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Unformatted text preview: Mathematics 105
April 2005 4
1. (10 marks) Let : r 2 . Find the average value of over the interval 0 g :1; g 2.
a i :1; 2. (10 marks) The demand (1 for a product is related to the unit price p by the demand
equation I) : D(q):10(1+ 4(1)“?
Find the consumer surplus if p : 2. Please simplify. Note that if p : 2 then (1 : 6. 3. (10 marks) Let A be the part of the ﬁrst quadrant that lies on or below the line y : 3 and
on or above the curve y : x/l +173. Use the trapezoidal rule with n : 2 to estimate the
area of A. A calclilator—ready answer is enough. 4. + 5 marks) The random variable X has probability density function where 0 elsewhere. . 4:574:53 if0§:1;§ 1;
1(1‘) * (a) Find the probability that X is less than or equal to 1/2. Please simplify.
(b) Find the mean (expected value) of X. Please simplify.
5. (8 + 2 marks) The Museum of Calculus is currently 50 meters from the edge of an eroding cliff. Geologists predict that at time t, the distance y, in meters, from the Museum to the
cliff edge will be changing at a rate given by 7 r
(1g 71208 0"”
(115 (1+ 4870502. Here If is measured in years, and t : 0 is now. (a) Find a formula for y in terms of t. (b) “ill the distance from the Museum to the cliff edge ever reach 0‘? Explain.
6. (8 + 2 marks) Suppose that y(0) : 1/5 and (121/ 2 72
— : ’1 (a /
(115 J for all t. (a) Find an explicit formula for y as a function of t. (b) Find Inﬁll/(t)- 7. (3 + 7 rnarks) Tho HighorPricos corporation (HP) has a lo :al monopoly on inkjot printors
and tho propriotary ink :artridgos for thorn. Tho solling prico y), in dollars, of a printor is connoctod to tho nurnbor :1; of printors
and tho nurnbor y of cartridgos sold por wook by tho oquation y) : 300 i Noto thoro
is no dopondonco on y shoppors do not think about tho cost of cartridgos whon buying a
printor. Tho cost to HP of a printor is $200. Tho solling prico 1](:1;,;1/), in dollars, of a cartridgo is connoctod to tho nurnbor :1; of printors
—‘. Tho cost and tho nurnbor y of cartridgos sold por wook by tho oquation 1](:1;,;1/) : 61 i
to HP of a :artridgo is $1. (a) Lot P(:1;,;1/) bo HP’s wookly profit if it solls :1; printors and y cartridgos. \Vrito down an
oxplicit oxprossion for P(:1;, y). (b) How rnany printors and how rnany :artridgos should HP soll to rnaxirnizo its wookly
profit? You nood not givo a justification that tho nurnbors calculatod produco tho 11111176111111”
profit. (10 rnarks) By ornploying :1; sorni—skillod workors and y skillod workors, a factory :an as—
sornblo (4:1;y + 1/2)” 2 custorn—built cornputors por hour. Tho factory pays oach sorni—skillod
workor $8 por hour, and oach skillod workor $20 por hour. Uso tho rnothod of Lagrango rnul—
tipliors (no crodit will bo givon for any othor rnothod) to dotorrnino tho rnaxirnurn nurnbor
of cornputors tho factory :an assornblo in an hour for a total labour cost of $720. Ploaso
sirnplify. You nood not show that tho answor you cornputo is actually tho 11111,:172111/11111. (2 + 8 rnarks) If ()2 7 40 > 0, thon tho oquation :1;2 + b:1; + c : 0 has two roal unoqual roots.
Lot f(b, c) bo tho largor of thoso roots. (a) \Vrito down a forrnula for f(b, c). (b) It is oasy to vorify that *1 is tho largor of tho two roots of :1;2 + + 4 : 0. Uso
a suitablo linoar approxirnation to f(b, c) to ostirnato tho largor of tho two roots of tho
oquation :1;2 + 5.03:1; + 4.06 : 0. (No crodit will bo givon for a rnothod that doos not uso
linoar approxirnation.) Ploaso sirnplify. 10. (10 rnarks) Taxation law allows a firrn to clairn a tax crodit for tho doprociation of an assot. ln straight line dcp’r'c1;’1i(1,t7i()’rl,, if tho assot is worth A at tirno t : 0, and is doprociatod
to r'aluo 0 at tirno T, thon tho tax crodit flows in according to a continuous incorno stroarn.
Tho rato of flow at tirno t is givon by : 2A(T i t)/T2 for 0 g t g T. This rnoans
that tho tax crodit gonoratod botwoon t and 15+ (115 is approxirnatoly (ft. Lot t bo rnoasurod in yoars, lot if : 0 bo tho prosont. and supposo that your cornpany owns
an oil woll now worth 100 (rnillion dollars). So A : 100. Supposo your cornpany is allowod
to doprociato tho woll to r'aluo 0 in 10 yoars, using straight lino doprociation. So T : 10.
Supposo also that tho provailing intorost rato is 5%., cornpoundod continuously. Find tho
prosont valuo of tho doprociation tax crodits for tho oil woll. ...
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This note was uploaded on 01/23/2011 for the course MATH 100,200,30 taught by Professor Dr.alejandrocortas during the Winter '10 term at The University of British Columbia.
- Winter '10