WTW134 DIE UNIVERSITEIT VAN DIE VRYSTAAT HOOF-HALFJAAREKSAMEN 2012

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1 (68) 25 May 202 HOOFKAMPUS WTW34 DIE UNIVERSITEIT VAN DIE VRYSTAAT HOOF-HALFJAAREKSAMEN 202 Assessore: Prof. D. M. Murray (Afrikaans) Dr. H. W. Bargenda, Prof. T.M. Acho (Engels) Mr. N. Sebastian (Qwaqwa) Moderator: Mnr. C. Venter Tyd: 3 uur Punte: 00 Hierdie vraestel bestaan uit twee bladsye, met Afrikaans en Engels rug-aan-rug. Skryf die naam van jou dosent bo-aan jou antwoordboek. Sakrekenaars mag gebruik word.. (a) Wat is die definisieversameling van n funksie f?geediedefinisie! [ 2 ] (b) Vind die definisieversameling van g(x) = x(x 3). [2] (c) Lê diepunt( 2, 5) op die grafiek van h(x) =+x 2? Gee n rede! [2] (d) Het k(x) = x n inverse vir <x<? Gee nrede! [2] 2. (a) Krimp die grafiek van y =+e 2x met nfaktor/2 en skuif dan die gekrimpte grafiek met 4 eenhede na regs. Vind die vergelyking van die uiteindelike grafiek. [ 2 ] (b) Die grafiek van n derdegraadse polinoom p(x) snydiex-as by x = 2 endiey-as by y = 6 en raak, maar sny nie, die x-as by x =. Vind die formule vir p(x). [ 3 ] (c) Het die grafiek van die rasionale funksie f(x) = n vertikale asimptoot? Gee n rede! x 2 + [2] 3. (a) Laat (a, b) die punt wees op die eenheidsirkel, met middelpunt by die oorsprong, sodat a>0enb/a = 3. Vind booglengte t, linksom gemeet vanaf die punt (, 0) na die punt (a, b), akkuraat tot drie desimale. [ 3 ] (b) Laat f(t) n sinusvormige funksie wees. Veronderstel dat 6 f(t) 6die waardeversameling van f(t) is,datf(0) = 6, f(4) = 0 en dat f(t) afnemend is oor 0 t 4. Vind die periode P en die formule vir f(t). [ 4 ] 4. Vind die afgeleides f (x) van die volgende funksies f(x). MOENIE jou antwoorde vereenvoudig nie! (a) f(x) =x 5 +4e x + arctan(kx) met k enige konstante [ 4 ] (b) f(x) =(tanx) (arcsin x) [2] (c) f(x) = 8x [4] sin(2x) (d) f(x) =(+ cos x) 6. [3] 5. (a) Laat v(t) =ln(+t 2 ) die snelheid van n bewegende voorwerp wees, met t 0 gemeet in minute sedert 3:00 vandag en v(t) in meter/minuut. Vind v (t) endie waarde vir die versnelling van die voorwerp teen 3:07 vandag. Gee eenhede by jou antwoord! [ 5 ] (b) Gebruik Implisiete Differensiasie om die helling van die raaklyn aan die kurwe y 3 + xy = 3 by die punt (2, ) te vind. [ 4 ] x2

2 6. Laat f(x) n funksie wees vir <x< met f (x) =x 2 (x 2). (a) Vind die twee kritieke punte van f(x). [ 2 ] (b) Het f(x) n lokale minimum? Gee redes (Tabel of andersins)! [3] 7. Vind die globale maksimum M en die globale minimum m van f(x) =4x x 4 op die geslote interval 0 x 2. Wys jou werk! [ 5 ] 8. n Blombedding het die vorm van n sektor van n sirkel, soos in die figuur, met r die radius van die sirkel en s die booglengte vanaf P na Q. DieoppervlakteA van die sektor word gegee deur A = 2 rs. Die oppervlakte A van die blombedding moet 25 meter 2 wees. (a) Wys dat die omtrek van die blombedding gegee word deur C(r) =2r + 50 oor 0 <r<. r [2] (b) Gebruik die formule vir C(r) in (a) en vind die waarde van die minimum omtrek van die blombedding. Gee redes (Tabel of andersins)! Gee eenhede by jou antwoord. [ 6 ] 9. (a) Vind n funksie f(x) waarvoor f(x)dx =0 x + C. [2] (b) Vind (x n 3 x + cos 2 x +2x ) dx vir enige konstante n =. [ 5 ] (c) Los op die beginwaarde probleem dy =4sint, y(0) = 5. [ 4 ] dt 0. (a) Formuleer die Fundamentele Stelling van Calculus. [ 3 ] 8 +x d (b) Vind die waarde van dx. [2] dx (c) Vind die waarde van die oppervlakte A onder die grafiek van f(x) =x 2 + van x =0totx =3. [3] h (d) Vind die waarde van lim h 0 h 0 +x dx. [3] 2. (a) Die uitdrukking P (t) =7 e 0,02t beskryf die grootte van die wêreldbevolking, met t 0 gemeet in jare sedert die begin van die jaar 200 en P (t) inmiljardmense. Letop dat f(t) =350 e 0,02t n anti-afgeleide van P (t) is. Vind die gemiddelde grootte van die wêreldbevolking vanaf t = 0tott = 0 jare, akkuraat tot een desimaal. Gee eenhede by jou antwoord. [ 4 ] (b) Volgense Relatiwiteitsteorie is die massa m(v) van n bewegende liggaam n toenemende funksie van die versnelling v van die liggaam, met v gemeet in 0 3 km/sek 290 en m(v) in kg. Wat is die praktiese betekenis van die vegelyking m (v) dv =5? Gee eenhede by jou antwoord! 0 [ 3 ] 2. Gebruik Integrasie deur Substitusie (GEEN ander metodes!) en vind (a) cos(ln x) dx x [3] (b) ( x 2 ) /2 xdx (Wenk: Stel w = x 2.) [ 3 ] π/2 (c) e sin x cos xdx, akkuraat tot drie desimale. [ 4 ] 0 3. Gebruik Deelwyse Integrasie (GEEN ander metodes nie!) en vind x sin xdx. [4] 2 Totale Punte Moontlik: 3 5 7

3 (68) 25 May 202 MAIN CAMPUS WTW34 THE UNIVERSITY OF THE FREE STATE MAIN MID-YEAR EXAMINATION 202 Assessors: Dr. H. W. Bargenda, Prof. T. M. Acho (English) Prof. D. M. Murray (Afrikaans) Mr. N. Sebastian (Qwaqwa) Moderator: Mr. C. Venter Time: 3 hours Marks: 00 This exam paper consists of two pages, English and Afrikaans back to back. Write the name of your lecturer at the top of your answer book. Pocket calculators are allowed.. (a) What is the domain of a function f? Give the definition! [ 2 ] (b) Find the domain of g(x) = x(x 3). [2] (c) Does the point P ( 2, 5) lie on the graph of h(x) =+x 2? Give a reason! [ 2 ] (d) Is k(x) = x invertible on <x<? Giveareason! [2] 2. (a) Shrink the graph of y =+e 2x by the factor /2 and then move the shrinked graph by 4 units to the right. Find the equation of the resulting graph. [ 2 ] (b) The graph of a polynomial p(x) of degree 3 crosses the x-axis at x = 2 and the y-axis at y = 6 and touches, but does not cross, the x-axis at x =. Findthe formula for p(x). [ 3 ] (c) Does the graph of the rational function f(x) = have a vertical asymptote? Give a reason! x 2 + [ 2 ] 3. (a) Let P (a, b) be that point on the unit circle centered at the origin such that a>0 and b/a = 3. Find the counterclockwise taken arc length t from the point P (, 0) to P (a, b), accurate to three decimals. [ 3 ] (b) Let f(t) be a sinusoidal function. Assume that 6 f(t) 6istherangeoff(t), f(0) = 6, f(4) = 0 and f(t) isdecreasingon0 t 4. Find the period P and the formula for f(t). [ 4 ] 4. Find the derivatives f (x) of the following functions f(x). Do NOT simplify your answers! (a) f(x) =x 5 +4e x + arctan(kx) with k any constant [ 4 ] (b) f(x) =(tanx) (arcsin x) [2] (c) f(x) = 8x [4] sin(2x) (d) f(x) =(+ cos x) 6. [3] 5. (a) Let v(t) =ln(+t 2 ) be the velocity of a moving object, with t 0measuredin minutes since 3:00 today and v(t) in meter/minute. Find v (t) and the value for the acceleration of the object at 3:07 today. Include units in your answer! [ 5 ] (b) Use Implicit Differentiation to find the slope of the tangent line to the curve y 3 + xy = 3 at the point P (2, ). [ 4 ] x2

4 6. Let f(x) be a function on <x< with f (x) =x 2 (x 2). (a) Find the two critical points of f(x). [ 2 ] (b) Does f(x) have a local minimum? Give reasons (Table or otherwise)! [3] 7. Find the global maximum M and the global minimum m of f(x) =4x x 4 on the closed interval 0 x 2. Show your work! [ 5 ] 8. A flowerbedhastheformofasectorofacircle,asshownby the figure where r is the radius of the circle and s the arc length from P to Q. The area A of the sector is given by A = 2 rs. Assume that the flower bed has a prescribed area A of 25 meter 2. (a) Show that the circumference of the flower bed is given by C(r) =2r + 50 on 0 <r<. r [2] (b) Use the formula for C(r) in(a)tofind the value of the minimum circumference of the flower bed. Give reasons (Table or otherwise)! Include units in your answer! [ 6 ] 9. (a) Find a function f(x) for which f(x)dx =0 x + C. [2] (b) Find (x n 3 x + cos 2 x +2x ) dx for any constant n =. [ 5 ] (c) Solve the initial value problem dy =4sint, y(0) = 5. [ 4 ] dt 0. (a) State the Fundamental Theorem of Calculus. [ 3 ] (b) Find the value of 8 d dx +x dx. [2] (c) Find the value of the area A under the graph of f(x) =x 2 + from x =0tox =3. [3] h (d) Find the value of lim h 0 h +x dx. [3] 2 0. (a) P (t) =7 e 0,02t describes the size of the world s population, with t 0measured in years since the beginning of the year 200 and P (t) in billions of people. Note that f(t) =350 e 0,02t is an antiderivative of P (t). Find the average size of the world s population from t =0tot = 0 years, accurate to one decimal. Include units with your answer! [ 4 ] (b) In Relativistic Physics, the mass m(v) ofamovingbodyisanincreasingfunction of the velocity v of the body, with v measured in 0 3 km/sec and m(v) inkg.what 290 is the practical meaning of the equation m (v) dv = 5? Include units in your answer! 0 [3] 2. Use Integration by Substitution (NO other methods!) to find (a) (b) cos(ln x) dx x [3] ( x 2 ) /2 xdx (Hint: Put w = x 2.) [ 3 ] π/2 (c) e sin x cos xdx, accurate to three decimals. [ 4 ] 0 3. Use Integration by Parts (NO other methods!) to find x sin xdx. [4] 2 Total Marks Possible: 3 5 7

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