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

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1 UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS VAN/SURNAME: VOORNAME/FIRST NAMES: WTW CALCULUS SEMESTERTOETS / SEMESTER TEST 1 STUDENTENOMMER/STUDENT NUMBER: HANDTEKENING/SIGNATURE: TYD/TIME: 135 min PUNTE / MARKS: 55 Eksterne Eksaminator / External Examiner : Me M P Möller Interne Eksaminatore / Internal Examiners : Dr R Kufakunesu Prof M Sango Dr J H van der Walt PUNTE MARKS LEES DIE VOLGENDE INSTRUK- SIES 1. Die vraestel bestaan uit bladsye 1 tot 12 (vrae 1 tot 8). 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 12 (questions 1 to 8). 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. 8. Aangeheg tot hierdie vraestel is n bylae wat sekere stellings bevat. In jou argumente moet jy na hierdie stellings verwys, waar nodig. Outeursreg voorbehou 7. All answers have to be motivated completely. 8. Attached to this question paper is an appendix containing certain theorems. You should refer to these theorems in your arguments, when necessary. Copyright reserved 0

2 VRAAG 1 QUESTION 1 Beskou die funksie Consider the function f(x, y) = 2x2 + xy y 2, y x x + y (1.1) Bereken f 1 (2, 3) en f 2 (2, 3). (1.1) Determine f 1 (2, 3) and f 2 (2, 3). [2] (1.2) Bepaal die vergelyking van die raakvlak aan die oppervlak z = f(x, y) by die punt (2, 3, 1). (1.2) Determine the equation of the tangent plane to the surface z = f(x, y) at the point (2, 3, 1). [2] (1.3) Gebruik jou antwoord in (1.2) om f(2.5, 2.5) te benader. (1.3) Use your answer in (1.2) to approximate f(2.5, 2.5). [1] 1

3 (1.4) Bepaal die tempo van verandering van f by (2, 3) in die rigting van die vektor v = 2i + j. (1.4) Determine the rate of change of f at (2, 3) in the direction of the vector v = 2i + j. (1.5) In watter rigting styg die funksie f die vinnigste by (2, 3)? (1.5) In which direction does the function f increase the fastest at (2, 3)? [1] (1.6) Bepaal die vergelyking van die raaklyn aan die kontoer kromme van f by die punt (2, 3). (1.6) Determine the equation of the tangent line to the level curve of the function f at (2, 3). 2

4 (1.7) Gebruik gepaste limiet wette om die volgende limiet te bepaal. Motiveer jou antwoord volledig. (1.7) Use suitable limit laws to determine the following limit. Justify your solution in full. lim f(x, y) (x,y) (a, a) (1.8) Hoe kan die funksie f(x, y) op die lyn y = x gedefinieer word sodat die nuwe funksie kontinu sal wees op die hele xy-vlak? (1.8) How can the function f(x, y) be defined along the line y = x so that the resulting function is continuous on the whole xy-plane? [1] 3

5 VRAAG 2 QUESTION 2 Laat u = f(x, y) waar x = h(r, t) en y = g(r, t), terwyl r = k(t). Aanvaar dat die funksies f, g, h en k almal kontinue parsiële afgeleides van alle ordes het. Let u = f(x, y) where x = h(r, t) and y = g(r, t), while r = k(t). Assume that the functions f, g, h and k all have continuous partial derivatives of all orders. (2.1) Skryf n gepaste weergawe van die Ketting Reël neer vir die afgeleides u u r en t. (2.1) Write down an appropriate version of the Chain Rule for the derivatives u u r and t. [4] (2.2) Aanvaar nou dat h 1 (r, t) = A en g 1 (r, t) = B vir alle (r, t), waar A en B konstantes is. Druk 2 u uit r 2 in terme van A, B en die parsiële afgeleides van f. (2.2) Assume that h 1 (r, t) = A and g 1 (r, t) = B for all (r, t), where A and B are constants. Express 2 u r 2 in terms of A, B and the partial derivatives of f. 4

6 VRAAG 3 QUESTION 3 Beskou die stelsel vergelykings Consider the system of equations xe y + u cos v = 2 u cos y + x 2 v y = 1 (3.1) Toon aan dat die stelsel opgelos kan word vir u en v as funksies van x en y naby die punt P 0 waar (x, y) = (2, 0) en (u, v) = (1, 0). (3.1) Show that the system can be solved for u and v as functions of x and y near the point P 0 where (x, y) = (2, 0) and (u, v) = (1, 0). (3.2) Bepaal nou ( ) u x by (x, y) = (2, 0), y waar u = u(x, y) en v = v(x, y) die oplossing van die stelsel vergelykings is. (3.2) Determine ( ) u x at (x, y) = (2, 0), y where u = u(x, y) and v = v(x, y) is the solution of the system of equations. 5

7 VRAAG 4 QUESTION 4 Gebruik die definisie van n limiet om aan te toon dat Use the definition of a limit to show that [ lim x 2 5y 6] = 0 (x,y) (0,0) 6

8 VRAAG 5 QUESTION 5 (5.1) Skryf die definisie van differensieerbaarheid van n funksie f (x, y) by n punt (a, b) neer. (5.1) Write down the definition of differentiability of a function f (x, y) at a point (a, b). [1] (5.2) Gebruik die definisie in (5.1) om aan te toon dat f(x, y) = 2x 2 + 2y 2 differensieerbaar is by (1, 1). (5.2) Use the definition in (5.1) to show that f(x, y) = 2x 2 + 2y 2 is differentiable at (1, 1). 7

9 (5.3) Bewys dat n funksie f (x, y) wat differensieerbaar is by (a, b) ook kontinu is by (a, b). Motiveer jou argument volledig. (5.3) Prove that a function f (x, y) which is differentiable at (a, b) is also continuous at (a, b). Justify your argument in full. VRAAG 6 QUESTION 6 Veronderstel dat die funksies f : R 2 R en g : R 2 R kontinu is by (a, b). Bewys dat die funksie Assume that the functions f : R 2 R and g : R 2 R are continuous at (a, b). Prove that the function kontinu is by (a, b). Motiveer jou argument volledig. h (x, y) = f(x, y) g(x, y) is continuous at (a, b). Justify your argument in full. 8

10 VRAAG 7 QUESTION 7 Aanvaar dat f kontinue eerste orde parsiële afgeleides het op R 2. Laat h, k > 0 en (a, b) R 2 gegee wees. In die vrae wat volg moet jou argumente volledig gemotiveer word. Assume that f has continuous first order partial derivatives on R 2. Let h, k > 0 and (a, b) R 2 be given. In the questions that follow, your arguments must be justified in full. (7.1) Beskou die funksie van een veranderlikke v(t) = f(a + th, b + k). Bewys dat (7.1) Consider the function of one variable v(t) = f(a + th, b + k). Prove that v (t) = hf 1 (a + th, b + k), t [0, 1] [2] (7.2) Bewys dat daar n θ 1 (0, 1) is sodat (7.2) Prove that there is some θ 1 (0, 1) such that v(1) v(0) = hf 1 (a + θ 1 h, b + k) [2] 9

11 (7.3) Laat u(t) = f(a, b+kt), en aanvaar die volgende: (7.3) Let u(t) = f(a, b + kt), and assume the following: (A) : Daar is / There is θ 2 (0, 1) sodat / such that u(1) u(0) = kf 2 (a, b + θ 2 k) Bewys dat daar θ 1, θ 2 (0, 1) bestaan sodat Prove that there exist θ 1, θ 2 (0, 1) such that f(a + h, b + k) f(a, b) = hf 1 (a + θ 1 h, b + k) + kf 2 (a, b + θ 2 k) 10

12 VRAAG 8 QUESTION 8 Beskou die beginwaarde probleem Consider the initial value problem u x + u t = 0 (E1) u(x, 0) = u 0 (x), x R (E2) met u 0 n differensieerbare funksie op R. with u 0 a differentiable function on R. (8.1) Aanvaar dat u(x, t) n differensieerbare funksie is wat die vergelykings (E1) en (E2) bevredig. Skryf n vergelyking x = f(t) neer vir die kontoer kromme van u deur die punt (a, 0). (8.1) Assume that u(x, t) is a differentiable function that satisfies the equations (E1) and (E2). Write down an equation x = f(t) for the level curve of u through the point (a, 0). (8.2) Toon aan dat u(x, t) = u 0 (x t) die vergelykings (E1) en (E2) bevredig. (8.2) Show that u(x, t) = u 0 (x t) satisfies the equations (E1) and (E2). 11

13 Stellings / Theorems Theorem 1 Let f and g be functions of two variables defined on an open subset D of R 2 containing the point (a, b). Suppose that (a) (b) lim (x,y) (a,b) f(x, y) = L lim (x,y) (a,b) g(x, y) = M Then the following hold: (1) lim (x,y) (a,b) [f(x, y) + g(x, y)] = L + M (2) lim (x,y) (a,b) [cf(x, y)] = cl for any c R (3) If f(x, y) g(x, y) for every (x, y) D, then L M (4) If M 0, then lim (x,y) (a,b) [ f(x,y) g(x,y) ] = L M Theorem 2 Suppose that f is a function of two variables with continuous first order partial derivatives on an open set D containing (a, b). Then f is differentiable at (a, b). Theorem 3 Suppose that f(x) is continuous on the closed interval [a, b], and differentiable on the open interval (a, b). Then there exists x 0 (a, b) such that f (x 0 ) = f(b) f(a) b a Theorem 4 Let F : R 2 R have continuous first order partial derivatives, and suppose that and that F 2 (a, b) 0. Then the equation F (a, b) = 0 F (x, y) = 0 defines y as a function of x in a neighborhood of the point (a, b). That is, we can express y as for some δ > 0, in such a way that (i) F (x, g(x)) = 0, x (a δ, a + δ) (ii) (iii) g(a) = b y = g(x), x (a δ, a + δ) g is a differentiable function on (a δ, a + δ) and dy dx = g (x) = F 1(x,y), x (a δ, a + δ) F 2 (x,y) 12