.- ~I J3 Professor Katiraie Calculus I Spring 2008 Test V Fonn A (chapters 1 -- 5) Name ----------- Total Possible Points = 140 Plus 10 Points Extra Credit 1) Find the equation ofthe tangent line to the curve y = 2xcosx, at the point (Jr, - 2Jr). y~ Z-CoS X - 2-)( Jin X (10 Points) ~ I== YY1.::= z,-co5 7T ~ 2IFJ1~';; ~ -2 o 2) A particle starts at the origin and moves along the parabola y = x 2 such that its distance from the origin increases at 5 units per second. How fast is its x-coordinate changing as it passes through the point (5, 25)? (X) j.. t) 1-1.. if ~D ~X +-X r (0) 0) D;; ~1:';-'~;-i" ;if)1t,: ; )(%[t- -t,i(-f '1 1) ~ J1~ (is:, (5) :::: 5 dx +- 2(5)3 ~. ;;Q. t ----,., dx _ s-j{:i<;).rv 0 5 Tn:t ~ - 2-5"S- - 0 j;z 3) The angle ofelevation ofthe Sun is decreasing at a rate of0.25 rad/hour. How fast is the shadow cast by a 400-foot-tall building increasing when the angle ofelevation ofthe Sun is 1r 6 ~;:- _O.~) ~ff~ 1.;
3) A plane flying horizontally at an altitude of 5 km and a speed of400 km/h passed directly over a radar station. Find the rate at which the distance from the plane to the J.Jf~ Jfoo~/HfCstationis increasing when it is 7 km away from the station. :PI'. 2 -'1.., fi~ X..f-J ;;; '.:..J 12x d,,: +- 0 tff' -JIJ/J1 tilj'"'t: lfiij(1=) ~7 ij -===3 dd..,- tfo-: fvf _ ~()O{[ ~;{J 4) If 2000 sq. cm ofmaterial is available to make a box with a square base and an open top, find the largest possible volume ofthe box. 2
=4@J 10) Given fl(x) = 5x-2, x> 0, f(2) = 3, f(4) = Find f(x) ~ ~~-~/t_-~:f+~~.-j - _:..J rl -I l' ('X).:: -, X +C ~ {X).::: - S- I~)( +c. 'X +D ~(1L)~1 -==9 ~ :=- --5"')",1.. + 2t:+-D A~ lc.-rd..:- ~+ s-ly\"l ~'1);.\) ~ 0 ;:: -$\li q. + lfc- +.0 ~~o... 1) ;= 51Y1 'f _L;t;t;::=~~;i:i 6-2-@;;I~~a~Z~J-C) A~O 11) If ff(x)dx=21, fg(x)dx=4, and'tg(x)dx=7 000 o a) Find the value of ff(x) *g(x)dx 3 ~~~ ~cl <>v- ~ O;v.e.- 't~ V" I'~./ :,"'" ~::... o b) Find the value of f(f(x) - g(x))dx 3 0 ); f (){)J 1'- - J~ 9(>0 Jy.. -7.( - -7 ~-'l(+7 3
~K~ 1--3. =~ i 'f 12) Given the function j(x)=1+x 2, -3~x~1 Estimate the area under the graph off{x) using 4 (hint: n = 4)approximating rectangles and taking the sample points to be: a~ht end~draw the appropriate rectangles and find the area) b) Midpoints (Draw the appropriate rectangles and find the area). \ f)v-~( Ct-t-",:)t,l-.J -3+L 1- -k ) J :c 6J b a 13) Given Jxdx = 30 and J5dx = -30 Find a and b h a b 0<- ;=03 0 5:"J1-=3~ ~ - -30 b~_ ().. 1, ~ bo qb-)~ ~ 30
14) Water flows from the bottom ofa storage tank at a rate of liters. ret) = 200-4t., Where 0 :::;; t :::;; 50 Ill1nutes mmute a) Find the amount ofwater that flows from the tank initially (at time t = 0). ~: (2-oo-lIt;) df ;-0 ~ b) Find the amount ofwater that flows from the tank during the first 25 minutes. ~S ~ \~f ) (~oo~tt)j+ ~ 2txJ-t-~ 0 o ===" 0-OQL7» _ z{u)j -0) 15) Find the area enclosed by the following curves: y =2x+x 2 and y=2x+9 2 Xf-y..\' ~ ZX*~ X1.:;-4 e ~X+V -(U+f'lj J~ 'J-;-;t1 ~ :=c Ls fj-)("-jd.t-. )-3 3 /1 = 9X-} -3 3\ =:: ~(1)- ;1)_ (?(-v - '4l--l ~ 16) Let!(X)=[l,/IdfJ+100 I~ _ ~ \'6'/':--3C] ~ '1.. Find the value of f' (10000) f(j<)~ LS2X.u:t.lJ +/()\l t: '0 ~ f/{x)--= & JZX)C2-j +,10 f/( (0000 --= _1-(200.",]
17) The velocity ofa particle moving along a line is (2-3( - 4 meters per second. Find the acceleration ofthe particle when the velocity ofthe particle is zero. D ~ 18) Find the value ofthe integral f 3x- - 5 dx c x (Assume C >0 and D > 0, and leave your answer in terms ofe and D) D L ~.:= D )IY\~~-=- (~C _ 11-- si" C) 1 F -~~ ~IC -:)I~.D -- ~ C ~ + S1n C-j 19) Determine by differen~ whether the fo.11mn.ing formulais.true..or.false._...,.' (Must Show Procedure) J du - 1 1 2u +a C'" --=-~-- n u 2 +a 2 2a u-a + - -~ ---,;;;,-.-- Z~+~(y.-o) -'JOe. -~... _ \f-o) 2.. \ t1~a.-... ~:.-.,.......,_,,6
Bonus Question: 5x 20) Let!(x)=l fu+2 du 2 u-l 2x 2-+.(-1-) == - 3 -z;.. 21) A closed box with square base is to be built to house an ant colony. The bottom and /{ top ofthe box will be made ofmaterial costing $1 per square foot, and all four sides are to be constructed of glass costing $5 per square foot. What are the dimensions ofthe box J, ;ofgreatest volume that can be constructed for $65? JrROUlld your answers to two decimal places) / "c"l ' t ~11 ~1.x1!1)('l+P'5'{fX)-)~P65 'f ~bj- 'tx 1. 'lx1- +- 2, X t ::;;-: GJ ~:;;" -?.fj X - ~ -\ ~..=: 3.tf X - o. (X I/~~ x~~, V:; X1. (3.2fi l _ D,IXJ V :;;:- I 3 3.. Z~X - 011 X ~.:; 'J.Z! -0.1 x'z- = 0