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

Transcription

1 UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS WTW CALCULUS EKSAMEN / EXAM PUNTE MARKS TYD / TIME: 180 min PUNTE / MARKS: 60 VAN / SURNAME: VOORNAME / FIRST NAMES: STUDENTENOMMER / STUDENT NUMBER: HANDTEKENING / SIGNATURE: GROEP / GROUP Prof Jordaan (Afr) Dr Ntumba (Eng 1) Dr vd Walt (Eng 2) Eksterne Eksaminator / External Examiner : Prof M Sango Interne Eksaminatore / Internal Examiners : Prof K Jordaan Dr P Ntumba Dr J H van der Walt Mr W S Lee LEES DIE VOLGENDE INSTRUK- SIES 1. Die vraestel bestaan uit bladsye 1 tot 15 (vrae 1 tot 12). Kontroleer of jou vraestel volledig is. 2. Doen alle krapwerk op die teenblad. Dit word nie nagesien nie. 3. As jy meer as die beskikbare ruimte vir n antwoord nodig het, gebruik die teenblad en dui dit asseblief duidelik aan. 4. Geen potloodwerk of enige werk in rooi ink word nagesien nie. 5. 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 15 (questions 1 to 12). Check whether your paper is complete. 2. Do all scribbling on the facing page. It will not be marked. 3. If you need more than the available space for an answer, use the facing page and please indicate it clearly. 4. No pencil work or any work in red ink will be marked. 5. If you use correcting fluid ( Tipp-Ex ), you lose the right to question the marking or to indicate work that had not been marked. 6. Geen sakrekenaars word toegelaat nie. 6. No pocket calculators are allowed. 7. Alle antwoorde moet volledig gemotiveer word. 7. All answers have to be motivated in full. 8. Aangeheg tot hierdie vraestel is n 8. Attached to this question paper is an bylae wat sekere stellings bevat. In appendix containing certain theorems. jou argumente moet jy na hierdie stellings verwys, waar nodig. You should refer to these theorems in your arguments, when necessary. Outeursreg voorbehou Copyright reserved 0

2 VRAAG 1 QUESTION 1 Bepaal of die funksie xy sin(x) x f(x, y) = 2 + y 2 as / if (x, y) (0, 0) 0 as / if (x, y) = (0, 0) Determine whether or not the function kontinu is by die punt (0, 0). is continuous at the point (0, 0). Motiveer jou antwoord volledig. Justify your answer in full. 1

3 VRAAG 2 QUESTION 2 Beskou die funksie f(x, y) = x y Consider the function (2.1) Bepaal f x (x, y) en f y (x, y). (2.1) Determine f x (x, y) and f y (x, y). [1] (2.2) Gebruik gepaste stellings om te verduidelik hoekom n raakvlak aan die oppervlak z = f(x, y) bestaan by die punt (2, 1, 1). (2.2) Use suitable theorems to explain why a tangent plane to the surface z = f(x, y) exists at the point (2, 1, 1). [2] (2.3) Bepaal n vergelyking vir die raakvlak aan die oppervlak z = f(x, y) by die punt (2, 1, 1). (2.3) Determine an equation for the tangent plane tot he surface z = f(x, y) at the point (2, 1, 1). [2] (2.4) Gebruik jou antwoord in (3.3) om f(2.1, 0.9) te benader. (2.4) Use your answer in (3.3) to approximate f(2.1, 0.9). [1] 2

4 VRAAG 3 QUESTION 3 Veronderstel dat f : R 2 R kontinue tweede order parsiële afgeleides het. Laat g(r, t) = tr + f(e rt, sin(t)). Assume that f : R 2 R has continuous second order partial derivatives. Let g(r, t) = tr + f(e rt, sin(t)). (3.1) Druk g t (r, t) uit in terme van r, t en die parsiële afgeleides van f. (3.1) Express g t (r, t) in terms of r, t and the partial derivatives of f. [2] (3.2) As g r (r, t) = t + te rt f x (e rt, sin(t)), druk g rt (r, t) uit in terme van r, t en die parsiële afgeleides van f. (3.2) If g r (r, t) = t + te rt f x (e rt, sin(t)), express g rt (r, t) in terms of r, t and the partial derivatives of f. 3

5 VRAAG 4 QUESTION 4 Laat D = {(x, y) : x 2 + y 2 16}, en beskou die funksie f : D R gegee deur f(x, y) = 2x 2 + 3y 2 4x 5. Let D = {(x, y) : x 2 + y 2 16}, and consider the function f : D R given by f(x, y) = 2x 2 + 3y 2 4x 5. (4.1) Gebruik Lagrange Vermenigvuldigers om die minimum en maksimum waardes van f op die rand van D te bepaal. (4.1) Use Lagrange Multipliers to find the minimum and maximum values of f on the boundary of D. [5] 4

6 (4.2) Bepaal nou die absolute minimum en absolute maksimum waardes van f op D. (4.2) Now determine the absolute minimum and absolute maximum values of f on D. [2] VRAAG 5 QUESTION 5 Die funksie f(x, y) = x 4 + 4xy + 2y 2 het kritieke punte (0, 0), ( 1, 1) en (1, 1). Bepaal of elkeen van hierdie punte n lokale minimum, lokale maksimum of saalpunt van f is. The function f(x, y) = x 4 + 4xy + 2y 2 has critical points (0, 0), ( 1, 1) and (1, 1). Determine whether each of these points is a local minimum, local maximum or saddle point of f. 5

7 VRAAG 6 QUESTION 6 Bepaal die gemiddelde waarde van die funksie f(x, y) = 6y + e x2 op die driehoekige gebied met hoekpunte (0, 0), (2, 0) en (2, 4). Determine the average value of the function f(x, y) = 6y + e x2 on the triangular region with vertices (0, 0), (2, 0) and (2, 4). VRAAG 7 QUESTION 7 Laat D gebied in die eerste kwadrant, binne die sirkel x 2 + y 2 = 2y en onder die lyn y = 1 wees. Druk die gegewe integraal uit as n herhaalde integraal in poolkoördinate. I = D xyda. Let D be the region in the first quadrant, inside the circle x 2 +y 2 = 2y and below the line y = 1. Express the given integral as an iterated integral in polar coordinates. [4] 6

8 VRAAG 8 QUESTION 8 Laat E die gebied binne die sfeer x 2 + y 2 + z 2 = a 2 en onder die keël z = a x 2 + y 2 wees. Let E be the region inside the spheer x 2 + y 2 + z 2 = a 2 and below the cone z = a x 2 + y 2 (8.1) Skets die gebied E op die stel asse hieronder voorsien. Dui alle kenmerke van al die betrokke oppervlakke duidelik aan op jou skets. (8.1) Sketch the region E on the set of axes provided below. Clearly indicate all relevant features of the surfaces involved on your sketch. z x y [4] 7

9 (8.2) Druk die volume van E uit as n herhaalde integraal in Cartesiese koördinate. (8.2) Express the volume of E as an iterated integral in Cartesian coordinates. (8.3) Druk die volume van E uit as n herhaalde integraal in silindriese koördinate. (8.3) Express the volume of E as an iterated integral in cylindrical coordinates. [2] (8.4) Druk die volume van E uit as n herhaalde integraal in bolkoördinate. (8.4) Express the volume o fe as an iterated integral in spherical coordinates. 8

10 VRAAG 9 QUESTION 9 Beskou funksies f en g van twee veranderlikes. Aanvaar dat f(x, y) = L en lim (x,y) (a,b) lim (x,y) (a,b) g(x, y) = M. Gebruik die definisie van n limiet om te bewys dat Consider function f and g of two variables. Assume that f(x, y) = L and lim (x,y) (a,b) lim (x,y) (a,b) g(x, y) = M. Use the definition of a limit to prove that lim [f(x, y) + g(x, y)] = L + M. (x,y) (a,b) Motiveer jou argument volledig. Justify your argument in full. 9

11 VRAAG 10 QUESTION 10 Beskou n funksie f : R 2 R. Veronderstel dat f x en f y kontinu is op R 2, met f x (a, b) 0. Laat C die kontoerkromme van f deur die punt (a, b) wees. In die vrae wat volg moet jy jou argumente volledig motiveer. Consider a function f : R 2 R. Suppose that f x and f y are continuous on R 2, with f x (a, b) 0. Let C be the level curve of f through the point (a, b). In the questions that follow, you must justify your arguments in full. (10.1) Bewys dat C n raakvektor het by (a, b). (10.1) Prove that C has a tangent vector at (a, b). (10.2) Bewys nou dat f(a, b) loodreg is op die kromme C by (a, b). (10.2) Now prove that f(a, b) is perpendicular to C at (a, b). [2] 10

12 VRAAG 11 QUESTION 11 Aanvaar dat f : R 2 R n kontinue funksie is. Laat D R 2 n geslote, begrensde en samehangende versameling wees, met area A(D) > 0. In die vrae wat volg moet jou argumente volledig gemotiveer word. Assume that f : R 2 R is a continuous function. Let D R 2 be a closed, bounded and connected set with area A(D) > 0. In the questions that follow, your arguments must be justified in full. (11.1) Toon aan dat daar punte (x 1, y 1 ) D en (x 2, y 2 ) D bestaan sodat (11.1) Show that there are points (x 1, y 1 ) D and (x 2, y 2 ) D such that f(x 1, y 1 ) 1 f(x, y)da f(x 2, y 2 ). A(D) D 11

13 (11.2) Bewys nou dat daar n punt (x 0, y 0 ) D bestaan sodat (11.2) Now prove that there exists a point (x 0, y 0 ) D such that f(x 0, y 0 ) = 1 f(x, y)da. A(D) D [2] 12

14 VRAAG 12 QUESTION 12 Beskou n kromme r(t) = x(t), y(t) in R 2 sodat r(0) = 1, 1. By elke punt P op die kromme, is die kromme loodreg tot die kontoerkromme van f(x, y) = x 4 + y 2 deur P. Bepaal r(t). Consider a curve r(t) = x(t), y(t) in R 2 such that r(0) = 1, 1. At every point P on this curve, it is perpendicular to the level curve of f(x, y) = x 4 +y 2 through P. Find r(t). [4] 13

15 Stellings Stelling 1 As g : [a, b] R kontinu is, en N is n getal tussen g(a) en g(b), dan bestaan daar n punt c [a, b] so dat f(c) = N. Stelling 2 Laat D R 2 n oop versameling wees. As f : D R kontinue eerste parsiële afgeleides het by (a, b) D, dan is f differensieerbaar by (a, b). Stelling 3 Laat D R 2 n oop versameling wees. As f : D R differensieerbaar is by (a, b) D, dan is daar n raakvlak aan die oppervlak z = f(x, y) by (a, b). Stelling 4 As D R 2 n geslote en begrensde versameling is, en f : D R is kontinu, dan is daar punte (a, b) en (c, d) in D so dat f(a, b) f(x, y) f(c, d) vir alle (x, y) D. Stelling 5 Laat F : R 2 R kontinue eerste orde parsiële afgeleides hê, en veronderstel dat F (a, b) = 0 en dat F x (a, b) 0. Dan bepaal die vergelyking F (x, y) = 0 x as n funksie van y in n omgewing van die punt (a, b). Dit wil sê, daar is n oop interval I met b I, n funksie g : I R en n getal δ > 0 so dat (i) Vir y I en (x, y) (a, b) < δ is F (x, y) = 0 as en slegs as x = g(y) (ii) (iii) g(b) = a g is differensieerbaar op I en dx dy = g (y) = Fy(x,y) F x (x,y), y I. Definisie 1 Laat C n kromme in R 2 wees. As P C en T is n raakvektor aan C by P, dan is n vektor N loodreg tot C by P as N loodreg tot die raakvektor T is. Definisie 2 n Versameling D R 2 is samehangend as vir enige twee punte P en Q in D daar n kontinue vektorfunksie r : [0, 1] D is so dat r(0) = P en r(1) = Q. 14

16 Theorems Theorem 1 If g : [a, b] R is continuous and N is a number between g(a) and g(b), then there is a point c [a, b] so that f(c) = N. Theorem 2 Let D R 2 be an open set. If f : D R has continuous first order partial derivatives at (a, b) D, then f is differentiable at (a, b). Theorem 3 Let D R 2 be an open set. If f : D R is differentiable at (a, b) D, then there is a tangent plane to the surface z = f(x, y) at (a, b). Theorem 4 If D R 2 is closed and bounded, and f : D R is continuous, then there are points (a, b) and (c, d) in D so that f(a, b) f(x, y) f(c, d) for all (x, y) D. Theorem 5 Let F : R 2 R have continuous first order partial derivatives, and suppose that F (a, b) = 0 and that F x (a, b) 0. Then the equation F (x, y) = 0 defines x as a function of y in a neighborhood of the point (a, b). That is, there is an open interval I with b I, a function g : I R and a number δ > 0 such that (i) For y I and (x, y) (a, b) < δ, F (x, y) = 0 if and only if x = g(y) (ii) (iii) g(b) = a g is a differentiable function on I and dx dy = g (y) = Fy(x,y) F x (x,y), y I Definition 1 Let C be a curve in R 2. If P C and T is a tangent vector to C at P, then a vector N is perpendicular to C if N is perpendicular to the tangent vector T. Definition 2 A subset D R 2 is connected if for every two points P and Q in D there exists a continuous vector valued function r : [0, 1] D such that r(0) = P and r(1) = Q. 15