Calculus and Differential Equations page 1 of 17 CALCULUS and DIFFERENTIAL EQUATIONS

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1 alculus and Diffrnial Equaions pag of 7 ALULUS and DIFFERENTIAL EQUATIONS Th following 55 qusions concrn calculus and diffrnial quaions. In his vrsion of h am, h firs choic is always h corrc on. In h acual am, h corrc choic could b in any posiion, and hr may b ohr minor changs o h problms.. Givn y= +, hn dy d is onsidr h funcion y=, is drivaiv is.... onsidr h funcion y =, is drivaiv is.. y.

2 alculus and Diffrnial Equaions pag of 7 onsidr h funcion y =, is drivaiv is +... ( ) + ( ) + ( +) + ( ) ( +) onsidr h funcion y =, is drivaiv is +... ( + ) ( ) + ( ) ( ) + ( ) ( + ) ( + ) ( ) 6. If dy y= and 0, hn is d. ( ). ( ). ( + )

3 alculus and Diffrnial Equaions pag of 7 dy 7. If y =, hn is qual o 5 d. ( 5).. ( 5) + ( 5) ( 5) ( 5) ( ) ( 5) = is qual o 8. If f ( ) cos, hn f '( ). sin + cos.. sin sin + cos sin + cos cos 9. If y= cosh, hn dy is qual o d... +

4 alculus and Diffrnial Equaions pag of 7 0. If y= sinh, hn dy is qual o d If y= + 5, hn dy a = 0.5 is d If y=, hn d y d a = is. If =, hn "0 ( ) y. -.. y quals

5 alculus and Diffrnial Equaions pag 5 of 7 If d/d = 0 and (0)=, hn () quals If d/d = and (0) =, hn () quals d is qual o ( + ) 8 d is qual o ( ) ( ) 8 5. ( ) ( ) 9 5

6 alculus and Diffrnial Equaions pag 6 of d is qual o ( ) ( ) + + ( ) 9+ + ( ) 9. d is qual o d is qual o +. an +. ln( ) + +. an ( ) + + an + + ln + + 6

7 alculus and Diffrnial Equaions pag 7 of 7. (cosh ) d is qual o. sinh +.. sin + +. (sinh d ) is qual o. cosh +. cosh d is qual o d is qual o

8 alculus and Diffrnial Equaions pag 8 of 7 d quals ln. ln. ln d quals d 7. quals L f ( ) d = sin π, hn f ( ) quals 0. π. π. 0-8

9 alculus and Diffrnial Equaions pag 9 of 7 9: If ( ) f ' = f ( ) for all ral and f ( 0) = 0, f ( ) = hn f ( ) quals... ln ln L L 0. If f ( ) is dfind in h inrval ( LL, ), h soluion o f ( ) = sin d is. 0. L. π L L cos kπ kπ L L kπ L L L. If f ( ) is dfind in h inrval ( LL, ), h soluion o f ( ) = cos d is.. L L sin kπ kπ. sin kπ 0 π L mπ nπ. If m and n ar ingrs, and m = n, hn cos cos d is L L L. L. L. π 0 L sinmπ mπ L L kπ L 9

10 alculus and Diffrnial Equaions pag 0 of 7 L mπ nπ. If m and n ar ingrs, and m = n, hn sin sin d is L L L. L. L cosmπ mπ. L + sinmπ mπ 0 π If a= and b=, hn dy a = 7 is d y. 5 6b..8 b. 0 b 9 undfind 5a 0a 5a -b -b If a = and b =, which of h following valus is closs o h drivaiv of h funcion a =?. -.5 y b 0. 6b 6 b b 0 0 a 8a a 6a 6. If a =, b =, =, and = hn yd is y 6b b b 5a 0a 5a -b -b 0

11 alculus and Diffrnial Equaions pag of 7 7. onsidr h funcion y() in h following figur y() Which of h figurs blow is closs o h drivaiv of y()? No ha all graphs hav h sam scal. dy/d dy/d () () dy/d () dy/d dy/d () (5)

12 alculus and Diffrnial Equaions pag of 7 8. onsidr h funcion y() in h following figur y() If Y() is h ingral of y(), which of h figurs blow is closs o Y()? No ha all graphs hav h sam scal. Y() Y() () () Y() () Y() Y() () (5)

13 alculus and Diffrnial Equaions pag of 7 dy 9. Th soluion o h diffrnial quaion = is d y. y= +. y= +. = y + y= + y= + dy 0. Th soluion o h diffrnial quaion y d = is. y=. ln( ) = y + y=. y= ln( ) y= + Th soluion of h diffrnial quaion ydy = d is. y =. + y = = y. y = + y = y dy y Th soluion o h diffrnial quaion = is d. y=. y= +. y= y= ln + ln y= +

14 alculus and Diffrnial Equaions pag of 7 y dy. Th soluion o h diffrnial quaion = is d. y= ln +. y= ln +. y= + y= ln + y= + Th gnral soluion of diffrnial quaion dy= yd is a family of. Lins passing hrough h origin. ircls. Parabolas Hyprbolas Ellipss Th soluion curv of y'. y= y= y= = y ha passs hrough h poin (, ) is y = 0.06 y= Th implici soluion o h diffrnial quaion dy = is d cos y.. sin y = + y= +. y= + sin y = cos + y= sin +

15 alculus and Diffrnial Equaions pag 5 of 7 7. Suppos f ( k ) d whr k + k + = is consan, hn ( ) 0 k.. + k. + k k f d is qual o = and 0 = is 8. A funcion f ( ) which saisfis boh quaions f ( ) f ( ) f ( ) f = +. ( ). f ( ) =. f ( ) = f ( ) f ( ) = = + 9. Th acclraion of an lcron is givn by vcor a = i + j. A im = 0 h lcron has vlociy V = i. Wha is h vlociy vcor a anyim?. ( = ) + ( ) V i j. = ( ) V i. = ( ) V j ( = ) + ( ) V i j ( = ) + ( ) V i j 50. An organ dcomposs a a ra proporional o h amoun of h wigh of organ prsn. If h organ's wigh dcrass from 0gm o 0gm in hours, hn h consan of proporionaliy is. ln.. ln 5

16 alculus and Diffrnial Equaions pag 6 of 7 A cup of coff a mpraur 80 F is placd on a abl in a room a 68 F. Th diffrnial quaion for is dy mpraur a im is = 0.( y 68 ). If y (0) = 80, hn h mpraur (in F) of h coff afr 0 d minus is A ball is hrown vrically upward from h ground wih an iniial vlociy of 6 f/sc if h only forc considrd is ha aribud o h acclraion of graviy (which is f s ), find how high h ball will ris According o Nwon s Law of cooling, h mpraur of an objc dcrass a a ra proporional o h diffrnc bwn is mpraur and ha of h surrounding air. Suppos a corps a mpraur arrivs a a moruary whr h mpraur is kp a 0. If h corps cools o 7 in on hour, hn is mpraur is givn by h quaion. T = T = 7+ 0 T = + T = T = 0+ 5 Th compl soluion of h diffrnial quaion d y = + is d y= y= +. ( ). y= + + y= + y= + + 6

17 alculus and Diffrnial Equaions pag 7 of 7 5 If dy =, and y = 5 whn =, hn d 9 +. y= 9+ and y= y= y= 9+ y = 9+ y = 9 + End of Qusions on alculus and Diffrnial Equaions 7

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