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

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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

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