Math Final Review. 1. Match the following functions with the given graphs without using your calculator: f 5 (x) = 5x3 25 x.

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1 Mat 5 Final Review. Matc te following functions wit te given graps witout using our calculator: f () = /3 f () = /3 f 3 () = 4 5 (A) f 4 () = (B) f 5 () = (C) (D) f 6 () = (E) (Ans: A, B, C, D, E, F) (F). Matc te graps to te given quadratic functions. Some graps are redundant. f () = ( 5) + f () = a( 3) + (a < ) f 3 () = b( + 3) + (b > ) f 4 () = ( + 5) + (Ans: 3, 4, 6, ) () = 5 () (3, ) (3) (5, ) (4) (3, ) (5) = 3 = 3 (6) 3. Find te inverse of te following functions and give its domain. a. f() = e (+5) b. g() = 5 ln() + 3 c. () = 3 + (Ans: = (5 + ln )/; >, =.5e ( 3)/5 ; < <, = ( + 3)/( ) ) 4. Wen te price p for a trip to te Baamas on a cruise sip is $8, te demand is, tickets annuall. However, wen te price drops to $4 per ticket ten te demand rises to, tickets. On te cost side, te cruise liner as a $,5, fied cost and $5 epenses per passenger. (i) Assuming te demand function p() is linear, find a formula for it. (Ans: p =.5 + 9) (ii) Find te cost function C(). (Ans: C = 5 +, 5, ) (iii) Find te revenue function R(). (Ans: R =.5 + 9) (iv) Find te profit function. (Ans: P = , 5, ) (v) B complete te square in te profit function, at wat level sould te compan set te price in order to maimize te profit? Wat is te maimal profit? (Ans: p = $475; $,,5) 5. Te eigt of a ball trown upwards from te top of a ft tall tower wit initial velocit 6 feet per second is given b te formula s(t) = 6t + 6t +. Assume tat te position of te ball is measured from te base of te building. (i) Complete te square in order to write s(t) in te form s(t) = a(t ) + k. (Ans: s(t) = 6(t 5) + 6) (ii) At wat time does te ball reac its maimum eigt and wat is tis eigt? (Ans: t = 5 = 6) (iii) At wat time does te ball it te ground? Wat is velocit at tat moment? (Ans: t = 5 v = 3)

2 6. An landscaper as 3 ft of fencing and wises to enclosed a five sided figure as sown. Find an epression for te area A enclosed in terms of. B completing te square, find te values of and tat maimizes te area A. Wat is te maimum area tat can be enclosed? (Ans: = 4ft, = 6ft, ma A = 6ft ) Te demand function D and suppl function S of a model of jeans for te cloting compan Luck Clover s Wear are given, in term of price p, b D(p) = e 4 p S(p) = e p+ a. Give a sketc of te demand and suppl functions in te aes provided. b. Find te equilibrium price and equilibrium quantit. p (Ans: p e =, q e = e ) 8. Imagine tat ou are offered two retirement plans. (a) Wen ou are 3 ears old $8, is deposited in to an IRA, earning an annual interest rate of 8% compounded continuousl, till ou become 65 ears old. (Ans: 8e.8(35) ) (b) Wen ou are 65 ou receive $,5,. Wic one will ou coose? 9. Suppose te alf-life of a radioactive substance is 5 ears. How long will it take for te substance to be reduced to % of its initial amount. t = 5 ln 5/ ln. Wat sould te annual interest rate (compounded continuousl) be in order an amount of mone earning tis rate doubles in ears?. Consider te rational function + 8 were. (i) To make a guess about te value of lim fill te following table: + 8 (ii) Use algebra to compute te eact value of te limit lim + 8, if (iii) Let f() = Use limits to determine weter f() is continuous?, if =. (Ans: 6). Te grap of te function f() is given in figure below. Find eactl or state tat it does not eist eac of te following quantities. If it does not eist eplain w.

3 (a) lim f()? = (b) lim f()? = (c) lim f()? = (d) lim f()? = f() f() (e) lim? = (f) lim (Ans:,, does not eist,,, does not eist) f( + ) +? = 3 4 f( + ) f() 3. For eac of te following function f() form te difference quotient, wic is also called te average of f from to +, and simplif it so tat te quantit in te denominator is canceled. (i) f() = 5 + (ii) f() = 3/ (iii) f() = 4. Let p() = ( )g(), q() = 3 5 +, and r() = g( ). Given tat g( ) =, g() g() = 3, g ( 3) = and g () = 4 find te values of p (), q (), and r (). (Ans: p () = 6, q () =, r () = ) 5. A block of ice wit a square base as dimension inces b inces b 3 inces. If te block of ice is melting so tat its surface area A is decreasing at a rate of in /sec, find te rate at wic is canging wen = inces. (Ans: = 68 ) 6. Te carbon dioide emissions of a small countr in Januar of 6 is 8% of its emission level in 996, and is reducing at a rate of.5% per ear. Let t be te time in ears measured from Januar of 996. (a) Use linear approimation to find an epression for te percentage P (t) of te emission level of te countr in te near future of 6, (b) use our answer in (a) to estimate te percentage of carbon dioide emission in June of Te grap of te derivative of f() is as sown. (a) Wat are te critical points of f()? (b) Find te values of for wic f() is (i) increasing, and (ii) decreasing. (c) Wat are te inflection points of f(). Determine te concavit of f()? (Ans: CP: =, = 3, Increasing: < < 3, Decreasing: <, > 3, Inflection pts: =,, ) f () 8. (a) Sketc te grap of f given tat: f()= f () =, f () < for < and f () > for >. f (4) =, f () > for < 4 and f () < for > 4. 3 (b) Determine were g() = e f() is () increasing, and () decreasing. Decreasing: <, Increasing > P 9. Te grap of te profit P () (in million of dollars) from te sales of a cemical substance ( is te amount sold in million of gallons) is sown in te 5 figure. a. Find te marginal profit at = (Ans: 3 ) b. Find te linear approimation of P () at =. (Ans: L() = 3 ) c. Estimate te profit wen =.5 (Ans: L(.5) =.75) 3. Find te eact value of (i) lim (3 + ) (Hint: Tink derivatives!) (ii) lim ln(8 + ) ln 8 (iii) lim e 4+ e 4 (Ans: (i)6(3) 5 (ii) 8 (iii) e4 ) 3

4 . For eac of te following functions: (i) f() = + 3 (ii) f() = 6 (iii) f() = + 3 (Ans: (i) VA: 3 HA: (ii) VA: ±, HA: (iii) VA: 3, HA: ) find vertical asmptote(s), orizontal asmptote(s), -intercept and its zero(s). Ten, sketc te grap.. Let g be a differentiable function suc tat its derivative is given b g () = ( 5)( ). Find were te function g as a local maimum and a local minimum, if an. (Ans: Min = 5, Ma = ) 3. Differentiate / wit respect to, and use te result to find an approimate value for /.5. (Ans: L(.5) = 4 ) 4. Te following table gives te values of te continuous function f(). For wic intervals on te -ais could ou be sure tat f() as a zero. 3 f() In economics, a utilit function u assigns u() units of satisfaction (utiles) to units of consumption. It is required to satisf te conditions: a) u () > (te more te consumption te more te satisfaction) b) u () < (eac additional unit of consumption gives less satisfaction) Sow tat if γ > and γ, ten u() = γ, is a utilit function. γ 6. For te function f() = find te critical points, local maima/minima, were it is increasing/decreasing and were is concave up/down. Also, find its maimum/minimum in te interval [, 5]. (Ans: CP:, 4, Increasing: < < 4, Decreasing: > 4, <, Concave up: < 6, Concave down: > 6) 7. For te function f() = e find its local maima/minima, were it is increasing/decreasing and were is concave up/down. Also, find its maimum/minimum in te interval [, ). Finall, sketc its grap utilizing its smmetr and its asmptotes. (Ans: f () = e 3 e, f () = e e e ) 8. A clindrical can witout a top is to be made to contain 88π ft 3 of liquid. Material for te bottom (no top) costs $6 per ft and material for te sides costs $ per ft. Find te dimensions tat will minimized te cost of te metal to make te can. (Ans: r = 6, = 8) 9. A Norman window is one wit a semi-circular portion mounted (eactl) on one side of a rectangle. If te perimeter of suc a window is 3 ft, wat are te dimensions tat maimizes te amount of ligt tat can enter te window? (Ans: r = π+4 3, = 5 6 π+4 π+4 5π ) 3. A man launces is boat from point B on a bank of a straigt river 4 km wide. He wants to reac point C, km downstream on te opposite bank as quickl as possible. Point D is directl across te river from point B. He will row is boat to point P between points C and D. If e can row at 6 km/r and run at km/r, were sould e land in order to reac point B as soon as possible? (Ans: 3 km down from D) 3. B using implicit differentiation, find an epression for d d were 4 + e = Find also te equation of te tangent line at (, ). (Ans: = 9 + 9) 3. In an econom, te capital per worker k (in dollars), and its output per worker q are related b te formula q = 48k /3. Currentl k = 8 dollars and it is canging at te rate of dollars per ear. Find te rate at wic te output is canging. (Ans: 4,) 4

5 33. Te grap of te velocit v(t) of a particle moving on a orizontal straigt line is given below. Let s(t) meters be te displacement of te particle after time t minutes. Assume tat s() =. Find te eact value of te following quantities and epress eac of tem as a definite integral. a. Te position of te particle after minutes. b. Total cange in position for te duration t 4. c. Total distance covered b te particle. (Ans: (a) ft, (b) /4 ft, (c) 3 ft) Let f() and g() be functions as sown below. Find te definite integral b a [f() g()]d. Compare tis answer to te area between te grap of f() and g() over [a, b]. (Ans:, Area = 6) Area= 4 = f() b a = g() Area= 3 Area= Te marginal cost of a product is given b MC() = 4 3 were is te number of items produced in units of millions. Find te total cange in te cost (in dollars) if production level canges from million to 4 million. (Ans: $39.75) 36. Te demand function D (in units of tousand) of a line of jeans D(p) = were p is te price in p + dollars. Sketc a grap sowing te function and te average demand for te jeans for te price level between $5 and $ Find te area enclosed between te grap of f() = ( ) and te -ais. (Ans: /3) 38. Mid point rule and trapezoidal rule wit n = 5 to compute te value of ln d. Find te eact value of te definite integral and ceck our answers. (Ans: ln ) 39. Evaluate te following indefinite integrals: (ln ) a. (e e + e + 3/ ) d d. d g. b. ( + ) 3/ d e. ( ) 5 d. 4 c. d f. + d i. 3 e 3 d ln d 6 d (Ans: a. e + e + e + 5 5/ + C b. 5 ( + ) 5/ + C c. 5 ( ) + C d. (ln ) 3 / + C e. f. 4sqrt/3 5 5/ g. 9 e 3 (3 ) + C. 9 ln 3 6/9 i. 5 ln( + 3) 5 ln( ) + C ) 4. Te rate of cange r() of te intensit of pollution on a 5 mile straigt road between two factories A and B is given b r() = were < 5 is te distance from A in miles. Wat is te total 5 cange in intensit of pollution eperienced b a man walking on te road from a point two miles from A to a point two miles from B. (Ans: ln 44 ln 49) 4. Compute d dt t 9 e.5 d. (Ans: t 9 e.5t ) 4. Te instantaneous rate of cange of a quantit Q(t) is given b dq dt = te t. Compute te total cange of tis quantit from t = to t =. (Ans: 5 5/e) 5

6 43. Sketc te grap of F () = f(t) dt if grap of f(t) is as sow below. Indicate clearl te concavit of F () and were it is increasing and decreasing. = f() Solve te initial value problem d dt = te.t, () =. (Ans: te.t e.t + ) 45. A function f() on an interval [a, b] is a probabilit densit function (pdf) if f() and b a f()d =. Find c suc tat f() = c 4 is a pdf on [, ]. Do te same for f() = ce. on [, ], and for f() = ce.5 on [, ]. (Ans: 5/, /( e ), /( e 5 )) 46. Suppose ou deposit $5, in an account paing 3% annual interest compounded continuousl, and do not make an furter deposit or witdrawals. Find te average amount of mone in te account during te first 5 ears. (Ans: 683.4) 47. Find te equilibrium quantit and price, consumer surplus and producer surplus for te demand curve D(q) = (q ) + and te suppl curve S(q) = q + for q. (Ans: q e = 5, p e = 6, CS = 5 3, P S = 5 3 ) 48. Wic of te following savings accounts ields te most after ten ears: (a) For te net ears, mone is deposited continuousl into Account A wic pas 6% interest compounded continuousl, at a rate of + t dollars per ear. (Ans: FV a=$39,743.89) (b) An account B wit principal $5, paing interest at 5% compounded montl. (Ans: FV b =$4,75.4) (c) On te first of Januar tis ear and for te net four ears (no more after), $6 is deposited into Account C earning 4% annual interest compounded continuousl. (Ans: FV c=$4,379.97) 49. If a continuous income stream flows into a saving account at a constant rate of $, per ear and earns 7% interest, compounded continuousl, find te time required for te balance to become $,,. (Ans: ln 8.7 ) 5. A retired person as $ million in an IRA paing an annual interest rate of 5%, compounded continuousl. Over te net ears se plans to witdraw mone continuousl at te constant rate of S dollars per ear. Find te value of S so tat tere is $ million left in er account at te end of ears. ( Ans: S =.5( e ) e or.e.5 million) e 5. Define te derivative of a function f() at and provide te different names tat people call it. 5. (a) Define te definite integral of a function f() over an interval [a, b] and provide its relation to te area under te grap of f(). (b) Find 7 d. (Ans: ) (c) Find te area between f() = 7,, and te -ais. (Ans: 4 ) 53. Describe te teorem tat connects integration and differentiation. 54. How do we compute te present and future value of an income stream? 55. Wat is calculus good for? 6