We will not spend too much time on these matters here, but there are a few aspects to highlight. statistic, we still need to compare it against the y? 5 Figure 9.7 GDC screen for solution to Example 9.4(b) (1.12,56.2) makes sense since we know the maximum height of the ball and we presume the ball does not go underground. IB Diploma Standard Level is a comprehensive textbook covering the 2019 curriculum. The function notation form of the chain rule offers a very useful way of saying the rule in words, and thus, a very useful structure for applying it. (0)) Show that T'= (@ = Or =5~ a7 (@) Find 47 i dr The lobster trap is designed so that the length of steel used in its frame is a minimum. Piecewise functions are often used to model situations that display behaviour that fits with one or more type of function. We are often interested in calculating the area of sectors, and we can use the method of Example 5.1 to calculate the area of any sector. (a) (6 4i)(6 + 4i) (b) (7 + 2i)(7 + 2i) () 3v3 +i3v3 1) . Use the rules of exponents to write each of the following in the form 27, wheren Z. (@) zvim%" ) 2&:5% 2/ZaisT (@ z\im% 47 4cis LT 16. The system itself is a nice example of circular causality. (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. It is possible, however, to find the explicit definition of the sequence. =51 In questions 28-29, find the length of the arc, s. Angles are in radians. 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). All points on the floor are mapped by the x- and y-axes. An investment of $450 is worth $560 after 7 years of earning an unknown annual interest rate under the simple interest model. 127 Geometry and trigonometry 2 5. . If s is the intensity of a sound (measured in watts per m? 30 13. In 2016, the population growth rate in Singapore was 1.3% and the size of the population was 5.6 million (source: World Bank). We have to do a great deal of work to find them through inference and abstraction. From the Pythagorean Theorem, we know that x? fsi.n(3x2) cos(3x?) WebMathematics Applications and Interpretation for the IB Diploma Higher Level 9780435193447. of terminal velocity? World consumption of tin is 0.22e * million tons per year, where t is the number of years since 2013. For example, f/x) represents the fourth derivative of the function fwith respect to x. (c) Atwhat time is the tennis ball descending at a rate of 10ms~'? +3 Solution The terms in the numerator and denominator are consecutive integers, so they take on the absolute value of R equivalent form. You will benefit by being able to quickly convert between degree measure and radian measure for these common angles. But of these only 99 have the disease. The physicist Paul Dirac said, Tt seems that if one is working from the point of view of getting beauty in one equations, and if one has really a sound insight, one is on a sure line of progress. We therefore need to revise some of the derivative formulae. (a) 88 6. u,=4n+27 ) 36 16 @ 11 7. Determine the number of significant figures in each value. A typical example of a problem in this category is how to solve a particular type of equation. Example 5.9 A circle of radius 9 cm has a sector whose central angle has radian 2 measure =~ 3 Find the exact values of: (a) the length of the arc subtended by the central angle. However, there is an important difference between the graphs of sine and cosine. 'The initial gift is the present value of the income stream of 20000 per year. (c) Interpret the values of v, and s, in context. Distance Eitore (ion)from 02 ) (0406 : (0810|12 14 Depth (m) 20 (19 [23 | 23| Figure 19.18 Scatter diagram for solution to Example 19.8(a) 16]|18]|20 22 | 28 | 33 | 40 | 46 | 5.1 Then, enter the data as two separate data sets into a GDC and calculate the least-squares regression line for each. Note that the function y is in terms of u, and the function u is in terms of x. Y 2by_. (d) Determine a reasonable domain for this model. Consider a continuous function f{x) defined over a closed interval [a, b]. The first term of an arithmetic sequence is 7 and the common difference is 3. In this section we review several key concepts and consider some applications. (a) (b) (0 12. Interest is earned on each payment at an annual rate of 2% per year, compounded annually. All solutions to this problem will be in this form. (iv) Write down an appropriate domain for your model in context. b Thatis (b @) fy =< | f0) de = (b @) fee with the assumption that b > a, this in turn is equivalent to b S =5 [ S 1 St 0 = 2.6 ml?. We trying out different pairs of odd numbers and checking that prove that if we number. Therefore, we can either use a sine model or apply a phase shift to our current model. Find the perimeter of a rectangle with length [ = 2.3 X 10! 12. 36 51858 6. WebAuthor: Jane Forrest Publisher: ISBN: 9780198426981 Category : Languages : en Pages : 672 Download Book. (c) Estimate the age ofa 25-cm long Gila Monster. We, the authors, have strived to thoroughly explain and demonstrate the mathematical concepts and methods listed in the course syllabus. So MT = v302 + 602 + 20> = /100(9 + 36 + 4) = MT =70m Midpoints of line segments in 3-dimensional space The coordinates of C, the midpoint of the segment AB, where A and B have the coordinates A(x,, y,, z,) and B(x,, ,, z,), have the mean value of the x-, y-, x +x, 9 Ty, z+22) and z-coordinates: C(x, y, z) = C( 2 T & R Example 4.20 Find the midpoint of the line segment defined by each pair of points. + 2.615x%))dx ~ 0.589 14. oo5627 15. For example, an earthquake of magnitude R = 4 is ten times as strong as an earthquake of magnitude R = 3. Now consider the diagonally opposite point where the ceiling, back wall, and opposite side wall meet. This is basically the form z = a + bi Example 6.17 A particular AC circuit has a resistor of 62, a reactance across an inductor of 11 and a reactance across a capacitor of 32. @) a=7b=2 10. It is assumed that the prey do not die from natural causes or that the predators do not come into existence except through the provision of food. Otherwise express accurate to 3 significant figures. According to the model, the population will exceed 7 million after 17.3 years. . _ Solution Since the fifth term is given, using the explicit general form: as=a;+((51d =11=5+4d =d= _32 This leads to the general term a,=5+ 3 E(n =al Therefore, a,y = 5 + %(19) = % Sometimes a real-life situation can be modelled by an arithmetic sequence, even though it does not follow the sequence exactly. (a) Show that every member of the family of functions fix) = %(].nx + k) is a solution to the differential equation xly +xy=1=0 d; (b) Write the equation in the form Ey = Fx, ), and illustrate part (a) by graphing two members of the family on the same axes. derivative changes sign atx = c, then (c, fio) is apoint of inflection. 37. There are several other applications of geometric sequences, linked to exponential growth or decay. 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! N 12, ? Save my name, email, and website in this browser for the next time I comment. This is quite a big difference from the 990 in 1000 that we expected intuitively. Given that the safe level of DDT is 40 units, for how long will the area be unsafe? The first series, 3251, does not have a limit and we say it diverges. Last updated: In this case, the law of acceleration due to gravity. and so on. 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. 1=x 3 (b) Solve the equation (fo g~! (c) Using the model, we can find the wind speed required to produce 300000 watts: P =29.1V>= 300000 = 29.1V>= V = /10309 = 21.8ms~" Since this is greater than the wind speed that will cause damage, it is not reasonable to expect this turbine to generate 300 000 watts. (b) Find the temperature of the potato half an hour after it has been placed in the oven. 1800 m 2aby %al (@) Inx+1k ) hx=k-1 (ii) (eh0) ok @l (b) P = 7.3dm> 86.2m + 234 () R = 0.944; the model accounts for 94.4% of the variation in precipitation using month as the explanatory variable. IB Math AA vs AI. (b) Indicate the number of degrees of freedom, v. (c) Test the claim at the level of significance of a = 0.05 5. (a) ~11x)m(#+x"1 L (@ MR by L T e 12 s =s [TeYdxm016 10. It seems that our mathematics fits the universe rather well. Beware! The wheel rotates at a constant rate, in an anticlockwise direction. We see that the x-coordinate is 1.12, so the ball reaches a maximum |y height at 1.12s. T o Rl (a) Write down the amount of electrical charge in the battery at t = 0 (b) The line L is the horizontal asymptote to the graph. Figure 9.18 Diagram for question 9 m Time | Temperature (minutes) [) 0 100 1 85 5 ? on the same set of axes. (c) At the top of Mt. (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. If =2 < C < 2, then whether 2 sinx + Cis positive will depend on the value of x. 1.7616 (iii) 2.3681 (i) 23701 3. If you did, then your mathematical intuition let you down - very badly. Parabola passes through (17, 37): y = 0.128x> = area = [ 012802 = 209.62 . To solve a math problem, you need to figure out what information you have. 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. (@ y=2x3 y==2x+3 (b =2x =3 2 4x =16 (d) y=2x3 2xy=6 (&) y=2x3 x=2y+3 (c) y=2x3 2= Y =3 (f) 2x6y=4 x3y=8 Solution (a) Intersecting: their gradients are different. Therefore, we must let a < 0 and so our model is: h=17 sin(?) Find its height above the ground at Q. Of course, it depends on how fast she can swim relative to how fast she can run. (a) C=08d+4 (h) x = 3log,y (i) x=2lny o -1 () 5505C @ F= %K 4504 () () 4594F (i) 273C 3. Finding the inverse of a tion If a function fis always increasing or always decreasing in its domain (i.e. Could we allow ourselves to be guided to truth in mathematics because of the beauty of the equations? Refer to Figure 5.21. 609 Further differential calculus Example 14 A supply of four metres of wire is to be used to form a square and a circle. The negative part of the graph isn't very useful, but you can use the Zoom Box function to examine the graph more closely. (a) Arithmetic 10. Many people think that ellipses and numbers and so on are there whether or not any people know about them; I think thats a confusion. It hard to come to rigorous conclusions about this kind of thing, but I feel its almost obvious that Platonism and Formalism are anti-educational, and interfere with understanding, and Humanism at least doesnt hurt and could be beneficial. (b) What are its coordinates when it has covered three-quarters of the total distance? It is the same Gauss who, as a young man, enabled astronomers to rediscover the minor planet Ceres after they had lost it in the glare of the sun, by calculating its orbit from the scant data that had been collected on its initial discovery in 1801 and then predicting where in the sky it would be found more than a year later. 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