FUNCTIONS (11 UNIVERSITY)
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1 FINAL EXAM REVIEW FOR MCR U FUNCTIONS ( UNIVERSITY) Overall Remiders: To study for your eam your should redo all your past tests ad quizzes Write out all the formulas i the course to help you remember them for the eam. Remember to brig your borrowed tet book with you to the eam.. Simplify. EXAM REVIEW: Ratioal Epressios Factorig (commo, differece of squares, triomial) Simplifyig Ratioal Epressios à Multiplicatio/ Divisio à Additio/ Subtractio Restrictios ( ) ( ) b) y y( y) y ( y ) + d) ( a+ b) ( a b)( a+ b) c) ( z )( z z ) + 5 f)( + ) ( ) ( ) ( + ) e) ( ). Simplify ad state ay restrictios. 8pq p q t + b) t + 6t c) d) + e) y 6y f) y 0 t t t t t t g) h) 8 ( y) y y i) j) a a a a a a a a k) a + 6ab+ 8b a + 5ab 6b a + 8ab+ b a + 9ab+ 0b a ab 6b a + ab 5b 5 l) + y + y 6 ) m m m + m m) a o) + b a b a b + a ab + b b
2 EXAM REVIEW: Quadratics Solvig quadratic equatios à By factorig where possible à Usig Quadratic Formula whe factorig is ot possible Simplifyig Radicals Types of solutios (Nature of the Roots). Simplify. 5 b) c) + 7 d) g) e) ( 6 + ) f) ( )( 5 + ) h) 5 i) j) 0 k) +. Solve each of the followig usig the most efficiet method. C (No decimals) e) y 7y = b) y y = 0 c) = f) 6 5 = g) 7 = 0 d) 7 + = h) = + 8 = 0 i) = j) = k) = Solve each of the followig iequalities. + b) ( m ) ( m). Fid the value(s) of k so that each fuctio has: i. two -itercepts ii. oe -itercept iii. o -itercepts f( ) = k 9+ 6 b) f( ) k 5> + c) = + + c) ( ) f = k q+ 5+ q 5. Determie the ature of the roots without solvig the equatio. (Type ad umber of solutios.) 5+ 7= 0 b) 5 + = 0 c) 0+ 5= 0
3 EXAM REVIEW: Trigoometry Evaluate Epressios Solvig Trigoometric Equatios à Special Triagles à Cast Rule à Factorig à Usig Radias Prove Idetities Eact Values. Evaluate the followig: csc70. o 7 π b) sec. If secθ =, make a diagram ad fid the other five ratios, if θ is acute. 5. If ad the approimate value of θ, cot π c) ( ) cotθ = ad θ is obtuse, fid the eact values of the other five ratios. Determie the eact value. siπ b) e) 7π cos 6 c) 5π ta 5π π 5π π cos cos si si f) 5sec0 ta If cscθ =, fid all possible values of 0 θ 60. d) π π π 7π si cos + ta ta 6 6. Solve each of the followig for 0 π. Where ecessary, roud to oe decimal place. si ( cos + ) = 0 b) cos = 0 c) cot θ = cotθ d) si ta ta 0 = e) cscθ = f) 5si 7si = g) j) cos si cos 0 6si cos 5 0 = h) + = k) si cos si = i) 6sec θ + 5secθ = 0 cos si = 0 6. Prove the followig idetities. ( )( ) si + ta = b) si + ta + cos ta = cos ta + ta c) e) g) + si cos si θ si θ = cos θ cos θ d) + = cos + si cos ta θ ta θ + = si θ ta cos si cos f) = + si cos + cos + + = h) secθcosθ + secθsiθ = + taθ
4 EXAM REVIEW: Trasformatios of Fuctios Domai / Rage Amplitude, Period, Phase Shift, Vertical Traslatio, Reflectio (Vertical ad Horizotal) y asib d c y = acosb d + c ) Graphig Trigoometric Fuctios ( = ( ) + or ( ) Mappig Notatio: (, y ) + d, ay + c b Fuctio Notatio Iverse Fuctio y - or f () Eplaatio of a trasformatio / Write a equatio uder give trasformatio. State the domai, rage, period, amplitude, vertical traslatio ad phase shift of each of the followig: y = cos b) y = si c) y = si d) y = si π e) π y = cos ( 6 π ) 5 f) y = 5cos +. Sketch two cycles of each fuctio i #.. Describe how the graph of each of the followig fuctios ca be obtaied from the graph y = f. ( ) y = 5f( ) + b) y f( ) ( ) = + c) ( ) y = f + y = f 6 9 d) y = f 7( 5) + e) y = f 8( + 7) f) ( ). Write the trasformed equatio if y f( ) = udergoes the followig trasformatio: It is reflected across the y-ais, stretched vertically by a factor of, compressed horizotally by a factor of ¼, shifted uits dow ad traslated uits to the left. f = 5+, fid: 5. If ( ) f ( 0) b) f ( ) c) f( ) d) whe f ( ) = 0 6. State the domai ad rage of each of the followig: y 6 = + b) ( ) y = + 5 c) y = Give f( ) = + 8, determie: f ( ) b) whe f( ) = c) f ( 6) d) whe f( ) = f ( ) 8. Determie the iverse of each of the followig fuctios: f( ) = + b) g( ) = + 8 c) h( ) = + d) k( ) 9. If f( ) = ad g( ) =, determie: f g( ) b) g f( ) c) y whe ( ) ( ) ( ) f cos y = 0 ad 0 y π = 7 + 5
5 EXAM REVIEW: Epoetial Fuctios Simplifyig usig epoet rules (i power form, radical form, ad ratioal form) Solvig epoetial equatios Epoetial Growth ad Decay. Evaluate. (No decimals) ( ) b) c) 7 d) 6 5 e) 8. Simplify: (leave your aswer i power form) ( ) + y a b ab + y + y 7 ( ) ( ) b) a a a c) g y a+ b a+ b ab. Simplify, ad the evaluate for a = -, b = :. Prove: a a+ a+ ( )( )( ) 7 a = ab ( 5 ab) (5 ab ) 5a b 6 5. Simplify each of the followig. Write the aswer usig positive epoets. p + p b) Evaluate (without calculators): Solve. C 6 = b) 6 0 c) m m b) ( ) 0. c) ( ) = c) f) d) ( )( + ) + 9 = d) = 6 e) h) + 0 = 8 6( ) 7 0 = * i) f) ( )( ) = 6 g) ( )( ) ( 6 ) 58 = = 8. I 976, a research hospital bought half of a gram of radium for cacer research. Assumig the hospital still eists, how much of this radium will the hospital have i the year 686, if the half life of the radium is 60 years? 9. There are 50 bacteria preset iitially i a culture. I mi., the cout is What is the doublig period? 0. I a recet dig, a huma skeleto was uearthed. It was later foud that the amout of C i it had decayed to ( 8 ) how old was the skeleto? of its origial amout. If C has a half life of 5760 years, 5
6 Arithmetic Sequeces / Series EXAM REVIEW: Sequeces ad Series = + ( ) = + ( ) t a d Geometric Sequece / Series t = ar Recursio Formula Pascal s Triagle s a d S ( ) a r = r. Fid t ad t for each of the followig sequeces: 9, 5,, b),,, 8,. Fid the sum of each of the followig series: b) c) d) Write the first term of each of the followig sequeces: t I = 6; t = t +5; > b) t = ; t = ; t = t t. I a arithmetic sequece, the rd term is 5 ad the 9 th term is. How may terms are less tha 00? 5. The sum of the first 6 terms is 97 ad the sum of the first 8 terms is 500. Determie the 5 th term if the sequece is arithmetic. 6. Isert terms betwee 6 ad 7776 so that you create a geometric sequece. 7. Isert terms betwee ad 97 so that you create a arithmetic sequece. 8. For ( ) : fid t b) how may terms are i the epasio? c) eplai where the umerical coefficiets of the epasio are comig from 8. Epad + b) Evaluate the sum. usig biomial theorem; take the coefficiets from Pascal triagle = 9. Epad (Leave your fial aswer i power form): z y 5 b) ( ) d d 6
7 Compoud Iterest: A= P( + i) R ( + i) Auity: A = i R + i Preset Value Auity: P = i EXAM REVIEW: Fiace ( ). I order to have $750 for tuitio i a year ad a half, you pla to deposit moey ito a savigs accout every three moths. If that accout pays 5.% compouded quarterly, what will your regular deposits be? b) how much iterest would you have eared?. Leo s parets wat him to be able to withdraw a allowace of $700 every 6 moths. How much should they ivest ito a accout today that pays 8.5% compouded semi-aually so that Leo ca withdraw his allowace for the et 5 years?. Determie: the amout $8500 will grow to if ivested at 8% compouded quarterly for te years. b) the pricipal that must be ivested ow at 6% compouded aually to be worth $0 000 i 5 years. c) The total accumulated amout of $500 ivested every moth at 7% compouded mothly for 8 years.. How log will it take, to the earest year, for $800 to grow to $000 i a accout that pays.5% aually compouded semi-aually? 5. If a savigs accout compouds daily, what would the aual iterest rate eed to be for $500 to grow to $650 i years? 7
8 MCR U Eam Review. Simplify ad epress with positive epoets:. Simplify: a b c a b c 5. Simplify: ( 6)( 6 + ). Simplify: If = y the y m = for what value of m? 6. Fid the lowest commo multiple (stated i factored form is acceptable) for the followig epressios: ( + + ) ad( + + ) ad( ) 7. A rectagular prism has the sides measurig, ad + cetimeters. Create a simplified equatio to represet R, the ratio of the surface area to volume. State ay logical restrictios o. a + 6a+ 8 a + a 6 8. Simplify ad state ay restrictios o a: a + a a 9 9. State the domai ad rage (i good math form) of: y = ( + ) + 0. Give that 5 ad - are the roots of a parabola ad P(, ) is a poit o the parabola, what is the equatio that models this, i stadard form?. Fid the solutio(s)/poit(s) of itersectio of the followig quadratic-liear system. Show/label steps appropriately: y + ² = 6 ad + y = 0. Solve usig both a algebraic ad graphical method.. Solve: 8= 0 usig both a algebraic ad graphical method.. For the followig equatio, describe ay ad all trasformatios for compared to y = ² y = + ( 5). Curret price is $00, curret sales are 5000 uits. For every icrease i price of $, sales decrease by 0 uits. Fid optimum reveue. 8
9 5. Solve for p, give: 6p = 0+ p 6. Height of a object i metres, based o time i secods, is give as: h=.9t + t+ 7. Determie time of ladig of object, maimum height ad whe it reaches the maimum. 7. Height of object i metres, based o time i secods, is give as: h= 5t + 9t+ 5, Determie time of ladig of object, maimum height ad whe it reaches the maimum. 8. Give the diagram of f() at right, draw the graph of: f(-), f(), f(-), f(), f(0.5), f() Determie the iverse of the fuctio: f( ) = + 0. If f( ) = 5+ ad g ( ) = f( ) +, determie the ew equatio g() i terms of Give: { (,), (,), (,), (,), (5,) }, State Domai, Rage, Is it a fuctio or relatio?, Iverse. Eamie the give equatios ad determie: Domai, Rage ad ay vertical asymptotes. A sketch may help i your determiatios.: f( ) = + ad f( ) = If cm cscθ = 9cm, determie θ.. Give r,, y, θ of a triagle, solve for the missig parts. a. r =, = 5; b. = 7, y = ; c. θ = ad r =. 5. Give: A is 0, b = 50 cm, for what possible rage of legths of side a would r ABC have two possible solutios (approimate to decimal places)? 6. Give: A is, b = 5cm, for what values of side a would yield o solutios at all? 7. Determie the eact trigoometric ratios for sie, cosie ad taget, give that the poit P ( 5, ) is o the termial arm. Fid the pricipal agle, to decimal place. 9
10 8. Determie the eact trigoometric ratios for sie, cosie ad taget, give: θ = Determie the eact trigoometric ratios for sie, cosie ad taget, give: θ = Give: ta θ =.6, 0 θ 60, solve for θ.. Give: sec θ = 6., 0 θ 60, solve for θ.. Give that some fuctio, f(), is periodic, with a period of ad has the followig kow values: f() =, f() =, f()=. Determie f(795), f(-), f(0).. Give r ABC has A=79, B=7, AC=cm ad AB=5cm, determie the legth of BC.. Give r ABC has A = 5, b = cm ad c = 59cm, determie a. 5. Sketch y= si( 90) + o top of this graph of y = si( ) 6. Prove: csc ta + cot = (usig basic idetities). cos 7. Determie the eplicit formula for 5 7 8,,,, b),,,,,... 0, Determie the formula for the th term of the sequece: 0, 0, 0, 5 9. Determie the formula for the th term of the sequece:, 7, 0, 5 0. Give: t 5 =9 ad t 9 =5, determie the sum of the first 5 terms of this arithmetic series.. Determie the sum of the first 8 terms i the series: 8, +. Give f()=, sketch g() = f(), h() = f(+), j() = f(-)+, ad determie the y-itercept ad horizotal asymptote of each.. Epad ( ) 5 usig Biomial Theorem.. What is the 7 th term i the polyomial made by epadig ( ) 8? 5. I Pascal s triagle, if we eamied the 7 row, ad looked at the th diagoal colum (coutig 0 th, st, d, etc). What umber would appear i that spot? 0
11 6. A coutry has a growth rate of.%. Curret populatio is millio, populatio i 5 years will be what? 7. A coutry has growth rate of 8.%. Curret populatio is 5 millio, populatio i years will be what? 8. Populatio doubles every 7 days. Startig populatio is. After days, what is the populatio? 9. A elemet has a half-life of 7 days. Startig amout of 7 grams. After days, what amout remais? 50. A house icreases i value by % per year. Values at $ i 005, what will be the value i 0? 5. A car depreciates by 8% per year. A $ 000 car will be worth how much after 9 years? 5. How much eeds to be deposited, today, at 5% mothly iterest, i order to have $500 i 8 years? 5. You deposit $00 per moth, for 5 years, at 9% aual iterest compouded mothly, how much will you have? 5. You pla o withdrawig $000 per moth, for 5 years, at 9% aual iterest compouded mothly. How much will you eed i the bak at the start of the 5 years?
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