THE INTERNATIONAL UNIVERSITY IU VIETNAM NATIONAL UNIVERSITY HCMC ANSWER TO FINAL EXAMINATION CALCULUS I Jan 2008 By N Dinh N N Hai and J J Strodiot SECTION A Question A1 10 marks Find the volume generated by revolving about the x axis the region bounded by the x axis and by the curves y sin x and y cos x for 0 x 2 Answer On the interval 0 intersect at the point 4 2 the two curves y sin x and y cos x 2 2 see the gure in the next page So the volume of the region bounded by the x axis and by the curves y sin x and y cos x is 4 Z Z cid 20 0 0 4 V sin2 xdx 1 cos 2x 2 x sin 2x 2 2 cid 21 4 0 2 4 2 Z 2 4 2 cid 20 cos2 xdx Z 2 4 x sin 2x 2 2 cid 21 2 4 dx 1 cos 2x dx Question A2 10 marks Suppose that f x 0 and f00 x 0 for all x a b where a b Which of the following cases is true of a trapezoidal approximation T for the integralR b ii T R b i T R b a f x dx a f x dx Explain a f x dx iii Can t say which is larger Answer Since f00 x 0 on a b f is concave on a b When using the trapezoid formula to approximate the integral for each i Si is smaller than the area Ai of the corresponding trapezoid i e Si Ai b B h f xi 1 f xi xi xi 1 1 2 1 2 1 f x dx f x dx Si Ai T nX i 1 nX i 1 and hence Z b a Therefore ii is true Z xi xi 1 nX i 1 Question A3 15 marks Evaluate a R e 1 ln t t2 dt b R e 1 dx ln x x Answer a 7 marks We use integration by parts with Then du 1 t dt Take v 1 1 t2 dt u ln t dv cid 20 t We get ln t t cid 21 e cid 20 1 Z e cid 21 e 1 I 1 t 1 t dt 1 t 1 e 2 e the given integral is improper By 1 0 26424 1 b 8 marks Since lim x 1 1 ln x x de nition Z e dx ln x x 1 lim t 1 dx ln x x Z e t 2 Z 1 Z e ln t Now we make the substitution u ln x Then du 1 ln t and when x e u 1 Thus x dx When x t u Z 1 ln t u 1 2du cid 2 2 u cid 3 1 ln t 2 1 ln t dx ln x x du u Z e t Consequently dx ln x x 1 lim t 1 2 1 ln t 2 Question A4 15 marks Find the eigenvalues and their corresponding eigenvectors for the matrix Answer We have det A I and the characteristic equation of A is cid 12 cid 12 cid 12 cid 12 2 3 2 cid 21 A cid 20 3 2 1 0 cid 12 cid 12 cid 12 cid 12 3 1 2 0 2 3 2 0 The solutions of this equation are 1 and 2 and these are the eigenvalues of A Case 1 1 To nd the eigenvectors corresponding to the eigenvalue 1 set 1 in the augmented matrix corresponding to the system A I X 0 This system is reduced to an equation 2x1 2x2 0 or equivalently x1 x2 0 This equation has the general solution X t t t R Therefore the eigenvectors corresponding to the eigenvalue 1 are X t t t 6 0 Case 2 2 The system A I X 0 is reduced to an equation x1 2x2 0 whose general solution is X 2t t t R Thus the eigenvectors corresponding to this eigenvalue are X 2t t t 6 0 Question A5 10 marks Find F 0 x if Z sin x F x tan t dt 0 3 Answer Note that if g x R x tan x It is easy to see that F x g u x where u x sin x So by the chain rule we get 0 tan t dt then g0 x tan u u0 x ptan sin x cos x F 0 x g0 u u0 x Question B1 barrels per hour by 20 marks An oil tanker is leaking at the rate given in SECTION B L0 t 80 ln t 1 t 1 where t is the time in hours after the tanker hits a hidden rock when t 0 a Find the total barrels that the ship will leak on the rst day b Find the total barrels that the ship will leak on the second day a 10 marks The total barrels that the ship will leak on the rst day Answer is Let us use the substitution u ln t 1 Then du 1 u 0 when t 24 u ln 25 So t 1 dt When t 0 Z 24 0 Z 24 0 cid 20 N1 L0 t dt 80 ln t 1 t 1 dt N1 u du 80 Z ln 25 0 40 ln 25 2 414 units cid 21 ln 25 u2 2 0 The oil in barrels that the ship will leak on the rst day is approximate 414 barrels b 10 marks On the second day i e t 24 to t 48 the number of barrels of oil that the ship will leak are evaluated by N2 80 ln t 1 dt 24 t 1 Z 48 4 Using the same substitution as in Question a we get Z ln 49 cid 20 cid 21 ln 49 ln 25 u2 2 N2 u du 80 ln 25 191 units The oil in barrels that the ship will leak on the second day are N2 191 barrels Question B2 20 marks The speed V in meters per second of a body moving in a viscous medium at time t in seconds satis es the equation Z 100 t dv V v 1 kv a Obtain the expression that gives t explicitly in terms of V b Find the time taken for V to become 50 c Find the expression of V explicitly in terms of t d Find the value of V when t becomes very large i e tends to in nity Answer a 5 marks We have Z 100 cid 20 V ln t So Z 100 1 100k 1 kV k dv v 1 kv ln V v 1 kv v 1 kv dv V 1 v cid 21 100 cid 18 100 1 kV ln 100 V V 1 100k t ln cid 18 100 1 50k cid 19 50 1 100k cid 19 cid 18 2 100k 1 100k cid 19 t ln …