UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS

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1 VAN/SURNAME: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS VOORNAME / FIRST NAMES: WTW 128 SEMESTERTOETS 2 / SEMESTER TEST 2 STUDENTENOMMER / STUDENT NUMBER: HANDTEKENING / SIGNATURE: SELFOON NOMMER / CELL PHONE NUMBER: GROEP/GROUP: Groep / Group 1 Groep / Group 2 Groep / Group TYD / TIME: 120 min PUNTE / MARKS: 35 Wisk / Math 2-1 / van der Walt IT 2-26 & Muller / Pretorius Law Auditorium & North Hall / Ntumba Regs Auditorium & Noord Saal Eksterne Eksaminator / External Examiner : Dr SM Maepa Interne Eksaminatore / Internal Examiners : Dr SA Mutangadura Dr PP Ntumba Prof LM Pretorius Dr JH van der Walt PUNTE MARKS LEES DIE VOLGENDE INSTRUK- SIES 1. Die vraestel bestaan uit bladsye 1 tot 10 (vrae 1 tot 10). Kontroleer of jou vraestel volledig is. 2. Doen alle krapwerk op die teenblad. Dit word nie nagesien nie. 4. As jy meer as die beskikbare ruimte vir n antwoord nodig het, gebruik die teenblad en dui dit asseblief duidelik aan. 5. Geen potloodwerk of enige werk in rooi ink word nagesien nie. 6. As jy korrigeerink ( Tipp-Ex ) gebruik, verbeur jy die reg om nasienwerk te bevraagteken of om werk wat nie nagesien is nie aan te dui. READ THE FOLLOWING IN- STRUCTIONS 1. The paper consists of pages 1 to 10 (questions 1 to 10). Check whether your paper is complete. 2. Do all scribbling on the facing page. It will not be marked. 4. If you need more than the available space for an answer, use the facing page and please indicate it clearly. 5. No pencil work or any work in red ink will be marked. 6. If you use correcting fluid ( Tipp-Ex ), you lose the right to question the marking or to indicate work that had not been marked. 7. Geen sakrekenaars word toegelaat nie. 7. No pocket calculators are allowed. 8. Alle antwoorde moet volledig gemotiveer word. 8. All answers have to be motivated completely. 9. Aangeheg tot hierdie vraestel is n bylae wat sekere stellings bevat. Verwys na hierdie stellings, waar nodig. 9. Attached to this question paper is an appendix containing certain theorems. Refer to these theorems, when needed. Outeursreg voorbehou 0 Copyright reserved

2 1. Evaluate / Bepaal x 3 + x (x 2 + 1) 2 dx [4] 1

3 2. Find the length of the curve / Vind die lengte van die kromme y = ln(cos x), 0 x π 3. [3] 3. If a and L are real numbers, give the formal definition of the following: As a en L reële getalle is, gee die formele definisie van die volgende: 3.1) lim x a + f(x) = L [2] 3.2) lim x f(x) = L [2] 2

4 4. Prove that / Bewys dat ( x ) lim x = 9 2. [3] 3

5 5. Prove that / Bewys dat lim x 0 (x2 + 2x 3) = 3. [4] 4

6 6. Prove that / Bewys dat lim x 1 2x = 0. [3] 5

7 7. In 7.1 and 7.2 find the derivative of the given function without evaluating the integral. In 7.1 en 7.2 bepaal die afgeleide van die gegewe funksie sonder om die integraal te bepaal. 7.1) F (x) = x 0 1 t dt [1] 7.2) G(x) = 1+x 2 e t2 1 dt [3] 6

8 8. Suppose that a R and that lim f(x) = L and lim g(x) = M, with L, M R. x a x a Prove that lim [f(x) + g(x)] = L + M. x a Gestel a R en dat lim f(x) = L en lim g(x) = M, met L, M R. x a x a Bewys dat lim [f(x) + g(x)] = L + M. [5] x a 7

9 9. Prove that if f is continuous on a closed interval [a, b], then there exists a number p in [a, b] such that Bewys dat as f kontinu is op die geslote interval [a, b], dan bestaan daar n getal p in [a, b] sodat f(p) = f ave = 1 b a b a f(x)dx. (Make use of appropriate theorems from the list on the last page of this question paper.) (Maak gebruik van gepaste stellings uit die lys op die laaste bladsy van hierdie vraestel.) [3] 8

10 10. Prove that if lim x a f(x) = L and L > 0 then there exists a δ > 0 such that if 0 < x a < δ then 0 < f(x) < 2L. Bewys dat as lim f(x) = L en L > 0 dan bestaan daar n δ > 0 sodat as 0 < x a < δ dan x a 0 < f(x) < 2L. [2] 9

11 Theorems / Stellings Theorem 1 If lim h(x) = L. x a + Theorem 2 If lim h(x) = L. x a lim f(x) = L and lim x a + lim f(x) = L and lim x a g(x) = L and f(x) h(x) g(x) for all x > a, then x a + g(x) = L and f(x) h(x) g(x) for all x < a, then x a Theorem 3 Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) f(b). Then there exists a number c in (a, b) such that f(c) = N. Theorem 4 If f is continuous on a closed interval [a, b], then there are points c and d in [a, b] such that f(c) f(x) f(d) for all x in [a, b]. Theorem 5 If f is integrable on a closed interval [a, b] and m f(x) M for all a x b, then m(b a) b a f(x)dx M(b a). Theorem 6 If f is integrable on a closed interval [a, b] and a < c < b, then b a f(x)dx = c a f(x)dx + b c f(x)dx. 10

UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS

VAN/SURNAME: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS VOORNAME/FIRST NAMES: WTW 162 DYNAMICAL PROCESSES EKSAMEN