UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA FAKULTEIT NATUUR- EN LANDBOUWETENSKAPPE / FACULTY OF NATURAL AND AGRICULTURAL SCIENCES DEPARTEMENT WISKUNDE EN TOEGEPASTE WISKUNDE DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS WTW 114 - CALCULUS 114 SEMESTERTOETS 1 / SEMESTER TEST 1 Maart/March 01 TYD/TIME: 10 min PUNTE/MARKS: 0 VAN/SURNAME: VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER: HANDTEKENING/SIGNATURE: SELNOMMER/CELL NUMBER : MERK JOU LESINGGROEP / INDICATE YOUR LECTURE GROUP Dr Djoko Kamdem Mr WhaSuck Lee Prof Lou Pretorius Mr Harry Wiggins PUNTE MARKS LEES DIE VOLGENDE INSTRUKSIES 1. Die vraestel bestaan uit bladsye 1 tot 9. Kontroleer of jou vraestel volledig is. READ THE FOLLOWING INSTRUCTIONS 1. The paper consists of pages 1 to 9. Check whether your paper is complete.. Jy mag nie n sakrekenaar by jou hê nie.. You may not have a calculator with you. 3. Jy mag op niks anders as die vraestel skryf nie. 3. You may not write on anything other than the question paper. Outeursreg voorbehou Copyright reserved 1
AFDELING A : MEERVOUDIGE KEUSEVRAE - 0 PUNTE DOEN HIERDIE AFDELING EERSTE, AANGESIEN JY DIE MERKLEESVORM MET JOU ANTWOORDE NA EEN UUR MOET INGEE. 1. Gebruik n sagte potlood.. Vul jou persoonlike inligting in potlood in op die merkleesvorm se KANT 1. 3. Vul jou studentenommer in van bo na onder en kodeer dit. 4. Indien jou studentenommer verkeerd gekodeer is of nie ingevul is nie, sal jy nie punte vir afdeling A kry nie.. Beantwoord vrae 1 tot 10 op die MERKLEESVORM se KANT 1. 6. Indien Kant gebruik word sal dit nie nagesien word nie. 7. Jy mag nie verkeerde antwoorde uitvee op die merkleesvorm nie; omkring dus jou antwoorde eers in hierdie vraestel en merk die antwoorde op die vorm as jy seker is van jou keuse. SECTION A : MULTIPLE CHOICE QUESTIONS - 18 MARKS DO THIS SECTION FIRST, AS YOU HAVE TO HAND IN THE OPTIC READER FORM WITH YOUR ANSWERS AFTER ONE HOUR. 1. Use a soft pencil.. Fill in your personal information in pencil on the optic reader form on SIDE 1. 3. Fill in your student number from top to bottom and then code it. 4. If you make a mistake when coding your student number or if the student number is not coded, you will get no marks for section A.. Answer questions 1 to 10 on the OPTIC READER FORM on SIDE 1. 6. If side is used it will not be marked. 7. As you may not erase wrong answers on the optic reader form, first circle your answers in this paper and only mark the answers on the form when you are certain of your choice. 1
VRAAG 1 / QUESTION 1 Bepaal al die moontlike waardes van cot α AS α [0, π] en sec α = 3. Kies die regte antwoord. Find all possible values for cot α if α [0, π] and sec α = 3. Select the correct answer. (a) (b) VRAAG / QUESTION (c) of/or (d) of/or Bepaal die waardeversameling van die funksie y = + cos x. Kies die regte antwoord. Find the range of the function y = + cos x. Select the correct answer. (a) (, ) (b) (, ) (c) ( 1, 3) (d) [1, 3] (e) [ 1, 1] VRAAG 3 / QUESTION 3 Watter van die volgende funksies f is nie ewe nie en ook nie onewe nie? Kies die regte antwoord. Which of the following functions is neither even nor odd? Choose the correct answer. (a) f(x) = 6x 3 + 8x + 7 (b) f(x) = x x 4 + 1 (c) f(x) = x 3 x (d) f(x) = 3 VRAAG 4 / QUESTION 4 Gebruik die tabel om (f g)(3) te bepaal. Kies die regte antwoord. Use the table to evaluate the expression (f g)(3). Select the correct answer. x 1 3 4 6 f(x) 3 1 0 1 g(x) 6 3 4 6 (a) (b) 3 (c) 1 (d) (e) 6. VRAAG / QUESTION As 1 f(x) x + 8x + 8 vir alle x R, bepaal If 1 f(x) x + 8x + 8 for all x R, find lim f(x). Kies die regte antwoord. x 1 lim f(x). Select the correct answer. x 1 (a) 1 (b) 8 (c) 1 8 VRAAG 6 / QUESTION 6 (d) 1 16 (e) nie genoeg inligting / not enough information. Beskou / Consider f(x) = x x 6 as/ if x 3. x 3 Hoe moet f(3) gedefinieer word om f kontinu by 3 te maak? Kies die regte antwoord. How would you define f(3) in order to make f continuous at 3? Select the correct answer. (a) f(3) = (b) f(3) = 0 (c) f(3) = 1 (d) f(3) = (e) nie een hiervan / none of these
VRAAG 7 / QUESTION 7 Bepaal die limiet / Find the limit lim x x + 3 x 3 + x 1. Kies die regte antwoord / Select the correct answer. (a) (b) 0 (c) 3 (d) 3 (e) VRAAG 8 / QUESTION 8 x 9 Bepaal die limiet / Find the limit lim x x 6. Kies die regte antwoord / Select the correct answer. (a) 1 (b) 3 (c) 3 (d) 0 (e) bestaan nie / does not exist. VRAAG 9 / QUESTION 9 { cx + 7 if x Beskou / Consider f(x) = cx if x > Vir watter waarde van die konstante c is die funksie f kontinu op (, )? Kies die regte antwoord. For what value of the constant c is the function f continuous on (, )? Select the correct answer. (a) c = 1 (b) c = (c) c = 6 (d) c = (e) c = 7 VRAAG 10 / QUESTION 10 Beskou / Consider f(x) = x x + 4. f(a + h) f(a) Bepaal / Evaluate (h 0). h Kies die regte antwoorde / Select the correct answer. (a) h (b) a (c) a h (d) a + h 1 (e) Nie een hiervan / none of these 3
AFDELING B : 30 PUNTE. LEES DIE VOLGENDE INSTRUKSIES: 1. Beantwoord die volgende vrae op hierdie vraestel in ink.. Antwoorde wat in potlood of rooi rink beantwoord is word nie nagesien nie. 3. U mag nie korrigeer-vloeistof ( Tipp-Ex ) gebruik nie. 4. Doen alle krapwerk op die teenblad. Dit word nie nagesien nie.. As u nog ruimte vir n antwoord nodig het, gebruik die teenblad en dui dit duidelik aan deur n raam daarom te trek. 6. Toon alle stappe en berekeninge en verduidelik u antwoorde. SECTION B : 30 MARKS. READ THE FOLLOWING INSTRUCTIONS: 1. Answer the following questions on this question paper in ink.. Work done in pencil or in red ink will not be marked. 3. You are not allowed to use correction fluid ( Tipp-Ex ). 4. Do all scribbling on the facing page. It will not be marked.. If you need more space for an answer, use the facing page and indicate it clearly by drawing a frame around it. 6. Show all steps and calculations and explain your answers. 4
VRAAG 1 / QUESTION 1 Los op vir x R / Solve for x R : x 1 x + 1 > 1 [4] VRAAG / QUESTION Los op vir x (0, π) as sin x < sin x. Solve for x (0, π) if sin x < sin x.
VRAAG 3 / QUESTION 3 Skryf f(x) = x 1 sonder absolute-waarde tekens en teken die grafiek van y = f(x). Write f(x) = x 1 without absolute-value signs and sketch the graph of y = f(x). VRAAG 4 / QUESTION 4 Beskou / Consider f(x) = x + en/and h(x) = 4x. Bepaal n lineêre polinoom g(x) sodat g(f(x)) = h(x). Find a linear polynomial g(x) such that g(f(x)) = h(x). [] 6
VRAAG / QUESTION Bepaal die limiete / Determine the limits.1 lim x x x. lim x 3x x + 4 x + x 8 [].3 lim x a x a x a met / with a > 0 [] [] 7
VRAAG 6 / QUESTION 6 Bewys dat as f en g funksies is wat kontinu is by a dan is die produkfunksie fg kontinu by a. Prove that if f and g are functions that are continuous at a then the product function fg is continuous at a. VRAAG 7 / QUESTION 7 Bewys dat as f en g onewe funksies is, dan is die samestelling g f n onewe funksie. (Neem aan die definisieversamelings van f en g is R.) Prove that if f and g are odd functions then the composition g f is an odd function. (Assume f and g have domain R.) 8
VRAAG 8 / QUESTION 8 Bewys dat as f differensieerbaar is by x = a dan is f kontinu by x = a. Prove that if f is differentiable at x = a then f is continuous at x = a. VRAAG 9 / QUESTION 9 Bewys dat as lim f(x) = L (a en L is reële getalle) dan is lim f(x) = L. x a x a Prove that if lim f(x) = L (a and L are real numbers) then lim f(x) = L. x a x a 9