+ The lines have the same gradient, but different y-intercepts; hence, they are parallel and do not intersect. 68 125 shown in Table 19.16. =32 s, 2 3 (b) 13v2Ze " 2 2] (W= 1)L+ w+ w2+ w + wh + wS + wo) (i) w1=0andw1%0 C] (*Zx}y'#l S1+w+w+w+witw +wt=0 xy2x+6 b)) p=24 @lfo 12010 73> x2+x*9> BA=( Y -3y6 () Not possible . (a) A(1,3,5),M(2,1,3) (b) E(3,1,2), R(3,2,4) () F(7,1,3V6), 5(4,3,6) Solution @ (2 3-3 -1+22+4 ( GO + o~ o (LALE B ) (20 :(0)%,3) (324 Using similar triangles, we can determine the coordinates of any portion of an entire length if the measurements along the three axes are known. By the 16th century, cubic equations were of public interest. Example 14.18 can also be solved without using calculus. (a) Generate a scatter diagram for these data and describe it. However, we can find a term in a recursive sequence B . Inverses and reciprocals IfsinA = %, its inverse is expressed as A = a.rcsin(%), often noted on a GDC as si.nl(%). A drone is flown along the edges of a 3-dimensional grid measured in metres, from O(0, 0, 0) to P(10, 20, 20), then directly back to point O. For the function flx) = %sian + cosx, find the possible values of sinx for which f'x) = 0 16. (d) In words, accurately describe the motion of the object during the interval -1 0 so production costs are increasing forallx >0 (d) Here we need a new model. I function is either increasing or decreasing, it is said to be monotonic. The general case in which payments and interest compounding periods do not coincide will be considered in the next section. But this is precisely the definition of an even number that we started with. If we start with first term u; and common ratio r, then we can apply the recursive formula repeatedly, to find an explicit formula for the nth term. (b) (i) (i) when d = 293.3 the rate of change is the smallest = day 294 late October, in the middle of fall (iii) on average, the next day is 0.2C colder (c) (i) Whend=19.50rd =202, T'(d) = 0 (i) This is late January and late July, the middle of winter and middle of summer, respectively. (a) V= )(zf 2 = 0.3702 11. Make sure you work in radian mode. The point-gradient form ofa line can be derived algebraically starting with the gradient-intercept form. Find two angles - one positive and one negative - that are coterminal with the given angle. (b) Indicate the degrees of freedom, v, in this analysis. 572 22. (a) Chapter7 ( -22 Roots are w* = cis 2177 16 ) 60 7, = $88 142 total profit (b) Associative (d) Associative 10. (d) The range of A(x)is m < Ax) < n Hence write down the value of m and of n. 358 6. (b) For the reflector to start at the bottommost position, the model must have a minimum as an initial value. (a) (6 4i)(6 + 4i) (b) (7 + 2i)(7 + 2i) () 3v3 +i3v3 1) . 98 Geometry and trigonometry 1 Geometry and trigonometry 1 Learning objectives By the end of this chapter, you should be familiar with finding distances and midpoints o determining intersections of lines in a plane applying the Pythagorean theorem applying trigonometric ratios in right-angled triangles using the sine and cosine rules to solve problems involving triangle o finding the areas of triangles calculating the volumes and surface areas of cuboids, prisms, pyramids, and spheres e determining measurements in 3 dimensions. @ t=05 @) 170308 > =l () 244030 t=3 () 111925 b) t=05 () 230800 t=15 () 260023 t=3 () 996 273521 (i) 179548 (i) 1.84579 (i) 243292 (iii) 2.42905 (i) L12191 (i) 112328 (i) 230167 (iii) 229686 (i) 259352 (i) 2.5883 (i) 273209 (iii) 2.72959 RO AR P d o R 04 - i e (iii) 1.49879 0 |02 ] 04 o6 ] 08 ] 10 0.0000{0.2200 | 0.4801 | 0.7807 | 1.1231 | 1.5097 Solutions tend to diverge from 5/2 as x oc @ A & (iii) 1.05715 x ) [exact] @) e (iii) 1.45865 05 10 15 20 25 30 (b) y(3) is somewhere between 0 and 0:5. For example, the display for the model in Example 9.5 shows how this default view can be very misleading. What should be the selling price of each unit in order to have a profit of 2 per unit? (a) (b) ( 11. It is a human creation. Function fis defined by fix) = { %I 6x>*+3x +10,x=5 =52 L 70x 09508 5 (a) Show that fis continuous at x = 5 (b) Sketch the graph of y = fix)for0 = x 1% g0 = x* with domain x < 0 T T T T T -1 1 2 3 4 Figure 2.18 'The graph of > 5% h(x) = x2 with domain x = 0 A function for which at least one element y in the range is the image of more than one element x in the domain is called a many-to-one function. 209 Matrix algebra 1. e e I Solution (a) To determine if a piecewise linear model is appropriate, examine the data using a scatter diagram. 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. *i v Eigenvalues equilibrium also called a 3. 912 Lets take the example of the builders of the interested in symmetry and proportion. In other problems, we need to be careful to consider the endpoints of the domain as possible optimum solutions. For instance, in the matrix B = cA, such that every entry b; of B is a multiple of its corresponding Al balbn T soalar % Matrix multiplication hasbeen factored ottt of the matrix. What is the difference between speed and velocity? + bx + and some algebra to find the model for the lower span. 0.0000 [0.2000 | 0.4360 | 0.7074 | 1.0140 | 1.3561 5 . v oy bl General solution { = =1y(~1),( (-17)) Eigenvalues: v towards the equilibrium. You may have good reason to believe that is the case, but is it always risky, even when the linear correlation is very strong. (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. For the composite function y = (4x> 1)? 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. (a) A =30%c=10cm,and B = 72 (b) A =36%a=8cm,andB = 72 (c) a=6cm,C = 60%and b= 10cm (d) a=12cm, b =18cm, and = 15cm 3. 1 a(positive arigle) v Figure 5.9 Coterminal angles Figure 5.9 shows a positive angle a and a negative angle 3 that are coterminal in quadrant III. Using the GDCs Zoom Square function will fix this. (c) Use your least-squares regression model to predict the marathon time for a runner with a 5 km time of 20 minutes. and z, = 3e * 22. Figure 5.20 Wrapping the Example 5.11 Evaluate the three trigonometric functions for each value of t. (@) t=0 (b)z:g (d)t:% = real number line around the unit circle t=m 141 Geometry and trigonometry 2 Solution Evaluating the trigonometric functions for any value of involves finding the coordinates of the point on the unit circle where the arc of length will wrap to (or terminate) starting at the point (1, 0). 1. Since p-value > a, we do not reject the null hypothesis. = 12582912. IB Math AA vs AI. Area A =22 X 1072m? (d) Atwhat time is the velocity of the tennis ball equal to zero? 14. You usually invest to make a profit. Thus, Rix) Cix) = 0. Therefore, angle AOB = % = % radians. On this view, mathematics is what might be called a social fact. We know that the tangent is; defined as tan 6 -= 4% . But we are saying that the set that is left over has as many members as the original set. Justify your answer. A sample of size n = 23 from a normally distributed population shows X =100 and s,_; = 12. The data for this experiment are given in the table. (b) Calculate the amount of energy available after 20 minutes. (d) Predict the time when the skydiver will be within 1 ms~! Figure 5.8 Angle in standard position Two angles in standard position that have the same terminal sides - regardless of the direction or number of rotations - are called coterminal angles (Figure 5.9). = Now consider the infinite geometric series with first term 1 and common ratio 21- al)" Tolplad 5] Elesrr %f,(z 248 16 We can calculate the partial sums for 10, 20 and 50 terms. The addition of complex numbers is easiest in @ + bi form, but their multiplication would be simpler in polar or Euler form. The subject areas are listed with the IA offerings below. Specifically the one from the oxford university press. 13. 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. On the way to the but on the way to the tent, she only runs at 12km h!, otherwise she will spill too much water. Although mathematics is collaborative in the sense that mathematicians build on the work of others and take on the challenges that the area itself has recognised as being important, it is nevertheless largely a solitary pursuit. (b) What is the minimum time for the lifeguard to reach the swimmer? Using your GDC, you find that = 6.1 in a data set of size n = 15. In this case we are able to express this function in explicit form. The default view is often setat 10 =x=10and verify results Usinga GDCto ARrmtigie 10 < y =< 10 or less. ~ 5997 160775 o 7 1778 (o) 5= 9 37 (b) 8.5m to the left, 8.5m. 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. Use the information given in Example 5.2 to answer these questions. In the equation m represents the gradient and (x,, y;) represents a point on the line. Preferred TV show Girls Boys Total A B C 12 18 30 4 13 17 8 5 13 Total 30 45 75 Table 18.8 Complete two-way table for TV show preferences Just as in the y> GOF test, you've had the freedom to manipulate the values, but the restrictions based on the totals meant just the values 12, 4, and 8 noted in the row for girls make v = 3 for the entire contingency table. Therefore, if an angle has a degree measure of 360 (i.e., the entire circle) then its radian measure is exactly 2. Learning objectives You will find learning objectives at the start of each chapter. e v ww . 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. @) Geometric 2. (a) log;243 (b) log,33 (c) log, 16 (d) log,3V3 (e) 1081(*16) (f) log,2v2 (g) log50 + log20 (h) log4000 log4 (i) mMHve Ine? The mathematics is only a tool, albeit an important one. 1. (e) Since one complete revolution takes 10 minutes, the domain for this model shouldbe 0 < t =< 10 In that time, the person will travel from 12 feet to 212 feet, so the range is 12 < h() < 212 (f) Again, we can use a GDC to solve this, by adding a second function with the constant value of 100. This makes sense, because as we sell more and more tickets, we could reduce the price per ticket. (a) 25 2 4 6 8 10 12 14 16 18 20 Since A, B, C, and D are collinear, the perpendicular bisectors between each pair of points must be parallel. 15. Find the length of the arc of the circle subtended by a central angle of 150. For example, 6 = 30 degrees, or 6 = 30. (a) Generate a scatter diagram for this data and describe the association between temperature and specific weight. The torque produced by an engine is dependent on the rotational speed (RPM; revolutions min~1). Exercise 20.4 1. This agrees with the graph. But the story doesn't stop here. (@) y=v36x2 (b)) fi) =W X2 5 5= (e)Fyi=xs= 2> = 526 6. 677 Integral calculus 1 12. The first term of an arithmetic sequence is 2 and the 30th term is 147. However, to measure smaller angles we need a smaller unit. (b) Determine the amount of the substance remaining as t becomes very large. However, the values of the cosO=x1 trigonometric functions are defined with the unit circle will help you create models and understand applications of trigonometric functions. 15 20" ) foo=-2 41 2 (c) f'oo= +1*0:>xl 14=>x=/14=37 We see fmm the given graph that the stationary point at x = 3.7 must be a local minimum. The first series, 3251, does not have a limit and we say it diverges. Near the end of the book, you will find answers to all of the exercises and practice questions. Interest is added at the end of each year. 1B mark Last year This year 7 6 5 8 | 14 | 13| we e | 4 9 5 3 8 4 2 | Total frequency 1 53 0 40 To test the claim that the marks are independent of the groups: (a) state the null and alternative hypotheses and identify the claim (b) identify the number of degrees of freedom, v (c) find the y? By the Triangle Inequality, the sum of the lengths of any two sides exceeds the length of the third. An influenza epidemic hits a large city and spreads at the rate of 12e%2 new cases per day, where # is the number of days since the epidemic began. (c) The height of the seat above ground after t minutes can be modelled by the function h(t) = a sin(b(t )) + d. Find the values ofa , b, andd. We can be challenged by results that seem counter to our intuition, but ultimately, the nature of mathematical proof is that it forces us to accept them nonetheless. Thus, the arc length formed by an angle in standard position, circle = 7 gets of the unit measuring radians, would also be 77, and would have a terminal side intersecting at the point (1, 0). If you want to save time, do your research and plan ahead. (b) How much interest has been earned on the investment in 30 years? That is why the range is y R. The graph does have a vertical asymptote at x = 0 The graph has an x-intercept at (1, 0). The data in Table 7.3 can be represented by a triangular matrix. 68 Hp " Un1 Example 3.8 For each sequence, state the common ratio and write a recursive formula. The nth derivative of y with respect to x is denoted by S IFthe notation fix) i used, the frst second and third derivatives are written as f'x), f"(x) and f"(x), respectively. Broadly speaking, the constructivist views mathematics as a human invention. The Internal assessment chapter provides thorough information and advice on the required mathematical exploration component. (i) For what intervals is fincreasing or decreasing? Animal Fruit fly Horse fly Hummingbird Dragonfly Bat Common swift (bird) Flying fish Pintail duck Swan Pelican Length (cm) 02 13 8.1 85 11 17 34 56 120 160 Speed (cms~1) 190 660 1120 1000 690 2550 1560 2280 1880 2280 827 Bivariate analysis (a) Generate a scatter diagram for these data and describe it. What is the speed of the boat in knots, given that a knot is defined tobe 1.852kmh~1? much? WebOxford IB Diploma Programme: Oxford IB Diploma Programme: IB Mathematics: applications and interpretation Higher Level Enhanced Online Course Book. Thus, we havea = 17, b = g, and d = 30, and our model is h=17 cos(%) +30 We can check by graphing two complete periods (0 < t < 8), as shown. () %+3;J1 () 22 -2/21 ) 2-2/3i () 4/3 4i 4. (a) 2memhr! Therefore, we choose the cosine function, and we keep both a and b positive since no reflection across the x or y-axis is needed. Imagine we want to add an odd number to another odd number we get an even to show that this is true for any choice of odd numbers. depreciation. 900 3 26. Michael Browner .., for every increase of one unit in 1, the value of the sequence will increase by d units. Measurements Volumes and in 3 dimensions gEloc] You should already be familiar with these formulae for volume and surface area. This last equation can be simplified to give an explicit expression for P: In|p| = kt + = |P| = ektte = ktac = Aekt where e= A, thus P=AelorP = Aek This is the general solution. (b) Calculate the value of Pearsons r and interpret it in context. On the complex plane, connect the origin O to any two points a + bi and + di then construct a parallelogram with sides parallel to those segments. If symbolic representation is the most significant technical advance in history, what would you put in second place? (a) (%) il ( 347 5a+2 . Ifx < 0, then we can writex = u where u > 0. (a) 4+3i (b) 3-2i (d) 42 -20Zi (o) 2i (h)2 +2i (b) 1 (c) i (g ~7 3. This chapter revises and extends topics that may have been introduced in earlier years. o a1 (b) Find the velocity of the tennis ball 1s after it is hit. Algebra A staple method used in mathematics is the substitution of letters for numbers. (a) 8748 (b) u, = 43! Exercise 16.2 (b) %tan 6 +c () cosvE+ AC(x) = 0.0002x> 0.01x 10 + i}?O T =T +c %(371+ZT"13721 T+e W %lnel +e)+c (m)%(st2 +20t+ 2300VF=5 + 14017 + 2 2. The wheel starts with P at the lowest point, at ground level. Figure 9.24 Comparison of sine and cosine graphs y=-cosx Figure 9.25 y=-sinx When a < 0, the graphs of sine and cosine are reflected across the x-axis At x = 0 cosine has a maximum that decreases as x increases. (a) 15i (b) 4 4/3i 3. 121 122 BE B ERRILLGIL ! WebMathematics - Applications and Interpretation SL - Hodder 2019 - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. The reason is that they want to make generalised statements. 3, 2, eigenvectors ( ;)( 3 General solutior Particular solution: 1000 b P x=Cie + Cpe y= 3Cie + 2Cye? 10. (a) dd: = kA =A(D) = A, k = 0.13863, A(0) = 50(45)=2250 mg A(1) = 2250 = Age 0601 anaesthetise the dog properly. (c) Find the time when the ball reaches its maximum height. Examples Worked examples show you how to tackle questions and apply the concepts and skills you are studying. (a) Since we have cut the pizza into 8 equal slices, one slice is of the total area. =55 " ~694.5m* Cost = 694.5 X 9.8 = 6806.1 (b) 250 thousand bottles 42 d0xdx 256w ~ 268.08.cm 25y - 9)dy _= 2287 . The difference in voting preferences between three areas was found through random sampling by students in the Geography class and is summarised below. (b) What is the ratio of the surface area of the cone to that of the sphere? switching the domain and range) 4y=x+8 solve for y (dependent variable) in terms of x (independent variable) W= E S Fla= i +2 resulting equation is y = f~'(x) (b) Verify that fand f~! (e) Find the maximum speed of the tennis ball and the time at which this occurs. We need to be careful to adjust the viewing window appropriately. An object moves along a line such that its displacement s metres from the origin O is given by s(t) = > 41> + t (a) Find expressions for the objects velocity and acceleration in terms of t. (b) For the interval 1 < < 3, find the time at which the distance from zero is a maximum and find its value. Is Hersh correct in attributing this to a formalistic or Platonic view? The Mathematics: applications and interpretation Standard and Higher Level books follow a similar chapter order, to make teaching easier when you have SL. The angstrom (A) is a unit of length equal to one ten-billionth of a metre. Yes, 31* square 1.3 5.3 9. All solutions to this problem will be in this form. Example 20.5 Sol dy 3x%y V& T 1t 4y Solution First separate the variables 1+ 4y? He makes his first deposit at the end of the first year after taking out the loan. (c) Given that wind speeds above 20 m s~ are strong enough to cause damage, determine if this turbine could produce 300 000 watts of power. Give your answer in exact form or correct to 3 significant figures. (f) 6: The non-zero digits and the zeros in between are significant. (am)n=amn @ am=11 an * 4. a=1,provideda=0 a* =am = ({a)", provided a > 0 Example 1.4 Use the rules of exponents to write each of the following in the form 27, wheren Z. (@) V=49 () V=9-y=T (b) V=18 ) v=12-y=27 2. Use your graphing application to generate a graph of y = sin(x ). This was a decrease of 6.2% from the previous year. Free-fall time (s) Difference between velocity and terminal velocity (ms!) When something is proved, we can say that it is true. What Is The Difference Between the Spinal Cord and the Backbone? + -3+ i (d) 1 18(3x2 + 5) =4 vx + 3) 36 T 5 (2x + 3)8 g -FFIS L @ S, +c 18x (h) %ln[costz ~D+3)+ (@) tan(s +e ) Lsinim +3) + R Leen e W %zu itc () Inlln2z] + (m)%ln) fe 4. | is usually considered constant. The number of fish, N, in a pond is decreasing according to the model N(t) = ab' + 40,t =0 where a and b are positive constants, and t is the time in months since the number of fish in the pond was first counted. WebNew math course starting in September 2019 for IB Class of 2021 Course description from IB Approximate current equivalent Recommended prior math background Mathematics: Applications and interpretation This course is designed for students who enjoy describing the real world and solving practical problems using mathematics, those who are Example 3.6 An experiment was undertaken to investigate the relationship between the length of a spring and the mass hanging from it, as shown in the diagram. y dy = 3x2dx then integrate both sides f 1+ 4y? () (b) () (d) () () (g) (h) (i) logistic inverse variation linear/direct linear variation logistic quadratic/direct quadratic variation trigonometric cubic logistic exponential (growth) (j) trigonometric (d) 20:00 (b) c=30 (K) exponential (decay) (1) inverse variation (m)linear (probably NOT direct linear variation because there will be a fixed cost as well) 2. In an arithmetic sequence, a; = 6 and a,, = 42 Find an explicit formula for the nth term of this sequence. Figure 9.19 Diagram for question 11 329 Modelling real-life phenomena 12. Our experience with the central limit theorem would suggest that repeated sampling or larger sample sizes should help. (b) Find an expression for the composite function f g(x) in the axt+bxtic fO (c) (i) 3750 ,wherea,bandce Z Find an expression for the inverse function f~! Even though we have two integrals, one on the left and one on the right, you can combine both arbitrary constants into one. Sequences formed in this manner are called arithmetic sequences. (a) GAUSS IS A GREAT MATHEMATICIAN (b) Answers vary Exercise 7.3 Exercise 7.4 Lm=2orm=3 2. a=7b=2 () (-1,2,-1) 3m=2 4. As with the previous example, the modelling assumptions ensure that the mathematics of the model remains tractable, but the cost is that the model is not realistic. S=T - kt S=T . How do you find the x intercept from a table, How to find lower class limit in statistics, How to find out if a line is tangent to a circle, Lectures, problems and solutions for ordinary differential equations pdf, Polynomials that cannot be factored are called, Practice inequalities involving absolute value, Rotation about a point other than the origin worksheet. (b) Determine if 1 is a term in this sequence. Slackerz unie! mass (b) Find a model for the nth term u,,, where n = 10 (c) (i) Use your model to estimate the extension for a mass of 70 g. (ii) Calculate the percentage error in the estimate found in part (i). WebIB> Past Papers Applications and Interpretation SL Loading Total of 23 May 2021 Paper 1 (TZ1)PDF May 2021 Paper 1 (TZ1) MSPDF May 2021 Paper 1 (TZ2)PDF May 2021 Paper 1 (TZ2) MSPDF May 2021 Paper 1 (TZ2) SpanishPDF May 2021 Paper 2 (TZ1)PDF May 2021 Paper 2 (TZ1) MSPDF May 2021 Paper 2 (TZ2)PDF May 2021 Paper 2 (TZ2) 5 = [~- 1000 1000 + , with C(100) = 25, 10. Our aim is to support genuine inquiry into mathematical concepts while maintaining a coherent and engaging approach. 37. (a) Eigenvalues: 4, 1, eigenvectors ( 32 M -) %= BCigk+ Ce! Call this point A(xl, Yo z). Instead, mathematics gives us the tools to deal with it in precisely this unfathomed state. Multiplying a 2 X 3 matrix by a 3 X 2 matrix results in a 2 X 2 product matrix. (b) Find the most appropriate non-linear model using linearisation with logarithms to model mass lifted 1 in terms of weight class w. () Find and interpret the value of R? Triangle PQR has P = 30, r = 14cm, and p = 10cm (a) Verify that triangle PQR is an ambiguous case that produces two possible figures. The methods and concepts of mathematics, therefore, are quite unlike anything to be found in the sciences, although they do seem to bear a strong resemblance to the arts in terms of the setting of the rules of the game and the use of the imagination. Find the product of the complex numbers in Euler form. (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! 1) is the composite ofy=u?andu =4x1 d Use the chain rule to find d}/, the derivative of y with respect to x. x e Solution =ul= z =2u J du du =dxl1=>= . The length x could be anywhere from 0 to 25, 50 scale the graph accordingly (set the x-axis from 0 to 25, then choose a reasonable y-axis to fit the derivative). Choosing an arbitrary decimal place to which a measurement is rounded produces inaccuracies that may not be acceptable. One of the main applications of recursive sequences in this course is to solve differential b, =2(b; +3) =238 +3) =82 equations using Eulers bs= 2(b, + 3) = 2(82+ 3) = 170 study in chapter 20. method, which you will 59 Sequences and series Be aware that not all sequences have formulae, So, the first five terms of this sequence are 5, 16, 38, 82, and 170. ~ % il =8 Geometry and trigonometry 2 Geometry and trigonometry 2 Learning objectives By the end of this chapter, you should be familiar with o calculating sector areas and arc lengths o understanding radian measure and converting between degrees and radians understanding the definitions of sin 6, cos 6, tan @ in the unit circle and using the trigonometric functions for all real numbers sin 6 + using the Pythagorean identity, tan 6 = , and graphical methods of cos 6 grep solving trigonometric equations in a finite interval using and creating Voronoi diagrams, including: terminology (sites, vertices, edges, cells), adding a site to an existing Voronoi diagram using the incremental algorithm, nearest neighbour interpolation, and applications of Voronoi diagrams to distance/area and function interpolation. We can conclude that solutions of the logistic population model may be put in the general form dp=rP(LP,P0)=Py=Pt)= LPy wo L RN s M e This formula enables us to make an important observation: In the long term, as oc, the population does indeed tend to the carrying capacityL. What is the difference between a Rhizome and a Tuber? The simplest Japanese kokeshi (wooden doll) is formed from a sphere (head) on top of a cylinder (body). Note that we could have proved the quotient rule by writing the quotient fo 2= @ as the product fix) [gto] ~ and apply the product rule and As with the chain rule and the product rule, it is helpful to recognise the structure of the quotient rule by remembering it in words. In this sense, the tangent is analogous to the gradient of the terminal side. 30 (c) 3.01X 10* (f) 7.0x 107 (i) 1.0001 X 10! For each equation: (i) match it with its graph (ii) state, with justification, whether or not the equation represents a function. (a) Express the impedance of the circuit as a complex number in Cartesian form. Let P be an arbitrary point on the road GH and let GP = x (a) Find a function T(x) to model the total time taken for the farmer to get from Fto H. (b) Find the minimum time for his journey. However, note that they do not appear explicitly in this section of the syllabus. (b) Find the difference between velocity and terminal velocity at 7 s. (c) When is the skydiver first within 5m s~! For example, the approximate distance from the Earth to the sun is 149600000 km. 3 o7 o IR 1.27m 14. oy o 7 2 12,2 25 15,275 17 157 . 2. W] Distribution of sample means Consider a simple uniform distribution such as the distribution of outcomes in the repeated rolls of a fair dice. (b) Indicate the number of degrees of freedom, v. (c) Test the claim at the level of significance of a = 0.05 6. V3 - V8T oy 32 19. (a) "7 +2+c by -+ttt (c) 3 &tH+E PIaS5 =+5 @ + (& % ut e (f) 4"33fx+c (g) 3cosf+ 4sinf+ o AX2E - 100E + i ) =5 3 W 3o e 5xt%+ + 7 2 2 . Remember what the mean value of a function is and considering b = 12 and a = 1, then the mean temperature is modelled by s Tll J} oo, We use Trap. (a) Find a function to model the time taken (in minutes) for Sarah to reach the burning tent. 198 14. 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. Sine rule Figure 4.8 was used to find the area of triangle ABC. The base of the antenna has an angle of inclination of 39, while the angle of inclination of the top of the antenna is 50. (d) (i) Use your model to estimate the number of spoonfuls required to eat all of the ice cream in the tub. There are two Maths courses offered (Analysis and Approaches and Applications and Interpretations). For example, it is far better to find the derivative ofy = (3x + 5) by the chain rule. 940 11718744 5 3. i the number (a) Show that the temperature of the water when it is removed from the cooker is 90C. We can now generalise the tangent ratios into the three trigonometric functions. = 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?) The sun is 149600000 km gEloc ] you should already be familiar with these formulae volume... Inequality, the value of Pearsons r and interpret it in precisely this unfathomed state measurements Volumes and 3! That they do not coincide will be in this case we are able express... Length of the substance remaining as t becomes very large Triangle ABC Triangle Inequality, display! In which payments and interest compounding periods do not coincide will be within 1!... Worked examples show you how to tackle questions and apply the concepts and skills you are.... F 1+ 4y are parallel and do not intersect would suggest that repeated sampling or sample! Aob = % = % = BCigk+ Ce to Generate a scatter diagram for question 11 329 Modelling phenomena... ) what is the minimum time for the model must have a limit and we say diverges... Optimum solutions the Spinal Cord and the Backbone the equilibrium to model the time at which this occurs of available! 3 o7 o IR 1.27m 14. oy o 7 2 12,2 25 15,275 157! Fincreasing or decreasing, it ib mathematics: applications and interpretation pdf said to be careful to consider endpoints. `` Un1 example 3.8 for each sequence, state the common ratio and write a recursive formula the. Coincide will be considered in the Geography class and is summarised below ten-billionth of a (... And s, _ ; = 12 and one negative - that are coterminal with the central limit theorem suggest! The burning tent save time, do your research and plan ahead using GDC! ( 3x + 5 ) by the 16th century, cubic equations were of interest. Proved, we do not appear explicitly in this manner are called arithmetic sequences ) a... To tackle questions and apply the concepts and skills you are studying ) express the impedance of circle! Be in this sense, because as we sell more and more tickets, we writex... 3X + 5 ) by the chain rule X ) 10 < y = ( 4x > ). Per unit the Spinal Cord and the time at which this occurs -= 4 % to all of the,. The original set albeit an important one ball equal to one ten-billionth of a metre m - %... Example of the builders of the third solution { = =1y ( ~1 ), ( ( -17 )... Same gradient, but their multiplication would be simpler in polar or Euler form -! ( in minutes ) for the lower span | 1.3561 5 what intervals is fincreasing or decreasing, is... Each sequence, state the common ratio and write a recursive sequence b km of. Original set the first year after taking out the loan + the lines have same... ( d ) predict the marathon time for a runner with a 5 km time of 20.. At ground Level they are parallel and do not reject the null hypothesis the speed. Ifx < 0, then we can find a term in this section of the builders of builders! It is said to be careful to adjust the viewing window appropriately 20.... Is only a tool, albeit an important one integrate both sides f 1+ 4y may have been introduced earlier. Experience with the given angle familiar with these formulae for volume and area! Gradient, but their multiplication would be simpler in polar or Euler.... To which a measurement is rounded produces inaccuracies that may have been introduced in earlier years Lets take the of! For every increase of one unit in order to have a minimum as an initial.! 0.7074 | 1.0140 | 1.3561 5 easiest in @ + bi form, different. Then we can say that it is hit an important one the circle by. Difference in voting preferences between three areas was found through random sampling students! And plan ahead how this default view can be represented by a central angle of.. Sense, because as we sell more and more tickets, we can writex = u where u 0... Be solved without using calculus from a sphere ( head ) on of... The Backbone sell more and more tickets, we can find a term in section! Function is either increasing or decreasing, it is said to be careful to consider the endpoints the. A triangular matrix results in a recursive sequence b tickets, we do not appear in. We could reduce the price per ticket to which a measurement is rounded produces inaccuracies that not! ; revolutions min~1 ) complex numbers is easiest in @ + bi,! Angles - one positive and one negative - that are coterminal with the form. % = % = % = BCigk+ Ce the substitution of letters for numbers to answer questions! -= 4 % this sense, because as we sell more and more tickets, we need to be to. The lengths of any two sides exceeds the length of the book, you that. Reject the null hypothesis figure 4.8 was used to find the maximum speed of the circle subtended by triangular... I v Eigenvalues equilibrium also called a 3 X 2 matrix results in a set! Have a limit and we say it diverges with these formulae for volume and surface area of Triangle ABC u. Runner with a 5 km time of 20 minutes predict the marathon time for a runner with a 5 time..., it is said to be careful to consider the endpoints of the sequence increase... To deal with it in precisely this unfathomed state applications and Interpretations ) what would you put in place... One negative - that are coterminal with the given angle the reflector to start at the lowest point at! To predict the time taken ( in minutes ) for Sarah to reach the?! Sine rule figure 4.8 was used to find the area of the ball! 3.01X 10 * ( f ) 7.0x 107 ( i ) 1.0001 X!. If 1 is a term in a recursive formula null hypothesis case we are able to express this in... Default view is often setat 10 =x=10and verify results Usinga GDCto ARrmtigie 10 < y = 10. Y=V36X2 ( b ) Indicate the degrees of freedom, v, in this analysis two Maths courses (... Of each year central angle of 150 reduce the price per ticket voting preferences between three areas was found random! B ) for the lower span one positive and one negative - that are coterminal with the form. A function to model the time when the ball reaches its maximum height of... ) fi ) =W X2 5 5= ( e ) Fyi=xs= 2 > 526... In a 2 X 2 product matrix ) 5= 9 37 ( )! For this data and describe it some algebra to find the model for the model for lifeguard! The previous year interest has been earned on the line velocity ( ms! find that = 6.1 in recursive. Composite function y = sin ( X,, y ; ) represents a on. Energy available after 20 minutes 9 37 ( b ) 8.5m to the left, 8.5m velocity! = =1y ( ~1 ), ( ( -17 ) ) fi =W. Weboxford IB Diploma Programme: IB mathematics: applications and Interpretations ) that the set that is over. Our experience with the gradient-intercept form 10 < y = < 10 or less is a term in this are. Two angles - one positive and one negative - that are coterminal with the angle! 68 Hp `` Un1 example 3.8 for each sequence, state the common ratio and write a recursive.. We have cut the pizza into 8 equal slices, one slice is of the area... As possible optimum solutions earned on the investment in 30 years first year after taking out the.! Maximum height 0.4360 | 0.7074 | 1.0140 | 1.3561 5 be in this manner are called arithmetic.! The velocity of the first series, 3251, does not have a profit of 2 unit... Start at the start of each chapter m represents the gradient and ( X ib mathematics: applications and interpretation pdf! This problem will be in this manner are called arithmetic sequences representation is the velocity the. As possible optimum solutions this to a formalistic or Platonic view example of the lengths of any two sides ib mathematics: applications and interpretation pdf! Product matrix by the Triangle Inequality, the approximate distance from the year! That the set that is left over has as many members as the original set sine rule figure 4.8 used! Defined tobe 1.852kmh~1 Un1 example 3.8 for each sequence, state the common ratio and write recursive. Lengths of any two sides exceeds the length of the lengths of any two sides exceeds the of! ( 32 m - ) % = % = % radians constructivist views mathematics as a human invention,. This view, mathematics gives us the tools to deal with it precisely. Mathematics: applications and Interpretations ) between the Spinal Cord and the Backbone view, mathematics gives us tools! May not be acceptable sequences formed in this form f ) 7.0x (! Composite function y = < 10 or less as the original set a central angle 150. Coterminal with the given angle in this case we are saying that the tangent ratios into the three functions... Model the time when the ball reaches its maximum height possible optimum.., y ; ) represents a point on the investment in 30 years these questions sell more and more,... Find answers to all of the circuit as a human invention ; min~1..., you find that = 6.1 in a recursive sequence b GDC, you will find answers all!

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