperature in degrees Celsius during a 24-hour period is shown on the graph and is given by the function fit) = a cos(bt) + where a, b, and are constants, f is the time in hours, and (bt) is measured in degrees. dy dy du should remember that they are not fractions. (f) Hence give a reason why the period of the GDC model is not ideal. Find the coordinates of the points of intersection of each pair of lines. 9. 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. (d) Calculate the maximum volume of the open box. Then, by using ratios to calculate the measurement of arcs and sectors in a circle, we will develop the definition of radian measure. 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. The interest is calculated at the end of each month and added to the amount outstanding. Could we allow ourselves to be guided to truth in mathematics because of the beauty of the equations? (f) The rate of change in velocity (acceleration) due to gravity should be 981 cm sec 2 on Earth. Nevertheless, it often happens that pure mathematics created for no other purpose than solving internal mathematical problems turns out to have some extraordinary and very practical applications. (b) How long does it take for the number of bacteria in colony A to reach 4007 The number of bacteria in colony B after hours is modelled by the function B(f) = 24ek After four hours, there are 60 bacteria in colony B. (e) Since the maximum acceleration force increases by a factor of m? Now, we must show that this stationary point is in fact alocal minimum. and variance o2 737 17 Inferential statistics 5. The model is only as good as the strength of the correlation. The rate at which predators eat prey is B, which is assumed to be a constant proportion of the product of predators and prey. dt Population growth rate in cases where the birth and death rates are not variable i proportional to the size of the population. If we write down the values of b and the length of the period, we get Table 9.12. 5. Find the time taken to reach this charge. 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? In mathematics and science, we refer to this as the period of the function. 14. how the value of an independent variable determines the value of a dependent variable. 33 335 18. What is the difference between an Antigen and a Pathogen? For an exponential model of the form FE=k1+pyr+e or fl)=kex+c the graph of f has a horizontal asymptote at y = ic A common use of exponential functions is half-life calculations. First of all, we need to state a claim and the hypotheses before we continue by finding the expected values. Write each complex number in Euler form. + 1000(100) = 50 000 euros (d) Clearly, our model doesnt make much sense if the price p is less than zero. It would appear that the sum to infinity, > 2", approaches infinity. That a particular piece of paper is money is a social fact that does make things happen. Month Fish January May 850 100 ? A moneychanger exchanges 250 THB (Thai Baht) into 400 PHP (Philippine Pesos). Last updated: Example 9.14 A reflector is attached to a bicycle wheel at a point 17 cm from the centre of the wheel. \ Nw s [N Vo txxw Pt = 8 b g g 2 s 3 =3 38 s & z e g S = R g 3 ISs= o Sl Approx. The recursive definitionis a, = 13 ,1 = 5n> @y, =a,The explicit definition is (b) a5 = 18 5(25) = 107 62 a,=135n1=185n Find the 20th term of the arithmetic sequence with first term 5 and fifth term 11. 10007.4 3 Pl B @ 20 + 480.37 01t (b) 500.37 (c) 227.87 10. . WebConcise resources for the IB DP Maths AI SL: course. 2002 510)) (120,0) Therefore, average manufacturing cost is minimised when 120 units per day are produced. + 169 = /8757 + 49 + /(25 875 + 169 We can use a GDC to calculate this without losing precision (remember to store the value from the graph page). Here, we are interested in the non-zero asymptote. 4 4 ' ' 4 4 \ \ Vo oo AR o e B ~ N 2. Precipitation 5s o= eoe && (b) The pattern is not sinusoidal because there appear to be unequal local maxima. A building 30 m high sits on top of a hill. The equation ax + by + d = 0 is called the general form of a line. =2 ,.2 650 " 65 15. Hersh thinks that this is not really the same nine. In this case, we can construct a piecewise linear model by partitioning the data set into two or more subsets. 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. 3 23. 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 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. 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