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1 OLLSCOIL NA ÉIREANN, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA OLLSCOILE, CORCAIGH UNIVERSITY COLLEGE, CORK 4/5 Autumn Suppement 5 MS Integ Ccuus nd Diffeenti Equtions Pof. P.J. Rippon Pof. B. Hnzon D. S.J. Wis Time owed: One nd f ous. Reding time of fifteen minutes is pemitted pio to te commencement of tis exmintion. Mks my e ost if not you wok is cey sown o if you do not indicte wee ccuto s een used. Answe question nd two ote questions. Question is wot 6% of te mks. PLEASE DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO THEN ENSURE THAT YOU HAVE THE CORRECT EXAM PAPER Pge of
2 MS: Autumn Suppement 5 / ( (. (i) Compute te definite integ sin x + ) ) x dx. Evute ny tigonometic expessions in you nswe, wic soud e exct (i.e. no decim expn- 6 sions). sin t (ii) Evute dt. t (iii) Evute x + x dx using te sustitution u = + x. (iv) Evute cos(4) d. (v) Use pti fctions to compute te definite integ You nswe soud e exct, i.e. no decim expnsions. (vi) Sove te initi-vue poem x dy dx y = x5, fo x >, nd wit y() =. Expess you nswe in te fom y = f(x). x + 7 (x + )(x ) dx.. () Find te e of te egion ounded y te coodinte xes, te ine x =, nd te cuve y =. Hint: compete te sque. x + 6x + () Sketc te gp of te function y = x x 8, in pticu giving octions of te intecepts wit te coodinte xes nd te citic point(s). Use tis sketc to genete second digm tt s sketces of te gps of te foowing two functions: 4 f(x) = nd g(x) =. x x 8 Find te points wee te gps of f(x) nd g(x) intesect. Find te e encosed etween te gps of f(x) nd g(x). You nswe soud e exct.. () Appy integtion y pts twice to evute te foowing integ: I = e x sin(x) dx. () Let I n = sin n θ dθ fo ec intege n. Evute I nd I. Use integtion y pts nd tigonometic identity to sow tt fo ec n I n = n n I n n cos θ sinn θ. Hence, o otewise, compute te definite integs / sin 8 θ dθ. Pge of
3 MS: Autumn Suppement 5 4. Rec tt te ypeoic functions e defined s foows: Pove te foowing: cos x = (ex + e x ), sin x = (ex e x ). (i) cos x > fo evey e nume x; d d (ii) cos x = sin x nd sin x = cos x; dx dx (iii) sin x is one-to-one function tt mps te intev (, ) onto te intev (, ); (iv) sin x = n ( x + x + ) fo x; (v) x + dx = sin x + C. 5. () Te gowt of ctei in cetin cutue is popotion to te nume of ctei pesent. If initiy tee e, ctei, nd te nume tipes in 5 minutes, ow ong wi it tke efoe tee e 6, ctei pesent? () A tnk contins ites of pue wte. Bine tt contins.8 kg of st pe ite entes te tnk t te of ites pe minute. Te soution is kept toougy mixed nd dins fom te tnk t te of ites pe minute. Let s(t) denote te mount of st in te tnk fte t minutes. (i) Wite down te initi-vue poem stisfied y s(t) (in tis cse fistode ine diffeenti, pus te initi condition). (ii) Sove te initi-vue poem to find out ow muc st is in te tnk fte t minutes. (iii) How muc st is in te tnk fte one ou? (iv) Sketc te gp of y = s(t). Wt is te imiting vue of s s t? Pge of End of Exm
4 UCC Mtemtic Tes Ccuus f(x) x n n x cos x sin x tn x f (x) nx n x sin x cos x sec x e x e x cos x sin x sin x cos x tn x sec x f(x) f(x) dx x n+ x n (n ) n + n x x cos x sin x sin x cos x tn x n sec x e x ex f(x) ( > ) f(x) dx sin x x x + x tn x + x n + x + x n + x x x x x sec n x + x x Poduct ue y = uv dy dx = udv dx + v du dx du Quotient ue y = u v dy v dx = dx udv dx v Cin ue f(x) = u ( v(x) ) f (x) = du dv Newton-Rpson x n+ = x n f(x n) f (x n ) Integtion y pts u(x)v (x) dx = u(x)v(x) Voume of soid of evoution out x-xis V = x= x= y dx dv dx = u ( v(x) ) v (x) u (x)v(x) dx Tyo seies (cente ) f( + x) = f() + f ()x + f () x + + f () () x +!! Tpezoid ue A [ ] y + y n + (y + y + + y n ) Simpson s ue (n even) A [ y + y n + (y + y y n ) ] + 4(y + y + + y n ) y y y y y n
5 Tigonomety A cos A 6 sin A 4 tn A tn A = sin A cos A sec A = cos A cot A = cos A sin A cosec A = sin A cos( A) = cos A sin( A) = sin A tn( A) = tn A cos A + sin A = cos(a + B) = cos A cos B sin A sin B sin(a + B) = sin A cos B + cos A sin B tn A + tn B tn(a + B) = tn A tn B cos A = cos A sin A = tn A + tn A sec A = + tn A sin A = sin A cos A = tn A + tn A e inθ = (cos θ + i sin θ) n tn A = tn A = cos nθ + i sin nθ tn A Lengt/Ae/Voume Tinge Ae = sin C = Sine ue: sin A = sin B = c sin C Cosine ue: = + c c cos A B c A C Rigt-nged tinge sin A = c cos A = c Peogm tn A = c = + Ae = = sin C Tpezium ( ) + Ae = Cice Cicumfeence = Ae = C A c Ac/secto Lengt = θ Ae = θ (θ in dins) Spee Sufce e = 4 Voume = 4 Cyinde Cuved sufce e = Voume = Cone Cuved sufce e = Voume = θ
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