\uu t^)t\\ be g,$erl f'recetsoqg fi*otn**fg OS ngdgt ry' tr\r*'u) ' Relative Rate of. A Y Functtn. (le,t) (tr.r) A = Pert

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1 ef**ftkae n6+ eary*re*' on &tus&' *r6gurrui ld,bl lo.(g, to.-il ll,lr ll, (.or ll'tl l3,t{ t 14,, l5,b Math 202: Calculus for Business and Economics. Fall2013 Final Exam ' Friday, December L3,201,3 Name: Student ld: Math 202 Section- Version A Directions. Read all directions carefully.. You must show all work or justify your reasoning with meaningful evidence to receive full credit, regardless of whether it is explicitly asked for in the question.. Simplif,, completely wherever possible.. Cleariy indicate your final answer for each problem.. Label units and problems appropriately.. The back page is left blank for you in case you need more space to work through a problem. Please indicate on the exam if you tr;ge used the back page for a problem. Average Value of a Continuous- Forrnulas -@ auu.agerateof change (ro,q\ A Y Functtn fromx--atox-a+his L lo-.- f(a+h)-f(a) h=+^ b,-a),ll*)o* h ' ILrv (le,t) A....\ ' Cini Index of Income ( t{. O). Q lnstantaneous Rate of Change (to,h) Concenrration (6e1-gnSrUS\ atx = a is f(a+h)-f(a) Gini. rnd"ex =, Ir'W - f e)l d.x iim h+0 Con tinuous Compo undlnlerqlt Formuia (nof on &ml-\ (tr.r) A = Pert ' Relative Rate of (tr,r\ t??it;.**$*$. Elasticity of Demand (no,trrf A*$ (ll.r) E(p) = pf '(p) f (p) Error Bounds for Approximation of Area by Left or Right Sums error < lf(b) - (r?.{) (nn+ f (a)l ' b-a tr\r*'u) Future Value of a Continuous ^ ^\ income stream (tq, e) Crmtmf,noff fv :," lo' f(t)e-'t dt consumers' Surpius ( rl,o) Cno+u-S"sl) fr CS: I lo(*)-p)dx Jo pro ducers' surptus (t{,e) Cvro+ orr$ref) P.s = lo'rp - s(x)) dx Integration by Parts Formula (f q.e) " Iudu:*"- f uau-(tryt***$ ++ \uu t^)t\\ be g,$erl f'recetsoqg fi*otn**fg OS ngdgt ry'

2 : t, [5 points eachj Find the indicated derivatives. ll,e,ll,tla'f'(*)torf(x)-(3x'+!)(x-2)' +'(x) = (a*'*') (af*'o 6r)+text'rsf ll, b b. s'(x) for s(x): H qj(,ry (x+65 ll,'-+ c. h" (r) for h(x) -,x2+4x-3 ht(x)' af+tr" (ar++) N.1,A, rx,+f*o/o) + {ru*'(e**f

3 2. [6 points each] Evaluate the iimits. a ar, -Aitr 9o,* rrd ^ L %l xsoo 4* ASg.+=Do b. ai -\z \ ro. ty!# a (x+o\&:r)'+irnxre Q.g1 I,, T = l+& =$ ^ar ro.d, ro.b r- -:.- -E: =OIT^ r.!iq -1 = ieta Lg Cg ra.a d m#,k[,\= t F r rnx 0r [la 5O,* )i$'rnxa I x.r, I x'tl J

4 3. lo, { to. ll{"& c, (2 points each] Write TRUE if the statement is correct aud FALSE if the statement is incorrect, ll,'7 e. 6.,ttg The derivative can be interpreted as the average rate of change ^, for any gineffdnction, - b. fr, A function is concave upwards on the intervalfsj where f "(x) > o. A Gini index of 1 indiclles ab^sg]ute equality - all people share equally in the income. incon:. (not (no 6yr m t{ngll) Si,ts.L' lo,3 r, \ xz+x. ---.Lid, f (*) = T ir continuous at every value or*.(l#,"n$t1al) when demand is elas[ip, a price decrease will cause an increase * **r* CtlO+ On $t1g1!) 18, I 1' fnfr Thecriricalvaluesof afunctionf(x)occurswhere f '(*)= 0or where f '(*) is undefined, I.5 g. 6StP, tb, I h fn*, I A. &,,. S,f.. The absolure maximum or minimum of a function can only occur at endpoints of a closed interval. A function Fis an antid.erivatlve of a function/if F'(x) : f (x), The value of x where rhe rate of change of sales goes from increasing to decreasing is called the point of diminishing returns' l{.& I. le, I k. 14,o I lb.3 m. l. & ". The consumers' surplus represents the total savings to consumers who are willing to pay more than $p for $e product but are still able to buy tire product for $p. (norf on {ina.ll) f ft fi f '(x) = 2x,thenthere is exactly one anticienvaflve, a I lx) = x'. Given a probability density function/[xj, the probabiiity that a product will last between a and b years can be represented by the integral, ff-g [," it, o, (nof on Stra$).. { Dr, Z*( ",:* An inflecrion point on a graph of a funcrion is the point where positive to negative or the slope ofthe tangent Iines to the graph change from from negative io positive. i:,n: 7t({f -bo*.ag -N=b

5 la, lo.6 +. [5 points eachi Evaluate the integrals. a. [."., d., t""'otl-& _i r2r, _ru ", dtt"rwgdl J-rodlrt t"d* ra,b l"'(--, t sex - =)o- -ro =4 +b?*-3]:, e +*,*-*], -1.,Fe+d-6to\] -f ar'rry" -sal jt;+g;; :*.e -- rf- 6 e-- & It.3 C.c lo'*'r* o* (no{ Dn n t',o.$)

6 lq,b d, fxmx ax (nc't on Snaf\ I6.& 16,5 {"W)'o, Wr$nt, Aog lf,tj""j I J" t d;'{i* i* usr$no3z5 'S't ['L'odu"? q1? Ar ' - E'q'iui'*-L= ry It, t f. / (zx'-x-t)dx, 3ple9- *)dr 6

7 l,5 5. [3 points eachj Circle the most appropriate answer lo the following questions given/(x) : 2x3 I 3xz - l.zxdefined on the closed interval l-3,21. a. What is the critical value of the function f (x)? f t*), 6(]+1,ff- t i. x:0,2,_3,1p(xd+f!a O) el =r?:i, 's ^f'&);tz.*g)^g') f (r)'os '(=- r t('i b. What is the absoiute maximum point of the function on l-3, 2]?,-.. i r-3,-e) *G3)=aCtf+cC F r-z zoi iii. (0, 0) = _E*+O'? +e(9 t Y :" i ^ -1 S(-e)= ) d,o v iz,+t' it,5-, -r c. What is the absolute minimum point of the function on [-3,2]? i, (-3, -9) ii. (-2,20) iii. v, (2, +) *(5t)={ te,'[ [3 points eachj.1 ^. - 1., llx) d.x: -b, A l f(x\dx Jl ) \ ', Suppose the integrals are as defined below, :,o, dx : z, tr']/*f, olotffs' {i f (o a*.l c. J"'(Zg(x) - f (x)) dx = d i,'(:oi,ry" = e. titzf frl - 3s(x)) d"x:

8 Ir{, I 7. Find the area bounded by the graphs olthe given equa[ions over the given interval.!=x2-1, y=-x2-1-, -2<x<2 lq. & (,nof *a*,) 5,f,, ;5.'rPe+o)dr q+o{. 'ely + d(o)) - f*/-'f +eco)) = E*1*er#.l= B+B= ++ =% B. The price - demand and price - supply equation of a product are given by P : D t3;,:.?; " iiiol;:h',':'.1#;uantity and price b. [5 pointsj Calculate the consumers' surplus at the equilibrium price leve1. c. [5 points] Calcr-rlate the producers' surplus at lhe equilibrium price level, o

9 la,+ g. Summarize the pertinent information obtained by applying the graphing strategy to analyze and sketch the function x-2 y., fr'.'\ - I \^) x - 3 a. [2 pointsj Domain orl(r), (-Do,?) (J b. [2 pointsj x - intercept[sj: r = & 4, c. [2pointsJ Y-intercept:Y: E 'F(o)= A'9O d. [2 points) Vertica] asymptote[.], X 7 3 (A,06) f '(*\=w f'(r)-},3- YO C''# f'(*\"! (*-# f '(x)'-1x-# e. [2 pointsj H orizo ntal asymptote [sj : f'(*\=o ne UNtrun {'(r) Dt.E Xob t. (2 PointsJ f '/o)" /(x) is increasing on the interval: $t*) - : --, o i,?.'-' ) -l 4aP q f'(s)= * 4

10 g. [2 pointsj f (x) is decreasing on the interval: C' f"(xfo i i *,,,-+,,,'b T ) f "(x) --*8(x-o5e *f;t h, [2 points] /(x) is concave upward on the interval: + 3,. (2 pointsj /(x) is concave dornmrarard on the interval: (3, oo) f -g -*-91O'* "/o\ = -di -ga13 8 j (2 pointsl Does /(x) have an inflection point? ICircle answer] r.tr) o k. [6 pointsj Sketch the function. t t i --r,- I I t'')r-,'rlfi -i. ::=1"1 (pro) fl'l)=+z& +(u)= *. -t t i ""- i - xro 10