The use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.

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1 ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion consists of 15 questions. Only 8 out of the 15 questions hve to be nswered. Hence, in order to pss the exm, not every topic listed below hs to be mstered. Ech question is worth t most 3 points. In order to pss the exmintion score of 12 points is sufficient. In cse cndidte nswers more thn the bove mentioned number of questions, the points for the superfluously nswered questions re not dded to the totl. In such cses, the exminer decides which of the nswers will be regrded s superfluous. All necessry steps, formuls, substitutions, digrms or grphs leding to your finl nswer must be written down. Furthermore, questions contining the words solve, derive or clculte require n exct nswer; deciml pproximtion is not llowed. The use of so clled grphing clcultor or progrmmble clcultor is not permitted. Simple scientific clcultors re llowed. Content informtion In the exm you my find questions regrding the following topics: A: NUMBERS AND RULES OF CALCULATION 1. The sets N, Z, Q, R nd the opertions ddition, substrction, multipliction nd division. Nturl numbers: N = {1, 2, 3,...} Whole numbers: Z = {..., 2, 1, 0, 1, 2,...} Rtionl numbers: Q = { b Z, b N} (b + c) = b + c ( + b)(c + d) = c + d + bc + bd c + b c = + b c ( + b) 2 = 2 + 2b + b 2 ( b) 2 = 2 2b + b 2 b + c d + bc = d bd 2 b 2 = ( + b)( b) b c d = c bd b : c d = b d c = d bc 2. Absolute vlue x nd simple grphs corresponding with bsolute vlue. { x when x 0 x = x when x < 0 1

2 3. The power rules nd corresponding rules for logrithms. Use of rtionl nd negtive exponents. The next rules hold under the following conditions: p, q Q,, b, g > 0 nd g N, g 1 : p q = p+q p = p q q ( p ) q = pq ( (b) p = p b p ) p b = p b p p = 1 p 1 p = p g log p = p glog g log = log log g = ln ln g g log + g log b = g log b g log g log b = g log b Remrk: log is the commonly used nottion for 10 log B: STANDARD FUNCTIONS 1. Typicl properties of the following stndrd functions: polynomils, In prticulr: the liner function y = x + b where x,, b R the kwdrtic function y = x 2 + bx + c where x,, b, c R power functions y = c x n, where c R, n Q, In prticulr the function y = x with x 0 exponentil functions y = g x where g N nd their inverse functions y = g log x, where g N, g 1 nd x > 0, the function y = e x, where x R nd its inverse function y = ln x, for x > 0, the functions y = sin x nd y = cos x, where x R In prticulr the following properties: sin x = sin( x) = sin(π + x) cos x = cos x = cos(π + x) = cos(π x) sin x = cos( π 2 x) cos x = sin( π 2 x) sin 2 x + cos 2 x = 1 2. Use combintions nd inverses of the stndrd functions. For exmple: y = 3x 6 where x 2. 2

3 3. Sketch the grphs of the stndrd functions nd use the notions domin (ll possible vlues of x for which the function is defined), rnge (the set of ll vlues of y obtined s x vries in the domin), zeroes, scending nd descending function. 4. Sketch the grphs of the goniometric functions y = sin x nd y = cos x, using the notions period (= 2π), mplitude (= 1) nd equilibrium vlue (= 0). 5. Apply trnsformtions on grphs such s shifting nd stretching, nd describe the link between the trnsformtion nd the ltertion of the corresponding function. For exmple, given function f(x): trnsltion of f(x) long the vector (, b) results in y = f(x ) + b multipliction of f(x) with with respect to the x-xis results in y = f(x) multipliction of f(x) with b with respect to the y-xis results in y = f(x/b) C: EQUATIONS AND INEQUALITIES 1. Fctoriztion of qudrtic expressions into liner terms. For exmple: f(x) = x 2 2x 15 cn be fctorized s follows: f(x) = (x 5)(x + 3) f(x) = x 2 2x cn be fctorized s follows: f(x) = x(x 2) 2. Compute the solution of qudrtic eqution, using fctoriztion or the following formul: x 2 + bx + c = 0 hs solutions x 1,2 = b± (b 2 4c) 2 3. Hndle the following equlities nd their solutions: x = b x = b 1 = b g x = x = g log = log g log x = x = g log g = ln ln g 4. The solution of sin x = 0, cos x = 0, sin t = sin u, nd cos t = cos u: sin x = 0 x = kπ k Z cos x = 0 x = π/2 + kπ k Z sin t = sin u t = u + 2kπ or t = π u + 2kπ k Z cos t = cos u t = u + 2kπ or t = u + 2kπ k Z 5. Solve inequlities using sketch. For exmple: x 2 3x 10 > x + 5. Solving the eqution gives: x = 5 or x = 3. A sketch of the grphs supplies the nswer to the problem: x < 3 or x > 5. 3

4 D: CALCULUS 1. Recognition nd use of the different symbols for the derivtive of function. f (x), dy dx, df(x) dx nd df dx (x) ll denote the sme notion. 2. Compute the derivtive of the sum, the product, the quotient nd (in simple cses) of combintions of stndrd functions, s described in items B1 nd B2. Compute the derivtive of sum, product, or quotient of functions. Use the chin rule. Some exmples: sum: f(x) = x 2 + x f (x) = 2x x product: f(x) = (x 2 3x + 5) ln x f (x) = (2x 3) ln x + (x 2 3x + 5) 1 x quotient: f(x) = x2 5x+3 f (x) = (5x+3) 2x x2 (5) (5x+3) = x2 +6x 2 (5x+3) 2 chin rule: f(x) = (5x 2 + 7) 4 f (x) = 4(5x 2 + 7) 3 10x = 40x(5x 2 + 7) 3 3. Determine on which intervls function is constnt (derivtive = 0), scending (derivtive > 0) or descending (derivtive < 0). E: STRAIGHT LINES AND SYSTEMS OF EQUATIONS 1. Determine the formul of stright line either in cse two of its points re given, or in cse one single point is given, combined with the slope. (Also, see D5.) 2. Know the conditions for two prllel stright lines nd two perpendiculr stright lines. (Also, see D5) The two lines y = x + b nd y = cx + d re prllel when = c. The lines re perpendiculr when c = Solve system of two liner equtions with two unknowns. For exmple: { 2x + 2y = 2 3x + y = 5 { x = 2 y = 1 4. Sketch the solution of system with liner inequlities. F: COMBINATORICS 1. In connection with given combintorics problem, drw suitble visuliztion, e.g. tree digrm, network or grid. For exmple: when you re throwing with two dice, grid is very useful to rrnge the outcomes. 4

5 2. Use permuttions in cse the order of the chosen items is essentil. The number of permuttions of r items out of set of n is equl to r out of n = P n k = n! (n r)! For exmple: choosing chir-mn, secretry nd tresurer out of group of 5 persons, cn be done in 60 different wys. 3. Use combintions in cse the order of the chosen items is not importnt. The number of combintions of r items out of set of n is equl to ( ) n bove r = Ck n n = n! = r (n r)!r! For exmple: choosing committee of 3 persons out of group of 5 persons, cn be done in 10 different wys. 4. Given combintorics problem, determine whether the problem indictes tht repetition is llowed or not. For exmple: when choosing three letters from the lphbet, llowing repetition yields different possibilities. Choosing three letters without repetition yields possibilities. 5. Determine whether the multipliction rule for independent occsions cn be pplied. (Also see G5 nd G7.) Consider the exmple bove: when repetition is llowed, the drwings of the letters re independent. When repetition is not llowed, the outcome of the second nd third drwing is depending on the first outcome. So in this cse the three drwings re not independent. G: PROBABILITY 1. At experiments deling with probbilities, use the notions outcomes, set of outcomes, event, elementry event, impossible event nd contrdictory events. 2. Trnslte experiments regrding probbility to model with jr contining mrbles. Determine whether the order in which the mrbles re drwn is importnt or not, nd determine whether, fter ech drwing, the mrble hs to be put bck into the jr or not. When fter drwing the possibilities remin the sme, this is modelled by jr of mrbles where the mrbles hve to be put bck into the jr fter ech drwing. Simultneous drwings trnslte to model with jr of mrbles where two mrbles re being drwn without putting the first one bck. 3. Clculte probbilities by using symmetry nd by counting systemticlly. 4. Recognize combintoric spects when counting the number of elements of set of outcomes. Use of the following rule: Probbility = (number of fvorble outcomes) / (totl number of outcomes) 5

6 5. Clculte probbilities by using the multipliction rule for independent occsions. (Also, see F5 nd G7.) Multipliction rule: for independent events A nd B it holds: P (A nd B) = P (A) P (B) Counterexmple: throwing two dice; X represents the totl number thrown. P (X > 9 nd Xis even) = 4 36 P (X > 9) P (Xis even) = Clculte probbilities by using the sum rule or the complement of probbility. Sum rule: when events A nd B rule ech other out: P (A or B) = P (A) + P (B). Using the complement: P (A) = 1 P (non-a). 7. Use the notions of independent events nd conditionl probbility when deling with symmetricl nd non-symmetricl probbility spces. (Also, see F5 nd G5.) The events A nd B re independent when P (A nd B) = P (A) P (B) A conditionl probbility stisfies P (A B) = P (A nd B)/P (B). 8. Describe the set of vlues for discrete probbility vrible (in simple cses, with the probbility distribution). For exmple: throwing four dice. We re interested in the number of dice tht show six dots. We denote the probbility vlue by X, hence X = the number of dice tht show six dots. X cn ttin the vlues 0, 1, 2, 3 or 4. P (X = 0) = 5 6 = 625 P (X = 1) = = 500 P (X = 2) = 6 = 150 P (X = 3) = = 20 P (X = 4) = 1 6 = 1 Note tht P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3) + P (X = 4) = Clculte nd interpret the expected vlue of discrete probbility vrible when its distribution of probbilities is given. In the bove exmple, the expected vlue E(X) of the discrete probbility vrible X is equl to E(X) = = 864 = Apply the following rule: The expected vlue of sum = the sum vn the expected vlues. 6

7 H: REMAINING TOPICS 1. Use the formul N(t) = N 0 g t, which denotes exponentil growth. Exmple: A bcteril popultion (strting with bcteri) grows exponentilly with the hourly fctor of growth equl to 1, 5 (so, every hour 50% of the present number of bcteri is dded). This popultion is described by the formul N(t) = (1, 5) t. 2. Convert fctors of growth nd percentges of growth. Exmple: n hourly fctor of growth equl to 1, 5 converts to fctor of growth equl to (1, 5) 1 4 1, 1067 for every qurter of n hour. Hence, n hourly percentge of growth equl to 50% converts to growth of 10, 67% per qurter of n hour. 3. Mke clcultions involving compound interest Exmple: in 10 yers, cpitl of euro, tht is deposited t n nnul interest rte of 5% will hve grown to (1, 05) 10 euro. 4. Clculte the so-clled mesures of center : men, medin nd mode. The men (=verge) of the vlues 1, 1, 1, 2, 2, 4, 4, 7, 9, nd 9 is equl to = 4. The medin (the middle vlue, or the men of the middle two vlues, when the dt is rrnged in numericl order) is equl to = 3, nd the mode (the vlue tht ppers the most) is equl to Given set of dt, crete frequency tble, in some cses using clss (or group) intervls. 7