$37008.79 1L g(}n +7) 925 Answers 12. Compound interest is interest calculated both on the initial investment and also on interest already paid. The pipe makes one full revolution around the pivot point in a day. (b) To generate the two pieces, we partition the data into two parts, before and after the breakpoint we have chosen visually. As the value of x increases, the value of y also increases and there is no limit on how largey can become. Products. (a) 4,7,12,19, 357 9 L35 gas 60 (b) 2,5,8,11, 135 .9 Wasels 5. (c) Find when the population of India will exceed that of China. Raise the complex number to the power indicated and state the result in the form a + bi (@) 1 +04 (b) 1 i* 18. Solving algebraically, we have: 0= 5p2+1000p = p=0,p =200 Therefore the p-intercepts are 0 and 200. Suppose that the lifeguard is at the point L(0, 0) and a swimmer in distress is at the point (300, 50) (distances in metres). Stating this as a logarithm problem, we must solve the equation x 1 = log,10 or x = log,(10) + 1 We can use our GDC to solve the equation. If one grain of rice is placed on the first square, two grains on the second square, four grains on the third square, doubling the number of grains with 1o, (10)+1 each square, on which square would there first be more than a million (10) E1:[10 grains of rice? Now, we must show that this stationary point is in fact alocal minimum. (b) 94 years (b) $1650 15. o y = 0is a horizontal asymptote for the graph of y = fix) o The domain of fis x R and its range isy > 0 a fixed line, but does not meet it at any finite said to be an asymptote to the curve, A function fis an increasing function if * < x, implies fiv) < fixy). This description gives us the recursive definition of the arithmetic sequence. A goat is tied to the outside corner of a rectangular building, using a 10 m leash. Solution (a) Start with an accurate diagram. Use your graphing application to generate a graph of y = sin(x ). Stable system. (d) Explain what the y-intercept of the line C = d + 2 represents. Assume that the point is at the leftmost position at time t = 0 332 Solution (a) Since the radius of the wheel is 30 cm, and the reflector is 17 cm from the centre of the wheel, we know that the minimum height of the reflector is 30 17 = 13 cm. If you generate many more sets of rolls and record their means, you will eventually have a distribution of sample means. What are the strengths and weaknesses of mathematical humanism? /3-8 15.972. Resource Request title. 5. This chapter will look at some general characteristics and properties of functions. 6. when y = 10. 5 is 27 (g) 4cis3 15. Note that value 5E+4 for the y-coordinate of the vertex is the GDC's version of scientific notation (or standard form). 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 changing strength of gravitational force is a less important factor for normal wells. Month Mean temperature (C) Jan | Feb | Mar | Apr | May | Jun | 06 | L1 | 50 | 100 | 148 | 181 Month Mean temperature (C) jul | Aug | Sep | Oct | Nov | Dec 199 | 195 | 152 | 10 | 48 | 10 (a) Generate a scatter diagram for mean temperature versus months since January, such that January is month 0. To do this, we can use a GDC to graph the model and look for a maximum value. 3. Address: Be the first to receive exclusive offers and the latest news on our products and services directly in your inbox. The potential differences across a resistor (V; = 6 V), an inductor (V; = 11.5V), and capacitor (V. = 3.5V) are individually measured in a series circuit. Example 9.5 The demand for a certain style of jeans is modelled by the function D = 1000 5p where D is the number of items sold and p is price of the jeans in euros. Substituting m = *% and (1, 4) gives: S2(x y4 1 The equation can now be rearranged into general form 3x2y+5=0 Piecewise functio . All DP mathematics courses serve to accommodate the range of needs, interests and abilities of students, and to fulfill the requirements of various university and career aspirations. (b) A preference survey is taken where respondents are offered six choices, each of which is initially considered to be as attractive an option as the rest. A tent flysheet is to be tethered with string to a nearby tree on level ground, 1 m away. EINSTEIN'S L FIELD EQUATION i) LR Figure 5 Einsteins field equation 1. Inm Hence, fle) > fim) = =Ine _>0 . 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. () () 1 (i) 7 (iv) 47 v -1 ) @) 1-2 (i) 4x 9 3. IB Applications and Interpretations SL and HL Resources. (f) Find and interpret the value of R2 (g) Use your model to predict the blood concentration of this drug 4 hours after it is given. 829 Bivariate analysis (d) The air temperature in a cave does not vary and is usually equal to the mean temperature of the region. 9t 3 scientific notation Scientific notation is used to represent very large and very small measurements without having to count decimal places. Before | 105 | 155 | 200 | 110 | After 95 | 115 | 185 | 145 | 110 | 100 | 120 | 170 | 110 120 | 180 | 210 | 135 | 110 | 135 | 210 | 170 | 100 | 110 | 140 | 160 | 130 L} Solution Hy:d =20 H,:d GOF test, if we were randomly entering values while being mindful of the total number of respondents, once the first three numbers such as 30, 17, and 13 are placed in the bottom row with 75 as the total, the last value in the row is pre-determined. In practice, the rate of inflation can affect the future value of an investment. (b) What is the minimum time for the lifeguard to reach the swimmer? (@) :kP$whcnt:0,k7T Q(t) = CV(1 e~1k%) 0'07823 Q) = CVet-tyke .01081 (b) P(t) = 7.6 000811 = 60010816 = 18,64 billion 1n7.895 (c) 60 = 7.6e001081 = { = 0.01081 ~ 191 years, in the year 2208. dN _ NN ~1n0.65 ~ 3543 years. Give your answer to the nearest dollar. 30 13. Potential difference across a resistor is noted as Vy, across a capacitor, V., and across an inductor, V;. Five minutes later, the angle of depression of the boat is 12. As such, the value of the annuity is the sum of 30 terms of the geometric series with first term $1200(1.02) and common ratio 1.02. Thi s the average temperature. The hero of The Glass Bead Game, a novel by the German writer Hermann Hesse, must learn music, mathematics, and cultural history to play the game. Find the value of b. @ () o() v3-4r=4 sl e @ a- (g)rn = area of original triangle = (3] of+2-43) ol o (@ n|= the y = xand y = xare multiples of the original wie S| Sl b+2) c+l (b) As:( a+ bd Za+bc) c+d 3 (b) Bigenvectors: K} and ( K. 1mages of vectors along vectors. The statement John is married is definitely either true or false. 1. Arc length and area of a sector A sector can be thought of as a pizza slice. This textbook has been designed so that the chapters proceed in a manner that supports effective learning of the course content. The tables give data for a Chevy 350 engine. Time (years) Amount in the account ($) 0 2500 1 2500 + 2500 X 0.03 = 2500(1 + 0.03) 2 3 2500(1 + 0.03) + (2500(1 + 0.03)) X 0.03 = 2500(1 + 0.03)(1 + 0.03) = 2500(1 + 0.03) 2500(1 + 0.037 + (2500(1 + 0.03)2) X 0.03= 2500(1 + 0.032(1 + 0.03) = 2500(1 + 0.03) 4 2500(1 + 0.03)* + (2500(1 + 0.03)%) X 0.03 = 2500(1 + 0.03)(1 + 0.03) = 2500(1 + 0.03)* Table 3.3 Compound interest This appears to be a geometric sequence with 5 terms. In fact, mathematicians use letters for many sorts of mathematical objects, not just numbers. The domain of f! So, we must be able to find its integral. WebApplication and Interpretation Formula Booklet . 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. 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. This is called the common ratio of the geometric sequence. Figure9.22 Phase shift ofa periodic function What does d do? For example, the number mapped to the point (1, 0) because 7 is half of the circumference circle. much? This is to emphasise the fact that the series contain an infinite number of terms. For example, consider the fishing line data from earlier in this section. Another way of defining a sequence is the recursive (or inductive) definition. = u(wdx 5ltb0) = Jau=u+e ) %{sin(u)} = cos(u)u'(x) = d(sin(u)) = cos(u)u'(xdx f cosudu = sinu + %(cos(u)) = sinw)u'(x) = d(cos(u)) = sin(uu'(x)dx fsinudu = cosu+c e Alm) = T A wwn= -1- = 4] = wwds | fwdu= 00+ ane - -1 tan = L) d ey (o) == = dtanw = w()de euy e(n) = dle W) == e ety () de inlul) == 29 Juce = dlinful) ul) == ioful) () ) ok du=tanu+ U gy = et+c Jerdu=et dx [i du=1nlu| d=tulud++ Table 16.4 Integration formulae Example 16.6 Evaluate: 672 [(sx + 2)8dx (@) [V6x+ 11 dx () [t VX' 8x+13 (d) [xsint(3x?) For certain applications taking the derivative of the 2 derivative - that is, the second derivative denoted by f"(x) or E); - provides us with useful information. 138 The trigonometric functions in the unit circle To generalise the definition of the trigonometric ratios, consider an angle of 45 in standard position in the unit circle, as shown in Figure 5.17. 2,4, 32256 " 33. (c) Interpret the values of v, and s, in context. 81 Sequences and series The sum, ,,, of1 terms ofa geometric series with common ratio r# 1 and first term a, can be expressed explictly as _all=r Sn Su -1 or _ar -1 r1 Ttis better to use the first version of the formula whenr < 1 and the second version when 7> 1. How much money will there be in the account at the end of 4 years? This is a position taken by surprisingly many mathematicians. Trap. Although you may be more familiar with degree measure, as you progress further in mathematics - and especially in calculus ~ radian measure s far more useful. Sign analysis shows that 77"5 is positive to the left ofr = 3.1 and negative to the right, so the volume is maximized at r = 3.11 m. (i) V=3117G.117 %m&m! Table 4.2 Volume and surface area formulae Example 4.16 A triangular pyramid sits on top of a triangular prism. Solution The gradient of AB is 76;225 = 1, so the gradient of the perpendicular to - _ [AB] is 1. Concavity is not defined foraline - it is neither concave up nor concave down. Since r = (ii) For the distance C to be formed into a circle, it must satisfy C = 2mr= 57 = 2mr= r = (i) P2+ H2= P (;) th=5 o - = (d) T 5=1 we have 1 2 5. li}cmby63cm 7. x=5/27 e = 5 77 e get 3927 = %rrlz .7 byzz V5 5 ! During 1993 Atthe very end of 1998 During 2007, CD sales decreased by 158 million. If the impedance is X, = 4 + 2j Q, what is the magnitude of the potential difference? If we try x = 0, then equation (1) gives us: 02:020+2=0,ie.2=0 which is clearly not true. The interest is compounded half-yearly. 5. (a) A bank offers loans of $P at the beginning of a particular month at a monthly interest rate of I . = du = cos(3x? First solve for x in terms of y and then interchange the domain (x) and range (). (b) Express the impedance in Euler form, with 6 given in degrees, correct to 3 significant figures. Substitute y = vx and dy = vdx + xdv into the given equation (%2 y? _eeeeeeeeeee Solution (a) MCis the rate of change of cost, thus, cost itself is an antiderivative of MC. Starting with the complex number z = 1 + 0i, show how Eulers formula e' + 1 = 0 can be found by expressing in Euler form. (c) Hence, use your GDC to find an appropriate model for temperature T in terms of hours since midnight h. (d) Based on your model, write down the average temperature in the growing chamber. For thetrigonometric models y = a sinfbtx ) + dand y = a coslbx ) + d + amplitude = o + The amplitude i equal to halfthe wave height. () ? (a) 2v2 +2V24 (b) 4/2 4/2i +V2i T T 3 G |z = V26 3w 4 37 2 2k okar 2km 2 A Exercise 6.5 L) Z=12+9; 2.Z=4-3j 4. It is immensely powerful because it covers an infinite number of situations. IB Mathematics: Applications and Interpretation SL in 70 Pages - May 11 2020 This revision guide will be a valuable resource and reference for students, assisting them to understand and learn the theory of IB Mathematics: Applications and Interpretation Standard Level. We will consider impedance in AC circuits here. 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 . Mathematics - Applications and Interpretation SL - WORKED SOLUTIONS - Hodder 2019 - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. Hersh defends this view on the Edge website: [Mathematics] is neither physical nor mental, its social. The first has a resistor with R, = 80 Q and an inductor with X; = 60 , while the second has a resistor with R, = 12Q and a capacitor with X, = 5Q. Features year we have n periods, therefore for years, we have nt periods. A sphere is carved out of a cube. @ z,=5 ds%ra.nd 2= 3cis 2 () z, = 4C152Tn-and z= 2cisT z 3. 104 (b) It appears that over time, the X drawn on the ground has been redrawn and moved! x=e 426 y= -3+ e Oy NN P v vy by NN NN N Ny ( Bigenvalues: 1, 4, eigenvectors (~ i)(g) x=Cief+2C,e" General solution: y=Cie '+ 3Cet Trajectories approach equilibrium and then move away. (c) Usea GDC to find a model for this data in the form T = asin(bM ) + d, where T is the mean temperature for month M, January is month 0, and a, b, , d are constants to be determined. Suppose $2000 is invested in an account paying simple interest at a rate of 5% per year. (0, 0), (3, 0), (3, 1) reflection iny = x. 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