5. 1 34. (a) log, 16 (b) log;2 (c) logz16 (d) log,v2 (e) log,(16) (f) log,2/2 (g) log2/2 (h) log, 52 (i) log4 + log25 log300 (j) log3 0 ) 1n() ) Inve (a) log,a (b) log,Va (c) logava (d) log;Va (e) log,:a (f) log,.va (g) log,:Va (h) log,-a/a (i) log,a >+ log,a* (j) log,:a=+log,.a* (k) log,a log,a> (1) log,a log,/a Simplify each expression. 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. WebThe IB DP Mathematics: applications and interpretation course recognizes the increasing role that mathematics and technology play in a diverse range of fields in a data-rich world. Example 5.8 The diagram shows a circle with centre O and radius r = 6 cm u /10\ Angle AOB subtends the minor arc AB such that the length of the arc is 10 cm. Likewise, you may seey,y,y"" o j, j, j for first, second, and third derivatives, respectively. (@) x=-1+3i (b) x=2*3i 1 L) 13. (6 (g) . Time (years) Amount in the account () 0 2000 1 2000 + 2000 2 2000 + 2000 X 0.05 + 2000 X 0.05 = 2000 + 2000 X 0.05 X 2 3 2000 + 2000 X 0.05 X 2 + 2000 4 2000 + 2000 X 0.05 X 3 + 2000 X 0.05 = 2000 + 2000 X 0.05 X 4 X 0.05 > 0.05 = 2000 + 2000 X 0.05 X 3 Table 3.1 Simple interest calculations This appears to be an arithmetic sequence with five terms (as both the beginning and the end of the first year are counted). 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This strangeness is characteristic of infinite sets (indeed it can be used to define what we mean by infinite). To do this, we can use a GDC to graph the model and look for a maximum value. 1. Parabola passes through (17, 37): y = 0.128x> = area = [ 012802 = 209.62 . To do this we simply divide the side length by the length of the hypotenuse, v2. sinx = % = (i) P @7 Car:acc = 4,v = 4t,s = 2% Truck: s = 1500 +13.89 2 =1500 _[4 15'(7{0 +13.89f t ~31s,s~ _f4t t=2 {8 1270 10. s Major and minor arcs Ifa central angle s less than 180, then the subtended arc is referred to as a minor arc. If this is right, then it is natural to ask about the purpose of this particular map. Other than the final two chapters (Theory of knowledge and Internal assessment), each chapter has a set of exercises at the end of every section. Why is it called least- squares regression? Learning objectives By the end of this chapter, you should be familiar with different forms of equations of lines and their gradients and intercepts parallel and perpendicular lines different methods to solve a system of linear equations (maximum of three equations in three unknowns) Key facts Key facts are drawn from the main text and highlighted for quick reference to help you identify clear learning points. The function A = Age can be used to describe the amount A of DDT left in an area t years after an initial application of A, units. @ =) & 2y, Example 4.2 Find the coordinates of the midpoint of the line segment joining each pair of points in Example 4.1 (a) B(8, 1), N(4,4) (b) C(12,6), P(3,2) (c) E(3,5),R(3V3,3) Solution o (54 ) o (27255~ (2323 o Coordinate geometry in a plane lines and intersections Given two lines in a plane, three possibilities exist: o The lines have the same gradient and the same y-intercept; hence, they are the same line (coincident lines). 7 1600 = 800 g 400 = Non-linear regression is useful for many types of models. (b) Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car. Using the value of a found in part (b), (i) find an expression for g~! 5 2. But, since x = cos (=1,0) Figure 5.21 The Pythagorean Identity and y = sin (), we have xX22=1 (cos 0)2 + (sin )2 = 1 cos?f +sin?6 = 1 This is known as the Pythagorean Identity. Determine the equation of the line parallel to y = 7%){ + 4 that passes through the point (6, 1) 7. If mathematics really is so other-worldy, how come it has so much to say about this one? Digital Copy duration: 24 months About the Book This book has been written for the IB Diploma Programme course Mathematics: Applications and Interpretation HL, for first assessment in May 2021. (a) () 26 (i) u,=4n6 (iii)u, = u,, + dandu, = 2 () () 923 (ii) u, = 1019 0.12n (i) u, =, , 0.12and u, = 10.07 ( () 79 (i) u, =103 3n (iii) 4 =, 3and u, = 100 @ - @u,=2-2n (i) 4y = tyey %a.nd w=2 3.a,=4n 14 4.q, L;n =51 5. The difference in voting preferences between three areas was found through random sampling by students in the Geography class and is summarised below. 11. Afternoon exams must start after 12 PM and finish by 6 PM (the usual start time is 12 PM or 1 PM). A taxi driver charges $4 to pick up a customer, plus $0.80 per kilometre. The table shows the extension of the spring (cm) for each mass (g). Therefore, when x = 8.75m, the length of the rope is T = 32.0m ormeicen Figure 14.16 GDC screen for the solution to Example 14.18 By usinga GDC to calculate and graph the first derivative, we were able to perform the first derivative test graphically. (d) There is a large gap between Mars and Jupiter. Featuring a wealth of digital content, this concept-based Print and Enhanced Online Course Book Pack has been developed in cooperation with the IB to provide the most comprehensive support for the new DP Mathematics: applications and interpretation HL syllabus, for first teaching in September 2019. For thetrigonometric models y = a sinfbtx ) + dand y = a coslbx ) + d + amplitude = o + The amplitude i equal to halfthe wave height. Platonists would certainly argue that mathematics is out there in the universe, with or without human beings. For those who struggle with math, equations can seem like an impossible task. W= Xr wy=u, Xr=u XrXr=uXr W=y X r=w XX r=u X1 and so on. There is a subtlety here because we use different letters j and i for the whole numbers in the expressions above because we want to allow m and 7 the possibility of being different odd numbers. In one experiement, a patient is given a drug and the patients blood is tested at regular intervals to determine the concentration in millimoles per litre (mmol L!). Solve My Task Do math equations The distribution of the outcomes may or may not look uniform in this first experiment; however, in theory, if you keep rolling the dice another 6 times, then another 6 times, and so on, the cumulative distribution will begin to look uniform. This is shown in the diagram along with a possible route APB. ) (b) Find the missing side(s) and angle(s). Ifpossible, resolve the resulting equation with respect to y, to obtain your equation in explicit form y = f(x) Example 20.6 Find the general solution of the differential equation & B y = R e 0 =0 S Solution The equation is separable because you can rearrange the equation as: d d; a} - %(x:z 1) which is in the form EV = pEqy) Now separate the variables and integrate: ! 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. Area of a sector As in section 5.1, the area of a sector is a fraction of the area of the entire circle (a pizza slice). fighe the model should be quadratic. 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. (a) 4,7,12,19, 357 9 L35 gas 60 (b) 2,5,8,11, 135 .9 Wasels 5. In the meantime, Fior had made little progress with Tartaglias problems and it was obvious who was the winner. Table 4.2 Volume and surface area formulae Example 4.16 A triangular pyramid sits on top of a triangular prism. quickly with a GDC. This is best done application such as GeoGebra that allows you to observe each parameter has on the shape of the graph. (Ignore any currents in the water.) Like other areas of knowledge, it possesses a specialised vocabulary naming important concepts to build this map. Solution The gradient of AB is 76;225 = 1, so the gradient of the perpendicular to - _ [AB] is 1. Example 18.10 Thirty girls and forty-five boys were surveyed to determine if there is any difference in their preference amongst four TV crime dramas. The area of a sector of a circle with a central angle of 60 is 24 cm. The diameter of the wheel is 8 m. The centre of the wheel, 4, is 2m above the water level. The French mathematician Pierre de Fermat wrote the conjecture in 1627 as a short observation in his copy of The Arithmetics of Diophantus. () ? 2. (a) 5.227 X 10* @5 ( 3 (b) 6110 (e) 0.00305 (o) 2 (h) 22 m2 @2 w2 () 124000 (f) 400 ) 2 @ 2 W2 2 @2 (b) 1.31401 X 10* () 6.04% 10% (d 9% 10 (a) 1.00 X 108 () 100X 1095 (a) 5=log,243 (b) 1.52 X 10! This description gives us the recursive definition of the arithmetic sequence. Use your graphing application to generate a graph of y = sin(x ). (ii) Giving a reason to support your answer, explain if you think this is an overestimate or an underestimate. P(Y>40 1) ! Here, 0 < h(t) < 56.2(m) On a GDC, we need to use x in place of . Instead of dividing a full revolution into an arbitrary number of equal divisions (e.g. Students have the option to choose between two different courses: IB Math Analysis and Approaches (AA) and IB Math Applications and Interpretation (AI). At x = 0 sine begins on the principal axis and then increases as x increases. The proof is straightforward. Yates continuity correction, which reduces the positive difference between each pair of observed and expected values by 0.5 before squaring, then dividing by the expected value, has been considered for quite some time, although some statisticians feel that it over-compensates. (a) Generate a scatter diagram for this data. mAn=20j+i+1) But j, i and 1 are all whole numbers so j + i + 1 is also a whole number. (e) Show that the value of r for which T'is a minimum is 0.719 m, correct to 3 significant figures. For each equation: (i) match it with its graph (ii) state, with justification, whether or not the equation represents a function. We can simply write log y. In general, in order to receive the IB Diploma, a student will have to score at least a4 in each subject, or 24 points or more in total. 100000 10000 1000 100 t 10 Ve Mercury 100 1000 Mean distance from Sun (millions km) (a) Estimate the average distance from the Sun and orbital period for Jupiter. Develop a quadratic model for the lower span such that one end of the span is positioned at (0, 0). In section 5.1 we used degrees to measure angles. 3. Although, explicit. (a) What is the volume of the pyramid? (g) 1: Only the 2 is significant. In 99% of these cases the test does its job and records a negative result. This is what we were doing when we started this chapter by counting cows. 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. e == Z BT % A0=0 (b) Q) = 120s(1 e~ %) 25 () 120 dp = = (In2)P & 140000 = P ~ 201977.31 1977.31e!n2 =201977.31 1977.31 X2 0=t=6.67 weeks 1003 Index A absolute value 181 absolute value function 24 ACcircuits 192-7 acceleration 560-1,588-90, 715, 718-24 due to gravity 589, 719-21,900-1 position and velocity from 718-19 accuracy 151,321, 598, 599 diagrams 602-3, 604, 606, 610 Eulers method 863,864 rounding and significant figures 2-5 addition 177,205-6, 276-9 addition rule (probability) 412,418 adjacency matrices 457-9,460-1 cost adjacency matrix 488, 492-3 strongly connected graphs 478-9 and walks 464-6 Affine transformations 255-60, 262-3 algorithms Toute inspection problem 5001 shortest-path 495-9 spanning trees 4856, 489-95 travelling salesman problem 503-8 Voronoi diagrams 155-62 alternative hypothesis 756-61,762, 770 amortisation 925 amplitude 192,330 angle of depression 106 angle of elevation 106 angles 105-9,129-38, 618 between vectors 289,292-3 central angle 125-9,132,134-5 components of 130 converting degrees/radians 132-4 known/unknown parts ofa triangle 110~ 14 measuring 130-1 negative/positive 130 phase angle 192-3,194-5 position angle 184-5 right-angled triangles 105-6 see also trigonometric functions annual depreciation 75 annuities 834, 90-2,94-5 Anscombes quartet 793 antiderivatives 666-78, 681, 684, 688 nonelementary 686 of sum/difference 669-70 approximation 2-5,63-4, 321,374 area undera curve 6789, 686-8 finding volumes 705 solution to differential equations 862-5 arclength 126-9,131,134-5 arccosfarcsin/arctan see inverse trigonometric functions area 693-705 along y-axis 698 between curve and x-axis 678-83, 686-8, 6936 between curves 696-8 economics applications 698-701 optimisation problems 61011 ofasector 124-5,127-9,135 surface areas 115-16,119 transformed image 254-5 of triangles 109-10, 112-13,220 velocity-time graph 717-18 area function 680-1 Argand diagram 181-5, 187-8, 190, 193 argument (complex numbers) 184-7 arithmetic mean (average) see mean arithmetic sequences 61-8 arithmetic series 77, 80-1 associated variables 783, 784-91 1004 linear regression 802-5, 808-14 non-linear regression 815-19, 8216 quantifying association 791-802 asymptotes 25,26 exponential models 326 logistic models 340, 341,342 tangent function 147 augmented matrix 230-5 average rate of change 526,529-32, 534-5, 589 average value of a function 682 axioms 910 axis of symmetry 304-5 B bargraphs 373 base vectors 277 bearings 107-8 beauty 912-15 Bernoulli trials 638-9 bestfit linear models 802-14 bimodal data 389 binomial distributions 63841, 644, 741 bipartite graphs 455-7, 471,475 bivariate analysis 782-832 describing association 783, 784-5, 787-8 estimating line of best it 786-91 linear regression 802-14 logarithmic linearisation 816-19, 822-6 measures of correlation 791-802 non-linear regression 815-19, 8216 scatter diagrams 782-96,797 bounded growth 33844 box plots 393-5, 740-3 caleulus 526-7 see also differential calculus; integration carbon-14 326 carrying capacity 845,846 Cartesian form ofa line 280-1 causal relationship 787-8 cells (Voronoi diagrams) 153-62 central angle 12579, 132, 134-5 central limit theorem 738 central tendency, measures of 385-9,397-9 ceteris paribus 699 chainrule 5536, 558, 559-60, 565, 5667, 568 implicit differentiation 614 and integration 669, 670 change of variables 670-3, 847 characteristic equation 239,243, 245-6 characteristic polynomial 239-40,245 chi squared test of goodness of fit 765-9 of independence 769-74 circles 124-9, 134-8, 168-70 seealso unit circle circuits (electric) 1927, 851-3 integrated circuits 819-20 circuits (graph theory) 463, 465-70, 500 claims, testing 752, 753-8, 762, 7634 classes 374-5 clock arithmetic 222 codes and coding 222-5,903 coefficient of determination 805-14,816, 818-19,821-2 coefficient of inequality 700-1 coin flipping 409-11, 639 collinearity 220-1 column vectors 204 combined distributions 649-53 combined events 418-19 common difference 614 common logarithm & common ratio 68-9 commutative matrix multiplication 208-9 complement rule (probability) 412 complementary graphs 455,458, 459 complete bipartite graphs 455,475 complete graphs 455,475 complex numbers 174-200 and AC circuits 192-7 adding/subtracting 177, 180 Argand diagrams 181-5, 187-8, 190, 193 conjugates 1789, 182 Euler form 189-91,196 polar form 184-7 powers and roots 184, 187-92 products 177,178, 1856, 191 quotients 177-8,186-7, 191 rootsfzeros 176, 178-9, 180 complex plane 181-5,193 composite functions 30-5, 36 derivatives 553-6, 560 composite transformations 47-50,259-62 compound interest 704, 688 annuities 83-4,90-2, 94-5 concavity 304, 587 conditional probability 42840 cones 115,116 confidence intervals 741-7, 752-4 congruence 222 conjugates 178-9,182 connected graphs 452,461, 46671, 4767 constant motion 283-8 constant of variation 344-5, 346 constructivism 904-5 contingency table 769-70 continuous data 364, 634, 636-8 continuous distributions 636,752-3 normal distribution 644-7, 649, 744-5, 760 continuous money flow 688-90 continuous random variables 634, 636-8, 701-2 normally distributed 644-7, 649, 744-5, 760 convenience sampling 369 convergent series 85-6 coordinate geometry 100-5,220-1 in three dimensions 116-18 coordinate systems 17 correlation 735, 784, 787-8, 804 coefficients 791802, 808 cosecant 105 cosine function 105-6,139-47, 564-5 cosine graph 145-7, 332 cosine model 332-4 cosine rule 113-14 cost adjacency matrix 488, 492-3 cost function 6734, 676-8 costmodels 673-5 direct variation 345 inverse variation 346 linear models 299-304 maximum/minimum values 586 optimisation problems 605 using functions 18,23 cotangent 105 coterminal angles 130 coupled differential equations 869-83 numerical solution 879 phase portrait solution 872-8 solving second order differential equations 879-81 cross product 290-1 Index cryptography 222-5,903 cube root function 35-6 cubic equations 898-9 cubic models 309-12, 351,352 cubing function 35-6 cuboid 115 cumulative frequency distribution 3756, 379-84,401 cumulative frequency graph 378-84, 394, 401 current 192-3,194,851-3 curve fitting 234-5 cycles 463,465-6 Hamiltonian 470-1, 502, 503, 506-7 cylinders 115, 705-6 cylindrical shells 71012 D data 363-72,784-5 collection 366-70 discrete/continuous 634-8 graphical representation 372-84 grouped 373-6,397-8 linearising 816-19 pairedfunpaired 761, 762-3 pooledjunpooled 762 ranking 796,797-8 summary measures 385402 transformed 398-9, 647-9 ungrouped 373,390 data encryption 222-5, 903 dataset 3634 de Moivres theorem 187,189 decay models 745,323 depreciation 75,323-4,326 population 842 radioactive decay 25-6, 326, 844 rate of cooling 324-5, 326, 8434 decreasing functions 25, 37-8, 540-2, 579 definite integral 644, 680-93 see also integration degree (angle measure) 130-1,132-4,195 degree (homogeneous functions) 847 degree (of vertex) 452,453, 460-1,464 degree sequence 460-1 degrees of freedom 752,753,755 chi squared analysis 765, 769-70, 771-3 pooled samples 762,763 demand 590-1, 698-700 dependent variables 16-17, 299301 bivariate statistics 782-3, 787-90, 804, 805-10 depreciation 68,75,323-4,326,795-6 derivatives 539-45 composite functions 5536, 560 derivative tests 57886, 586-91 exponential function 566-7 functions of form flx) = ax' 545-53 asgradients 596 graphical analysis 546-8, 584, 607, 609, 611 higher derivatives 560-1, 588 interpreting 540-2 natural logarithm 5678 numerical derivatives 584, 598-9, 609, 610 product of two functions 556-8 quotient of two functions 558-60 second derivatives 560-1, 584-5, 586-91 sign of 540-2, 579-81, 586 of asum/difference 54850 trigonometric functions 563-5 descriptive statistics 362-406 data 363-72 exploratory data analysis 372-84 graphical representation 372-84 groupedungrouped data 373-6,397-8 measures of centre 385-9, 397402 measures of spread 390-402 populations and samples 364-5, 373 sampling 366-70 statistic defined 365 variables 363-4,373,376 determinant 217-22, 229, 23940, 241 transformation matrix 254-5 diagonal matrix 204-5 diagonalisation 243-6,250 differential calculus 526-76, 624-32 finding equations of normals 595-601 finding equations of tangents 595601 finding maxima/minima 578-95, 60113 implicit differentiation 614-21, 837-8 instantaneous rate of change 52938, 539, 589 limits 527-9 moving objects 530-4, 561, 588-90 notation 539, 554-5, 561 numerical differentiation 546 optimisation problems 601-13 related rates problems 614, 616-24 rules 54560, 564-7, 568 see also derivatives; differential equations differential equations 834-86 coupled 869-83 electric circuit problems 851-3 Eulers method 862-5, 879, 8801 falling objects 855-7, 859, 862 initial-value problem 837, 840-1,842-3, 862-3 logistic 845-7 mathematical models 8417, 851-9, 869~ 72,901-2 and matrices 872-8,881-3 mixture problems 8534, 859 numerical solutions 860-8, 879-81 order of 834-5 phase portrait method 872-8 reducible to separable 847-8 second order 835,879-81 separable 838-44, 848 solutions 835-43,860-8, 87381 digraphs 454-5, 466, 478-81 Dijkstras algorithm 495-9 dilation 253-5, 261,264 Diracs equation of the electron 914 Diracs theorem 471 direct variation model 344-5, 347-9, 352 directed graphs see digraphs direction (scatter diagrams) 784,785,796 discrete distributions 63844, 649-50 discrete random variables 634-5, 637, 647-9 discrete variables 364 disjoint events see mutually exclusive events displacement 560, 588, 715-18, 722-3 distance 284-7, 590, 608-9 from origin 17, 181-5, 284-5, 590 in three dimensions 116-18, 2867 distance travelled 715-18, 7214 distance-time graphs 5314 distribution 364, 376-7 of sample means 738-9 see also probability distributions divergent series 86 domain (ofa function) 17,19 composite functions 31,32-3 endpoints 581-2, 604, 609-11 exponential functions 24-5 finding maxima/minima 581-3, 61011 interchanging with range 38-42 inverse functions 36-42 logarithmic functions 26 trigonometric functions 141,147 domain (of amodel) 304, 3057, 310, 311-12, 334-5,803 dot product 288-90, 291 drag 8557 drone models 284-7 dual key cryptography 903 E earthquakes 9,11 economics 673-5, 676-9, 698-701 see also cost models edges (graph theory) 451-7 adjacency matrices 457-9, 464-6 Eulers formula 475-7 incidence matrices 45960 planar graphs 473-5 spanning trees 4856 walkspaths/trails 463-73 weighted graphs 48894 edges (Voronoi diagrams) 1534 eigenvalues 238-46, 250-2, 262, 874-8, 881 eigenvectors 238-46,250-2, 874-7, 881 transformation matrix 261-2 Einstein's field equation 913-14 electric circuits see circuits (electric) electrical theory 192-7 elementary function 686 elementary row operations 230-1 elementary subdivision 477 empirical rule 645 endpoints 581-2, 604, 609-11 equal matrices 205 equally likely outcomes 411 equation ofa line 22-3,101-5,280-3 normals 102-3,595-601 tangents 550, 595-601,863 two-point form 221-2 equation solver (GDC function) 9 equilibrium solution 856, 870, 874-5 estimation 2-3, 321 Euler form (complex numbers) 189-91, 196 Eulerian circuits 46770, 500 Eulerian graphs 467-9,471-3 Eulerian trails 467-9 Eulerian walks 467-8 Eulers formula 4757 Eulers method 862-5 coupled systems 879, 8801 evaluation theorem 684-5 even numbers 908-10,911-12 events 409-27 expected values 412-14 binomial distributions 639-41 combined distributions 649-52 combined normal distribution 649 compared to observed data 76574 continuous distributions 636 discrete data 634-5 normal distribution 644, 649 Poisson distribution 641,642, 650 sample means distribution 738-9 transformed data 647-9 uniform distribution 647-9 experiments (trials) 247, 408-12, 644 Bernoulli trials 638-9 multinomial 765-9 explanatory variables 782-3,787-90, 804, 805-10 explicit series 59,612, 69 explicit solution 837,839, 840 exploratory data analysis 372-84 exponential decay 246, 74-5, 323-6, 566 1005 Index exponential functions 24-6, 320, 323, 324 antiderivative 667 derivative 5667 integration 669, 670-2, 686, 687 exponential growth 74-5, 319-20, 566, 8414 limited growth 33844, 844, 845 modelling see exponential models exponential models 319-30,351 continuous income flow 689-90 decay 323-6,352 developing 321-3 graphical interpretation 326 interpreting 323-6 non-linear regression 81516, 817-18 population growth 321-3, 352, 841-3, 845 probability distributions 636 supply and demand 699-700 exponential regression 815-16 exponents 6-7, 8-10, 36,191, 548 extrapolation 354-5,804 F F-test 762,763 falling objects 305-6, 352, 719-20, 900-1 differential equations 855-7, 859, 862 Fermat's conjecture 899 Fibonacci sequence 61,907,913 finite graphs 451 first derivative test 578-86 first fundamental theorem of integral calculus 6834 first order differential equations 834-5 five-number summary 392, 393,395 forces 289-90, 346-7, 595, 8557 form (scatter diagrams) 783,785, 795-6 fractals 262-5, 267 fractions 177-8,907-8 frequency distributions 372-84, 389 frequency graphs 378 function notation 31, 36, 554-5, 559 functions 15-56 antiderivatives 666-8, 675 average value 682 composition 30-5, 36 derivatives see derivatives domain see domain (ofa function) graphs 24-7, 30, 3740, 145-8, 542 homogeneous 847 increasing/decreasing 25, 37-8, 540-2, 579 inverse 3545 limits 527-9 as mathematical models 16-21,23,25-6, 27,636 monotonic 25, 38,7967 non-monotonic 40, 796-7 piecewise functions 234, 313-14 range see range (ofa function) representing 17,19-20 solutions of differential equations 835-8 Voronoi diagrams 162 see also specific ypes of function fundamental theorems of calculus 681, 683-5 future values 17, 689-90 annuities 90 compound interest 70-1 simple interest 64-5 G Gauss-Jordan elimination 2304 Gaussian distribution see normal distribution general form of aline 22-3 general solution 837, 839, 840, 841, 842 geometric sequences 6877 geometric series 80-7,90 1006 geometry 119-22 circles 124-9,168-9 coordinate geometry ina plane 100-5, 220-1 distancesin 3D 116-18 midpoints 101,103, 118 volumes and surface areas 115-16,119 Voronoi diagrams 15368 Giniindex 700-1 golden section ratio 913 goodness of fit (GOF) 765-9 gradient (slope) 22-3,101-2 function see derivatives linear models 299, 300, 301,303 normals 595-6, 598-9 perpendicular lines 102-3, 595 positive[negative 540-2, 578-9 and rate of change 530-5 tangent lines 532-5, 550, 595-9 gradient-intercept form 22-3 graph theory 450-524 definitions 451-63 degree sequence 460-1 elements ofa graph 4512 Eulerian graphs 467-9,471-3 Eulers formula 475-7 Hamiltonian graphs 470-3 handshaking theorem 453 homeomorphic graphs 477-8 and matrices 457-61, 464-6,478-81, 488, 492-3 planar graphs 473-82 route inspection 500-1 shortest path 495-510 travelling salesman problem 501-8 trees 4828, 48995, 506-7 types of graph 451-7 walks/paths|trails 46373, 483, 488,495~ 510 weighted graphs 479, 488-95,496-510 graphical analysis derivatives 546-8, 584,607, 609, 611 models 304-6,307-8, 312, 313-14 trigonometric functions 145-52 graphical display calculator (GDC) 585 accuracy 151,321, 598, 599 chi squared statistic 7667, 771 confidence intervals 743-4,745,7534,754 correlation coefficients 796 definite integrals 685-6, 687-8 degreefradian mode 133,195 descriptive statistics 386, 387, 392, 397-8 Euler form complex numbers 190,196 Eulers method 864 finding areas 61011, 696, 698 finding intersections 26, 151,322, 598 frequency graphs/histograms 377,378 generating random samples 740 graphing derivatives 541-2, 564, 566, 567-8, 584, 607 graphing lines 103 inferential statistics 739,740, 7434, 745, 746 interest calculations 65, 71-2 inverse normal (invNorm) 645-6, 744,745 linear regression 808,817 linearising data 816 logistic models 339, 342 matrices 219,232, 654-5 maximafminima 18, 581-3, 584, 611 model analysis 306, 307-9, 312,322 non-linear regression 815-19 numerical derivatives 584, 598-9, 609, 610-11 numerical solver 322,325 polynomial root finder 32, 583 probability density functions 636, 647-8 probability distributions 635, 638, 641, 644 sequences 60 solving equations 9,26, 32, 151-2,322 solving inequalities 65,71,74, 151 t-tests 7534, 755,757,759, 760, 762-3 Time-Value-Money (TVM) solver 72,73, 75,84,90-5 trigonometric models 149-51,334-5 vector products 290 viewing window settings 151, 307-8, 309, 312 graphs cosine graph 145-7, 332 cumulative frequency 37884, 394, 401 exponential functions 24-6, 326 inverse functions 3740 logarithmic axes 819-22 logarithmic functions 26-7, 30 logistic function 338, 339-40, 845 relative frequency 410-11 sinegraph 145-7,330-2, 563 tangent function 147-8 transformations 45-51 velocity-time graph 717-18 see also graph theory gravity 589, 719-21,900-1 great circle routes 1267 greedy algorithms 489-95 grouped data 373-6,397-8 H half-life 326 Hamiltonian cycles 470-1, 502, 503, 506-7 Hamiltonian graphs 470-3 Hamiltonian paths 4701 handshaking theorem 453 Hill's method 222-5 histograms 376-80, 389 homeomorphic graphs 477-8 homogeneous coordinates 259 homogeneous functions 847 homaogeneous systems 218 hypothesis testing 756-61, 762, 7634 multinomial outcomes 766-7, 769, 7704 power of the test 760 typeland Il errors 758-60 1 identity function 36 identity matrix 207-8, 209, 216,223,239 imaginary numbers 174-6, 181 impedance 194-7 implicit differentiation 614-21, 837-8 implicit solution 837-8, 839, 840, 841 in-degrees 454-5 incidence matrices 45960 income 688-90, 700-1 increasing functions 25, 37-8, 540-2, 579 incremental insertion algorithm 155-62 independence, test of 76974 independent events 419-23,435, 638, 641 independent variables 16-17,299-301, 304, 355, 604-11 bivariate statistics 782-3, 787-90, 804, 805-10 individuals 363 inequalities 65, 71,74, 151 max-min inequality 683 testing claims involving 754-5, 756-7, 762 inferential statistics 369, 734-50 confidence intervals 741-7, 7524 Index distribution of sample means 738-9 margin of error 743, 744-6 probability intervals 7403 reliability and validity 734-6 sample size 741, 742-5 unbiased estimators 736-8 see also statistical tests and analyses infinite series 78, 84-6 infinite sets 911-12 inflation 734 inflection point 340, 587-8 influence graph 480-1 initial side 130 initial-value problem 837, 840-1,842-3 Eulers method 862-3 instantaneous rate of change 529-38, 589 integral calculus applications continuous money flow 688-90 costs 673-5,676-8 finding volumes 70514 Lorentz curves 700-1 modelling linear motion 714-24 probability 636, 638, 701-2 supply and demand 698700 integrated circuits 819-20 integration 666-732 antiderivatives 666-78 area under function curve 678-83, 686-8, 693-8,717-18 area under velocity curve 717-18 average value ofa function 682 definite integral 680-93 fundamental theorems of calculus 681, 683-5 integration formulae 668-70, 672 maxmin inequality 683 notation 667-8 numerical integration 686-8 substitution rule 670-3, 685-6 see also differential equations; integral calculus applications interest 64-5, 704, 688 annuities 83-4,90-2,94-5 onloans 92-5 internal assessment 887-93 interquartile range (IQR) 391-2, 394, 395, 398-9 intersection lines in a plane 101-2 points 26, 151-2, 322, 595-6, 598-9 intersection (of events) 418-19 inverse functions 35-45 inverse matrix 215-18,224-5,229 inverse normal 645-6, 744,745 inverse trigonometric functions 105, 194, 274,292 inverse variation model 344, 346-9, 352 investments 17, 64-5, 704, 688-90 annuities 83-4,90-2,95 irrational denominators 177-8 isometries 256-8 K kinematics 283-8, 588-90, 714-24 Koch curve 264-5,267 Konigsberg bridge problem 451,468 Kruskal's algorithm 485-6 weighted graph 489-91,492,493-5 Kuratowskis theorem 477 L largest empty circle (LEC) 1634 least-squares regression line 802-5, 806-14, 817 Leibniz notation 554 limited growth 338-44, 844, 845 limiting (maximum) value 341-3 limits functions 527-9 of integration 680 series 78,856 line of best fit by eye 786-91 line segments 100, 116-18,153 midpoints 101,103, 118 linear combinations 276, 649-50 linear correlation 791-5, 798-802 linear equations 213-19,229-38 linear functions 22-3,28-9, 680-1 arithmetic sequence as 62 gradient 530-1,533, 540 linear models 299-304, 351,352, 802-14 inappropriate use 34950 linear motion 283-8, 590, 714-24 linear regression 802-14 lines 101-5,153,220-2 equation see equation ofa line loans see amortisation local extrema 57986, 587-8 log-lin graph 820 log-log graph 820-1 logarithmic functions 26-7, 30 logarithmic linearisation 816-19,822-6 logarithmic models 322-3,325,817-18 logarithms 812, 322-3, 325 logistic curve 338, 339-40, 845 logistic differential equations 845-7 logistic equation 845 logistic models 33844, 8457, 855 loops 452,454,460 Lorentz curves 700-1 Lotka-Volterra equations 86970, 9012 lower bounds 506-8, 581-2, 636 M majorarcs 131 many-to-one functions 38 mappings 17,31, 36,8967, 911 margin of error 743, 744-6 marginal costs 674-5 marginal price 699-700 Markov chains 246-52, 653-8 matrices 202-72 adding and subtracting 205-6 applications 215-38, 246-52, 6538, 771 augmented matrix 230-5 definitions and operations 203-13 determinant 217-22,229,239-40, 241, 254-5 diagonalisation 2436, 250 and differential equations 872-8, 881-3 eigenvaluesfeigenvectors 23846, 250-2, 261-2,874-8, 881 elementary row operations 230-1 Gauss-Jordan elimination 2304 geometric transformations see transformation matrices identity matrix 207-8, 209, 216,223,239 inverse 215-18,224-5,229 Markov chains 246-52, 653-8 modelling with 246-52, 479-81,653-8 multiplying 206-13,259-62 reduced row echelon form 230-2, 234 scalar multiplication 206 singular/non-singular 217,229 Solving systems of equations see matrix methods transpose 222,245 vectors 204 see also adjacency matrices; transition matrices matrix basis theorem 253 matrix methods 213-38 applications 220-5,234-5 Gauss-Jordan elimination 230-5 using augmented matrices 2304 using inverse matrices 216-19,229, 235 matrix multiplication 20613, 259-62 max-min inequality 683 maximum values 578-95, 6024, 610-11 limiting value 341-3 Maxwell's equations 913-14 mean 385-6, 387, 389, 390, 397-402 confidence interval for 745-6, 7534 hypothesis testing 756-61 and outliers 386, 388 asa parameter 386 population mean 736-8,739, 7456, 753-4,760 probability distributions see expected values sample mean 736-9, 745-6, 7614 standard error 745-6, 759,760 unbiased estimators 736-8 mean value of coordinates 101,118 measurements in three dimensions 115-19 measures of centre 385-9, 397-9 measures of spread 390-402 median 385, 386-8, 389, 391, 392, 3989 midpoint (class) 374-5 midpoint (line segments) 101,103,118 minimum distances between points 285-7 minimum spanning trees 489-95, 506-7 minimum values 578-95, 601-2, 604-9 minorarcs 131 mixture problems 8534, 859 mode 385, 388-9 modelling 16-21, 298-360, 899-903 assumptions 301, 350, 353, 834, 900-2 continuous change 526, 533-5, 539-41, 561 with differential equations 841-7, 8509, 869-72,901-2 extrapolation 354-5 with functions 16-21,23,25-6,27,636 interpolation 355,356 interpretation 303-4,323-6 linear motion 714-24. with matrices 246-52, 479-81,653-8 model choice 298-9, 349-53, 356, 602, 845 model development 299303, 306-9, 321-3,332-5, 3447 model limitations 301, 3034 model revision 300-1, 3324 optimisation problems 311-12, 60113 periodic phenomena 148-51, 330, 332-8 population see population models probability density function 636 related rates problems 614, 616-24 with sequences 634, 70-5 testingfevaluation 299-304,353 with trigonometric equations 148-51, 330-8 with trigonometry 106-9 types of model 298 with vectors 479-81 see also specfic types of model modular arithmetic 222 modulo operation (mod) 222,224,225 modulus (complex numbers) 181,185 modulus-argument form 185-7 monotonic functions 25, 38, 7967 multigraphs 452,453, 457, 458, 465-6 multimodal data 389 multinomial experiments 765-9 1007 Index multiplication byascalar 206,276 complex numbers 177,178, 185-6, 191 matrices 206-13, 25962 vector multiplication 288-91 multiplication rule (probability) 419-20,430 mutually exclusive events 412, 418,420 N natural logarithm 8, 567-8 nearest insertion algorithm 504-5, 507 nearest neighbour algorithm 5034, 505, 507-8 nearest-neighbour interpolation 162-3 negative angle 130 networks 451 Newton's law of cooling 843 Newton's second law 855-7 90% box plots 740-3 nominal rate 734 non-linear regression 815-19, 8216 non-monotonic functions 40, 796-7 non-random sampling 366, 369-70 non-singular matrix 217,229 nonparametric statistic 766 nonprobability sampling 366, 369-70 normal distribution 644-7, 649, 744-5,760 normal lines 102-3, 595-601 notation 907-9 antiderivatives 667-8 differential calculus 539, 554-5, 561 function notation 31, 36, 554-5 scientific notation 7-8 sigma notation 78-80 nth partial sum 80-3 nth term ofa sequence 58-60, 61-3, 6970 null graph 455 null hypothesis 756-61, 762, 766,769, 770-1 number theory 902-3 numerical derivatives 584,598-9, 609, 610 numerical differentiation 546 numerical integration 686-8 numerical solutions 860-8, 87981 numerical solver (GDC function) 322,325 o observed values (outcomes) 765-9, 7701 ogive see cumulative frequency graph one-tailed test 752-3,755, 756 one-to-one functions 37-8 open box problem 311-12, 602-4 optimisation problems 311-12,601-13 order (differential equations) 834-5 ordered pairs 17, 39,203 Ores theorem 471 origin, distance from 17, 181-5, 284-5, 590 out-degrees 454-5 outcomes 409, 411-12, 765-9, 76974 Markoy chains 246-50, 653-5 representing graphically 417-20 outliers 393,394,397 disregarding 794-5 and mean 386, 388 and median 388 and range 395 scatter diagrams 784-5,794-5,797 and standard deviation 397 P p-value 755,757,759, 766-7 page rank vector 479-80 paired data 761-2 paired t-tests 761-2 pairs of inverse functions 35-7 1008 of inverse operations 36 of points in 3D space 116-18 of points in a plane 100-1 parabolas 304-5, 597, 706, 711-12, 821-2 parallel circuits 195-6 parallel lines 101-2, 255 parallel testing 735 parallel trajectories 874 parallelepiped 710 parametric form ofa line 281 parametric representation 213-14 partial sum ofa series 78,80-3,85 particular solution 837, 841 paths 4634, 466, 470-3 Hamiltonian 470-1 shortest 495-510 trees 483 weighted graphs 488 patterns 894, 895-6,903 Pearsons r 791-5, 7967, 798-802, 808 percentage error 34,5 percentiles 379-80 period 145-7,192,331 perpendicularbisectors 103, 153,155-62 perpendicular lines 102-3 phase 192 phase angle 192-3,194-5 phase portrait method 872-8, 881-3 phase shift 331,334 phase space diagram 902 piecewise functions 234, 313-14 piecewise models 313-14,804-5 planar graphs 473-82 point estimates 737 point of inflection 340, 587-8 point-gradient form ofa line 22-3 Poisson distribution 6414, 649-50 polar form (complex numbers) 184-7 polygraphic systems 222-5 polynomial equations 898-9 polynomial functions 5534, 556-7 modelling with 299-319 pooled data 762 population (descriptive statistics) 364-5 population experiments 534-5, 541 population mean 736-8,739, 745-6, 7534, 760 population models 16-17, 74 basic growth model 841-3 coupled systems 86972, 901-2 decay 323,842 exponential model 321-3, 352, 841-3, 845 growth rate 321-3, 834,841, 869 logistic models 341-3, 8457, 855 predator-prey models 869-72,901-2 predicting maximum 341-3 population parameters 365, 386, 734,736-7, 752 position angle 184-5 position function 716-21,722 position vector 281 potential difference 192-5,851-3 power model 815,816, 817-18 power rule 545-53, 559,565, 669 powers (complex numbers) 184, 187-92 predator-prey models 869-72,901-2 price-supply curve 699 prime numbers 903,906 Prims algorithm 491-5 principal (investment) 64,70,72 principal axis 331 prisms 115 probability 408-48,915 combined events 418-19 concepts and definitions 408-17 conditional probability 428-40 continuous random variables 701-2 equally likely outcomes 411 events at discrete time intervals 653 independent events 419-23,435, 638, 641 mutually exclusive events 412,418,420 probability intervals 740-3 random variables 638 rules 412, 418-20, 430-1 small values of 641-2 tree diagrams 421-3,430,432-3, 434 trials 40811, 638-9 Venn diagrams 417-20,429, 431 probability density function (pdf) 636, 701-2 probability distributions 63464 binomial distributions 638-41,644, 741 combinations of 649-53 discrete/random data 634-8,701-2 long term projections 653-8 and matrices 653-8 mean sec expected values normal distribution 644-7, 649, 744-5, 760 Poisson distribution 6414, 649-51 rate of occurrences 6424 transformed data 647-9 uniform distribution 6479, 738 see also statistical tests and analyses probability intervals 740-3 probability model 409 probability sampling 367-9 product rule 556-8, 560, 565, 837 profit function 674 projectiles 304-6, 539-40, 5834 proofs 899,909-11 proportion 912-13 pyramids 115 Pythagorean identity 1434 Pythagorean theorem 116-18, 6067 Q quadratic equations 178-9, 308, 540 quadratic formula 176, 305-6 quadratic functions 179, 285, 304-9, 540-1 derivatives 546-7 quadratic models 235,304-9, 351, 353, 821-2 quadratic polynomials 178-9 qualitative variables 364, 373,376 quantitative variables 364, 376 quartiles 390-2,393-5,398-9 quota sampling 369-70 quotientrule 55860, 561, 564, 565, 568 quotients complex numbers 177-8,186-7, 191 derivative of 558-60 of two functions 558-60, 838 R radians 130-5 radioactive decay 25-6,326, 844 random sampling 367-9 random variables 634-8, 738-9 binomial distribution 638-9 continuous 634, 636-8, 644-7, 649, 701-2 discrete 634-5, 637, 647-9 normal distribution 644-7, 649 Poisson distribution 6414, 649-51 standard deviation 635, 645, 647-8, 701-2 see also expected values; variance range (data) 390-2, 395, 398-9 range (independent variables) 355 range (ofa function) 17, 19 composite functions 32-3 Index exponential functions 24-5 interchanging with domain 38-42 inverse functions 36-7, 38-42 logarithmic functions 26 range (ofa model) 305-7, 334-5 rank-order correlation 7958, 800-2 rate of change 351,526 average 526,529-32, 534-5, 589 by constant factor 319-21 composite functions 553 constant 299-300, 302, 350, 351 and gradient (slope) 530-5 income 688-90 instantaneous 529-38, 539, 589 integral of 684 interpreting 816, 817-18,821 maximum/minimum values 5789, 584-6 moving objects 304, 5304, 588-90 populations 534-5, 541 and quadratic models 304 of rate of change 5601 related rates 614,616-24 supply and demand 590-1, 699-700 temperature 8434 rate of occurrences 6424 raw scores 645-6 rays 130,153,157 re-expressed data 816-17, 818-19 real numbers 141-3, 174,175, 181 real rate of return 734 reciprocal trigonometric functions 105 recursive formula 69, 865 recursive sequence 59-60, 61,62 reduced row echelon form 230-2, 234,235 reducible differential equations 847-8 reflections 46,47, 48-9 transformation matrix 252-3,254-5,258 regular graphs 464 rejection region 752-5, 7567, 7589 related rates problems 614, 616-24 relative frequency 373, 408-9, 410-11,412, 413 relative frequency distribution 373, 375-6 relative frequency graph 410-11 relative frequency histogram 377-8, 701 reliability 366, 734-6, 741 representative sample 366 residuals 807-8 resistance 194, 851-3 resistant measure of centre_386 response variables 782-3,787-90, 804, 805-10 revenue function 6734 right hand rule 290 right-angled triangles 105-6,109-10,113, 17 rooted tree 4834 roots (of complex numbers) 188-9,191-2 roots (of equations) 178-9 roots (tree graphs) 4834 rotations 256-8, 259, 260, 264 route inspection 5001 row vectors 204,222 Rule of 70 (approximation) 321 s S-curve 338,339-40 saddle point 875 sales models 298, 304 marginal costs 674-5 price 3034, 346, 354-5, 586, 8024 revenue 304, 3067, 6734 supply and demand 5901, 698-700 sample mean 736-9, 745-6, 7614 sample size 741, 742-5 sample space 409-11,417-27 samples (descriptive statistics) 364-5,373 sampling 366-70 sampling error 367, 368 scalar multiples 206,276 scalar product 288-90, 291,292-3 scaling 254,261 scatter diagrams 782-96,797 scientific notation 7-8 secant 105 secant (of acurve) 531-2,533 second derivative 560-1, 584-5, 586-91 second derivative test 586-91 second fundamental theorem of integral calculus 684-5 second order differential equations 835, 879-81 sectorarea 124-5,127-9,135 separable differential equations 838-44, 848 sequences 5877, 96-8,907, 913 arithmetic 61-8 explicit 59,61-2 finitefinfinite 58 geometric_68-77 nthterm 58-60, 61-3, 69-70 recursive 59-60, 61,62 series 77-89,96-8 arithmetic 77, 80-1 convergent/divergent 85-6 geometric 80-7,90 infinite 78,84-5 nth partial sum 80-3 sigma notation 78-80 series circuits 193 sets 17,8967, 911-12 shearing 256 shortest path 495-510 shrink 253-5,264 Sierpinski carpet 262-3 sigma notation 78-80 sigmoid curve 338, 339-40 significant figures 4-5 similar triangles 118 simple event.

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