REVIEW. log e. log. 3 k. x 4. log ( x+ 3) log x= ,if x 2 y. . h

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Tyrone Henderson
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1 Math REVIEW Part I: Problems Simplif (without the use of calculators) ln log 000 e 0 k = k = k 7 log ( ) 8 lo g (log ) Solve the following equations/inequalities Check when necessar 8 = = log ( ) log (7 + ) = = = + log ( + ) log = = 00, = 7, 00 Graph (on separate aes) #7-7 = +, 0,if 8 9 = +, if = 0 = = e = ln( ) + Determine the domain of If f( ) = + 7, find + 9 f( ) = f ( + h) f( ) h If f ( ) = + and g ( ) =, + a) find ( f g ) and b) determine the domains of f, g, and ( f g ) If 7 If + f( ) = and g( ) =, a) find (g f) and b) determine the domains of f, g, and ( g f ) + g = where 0 0, a) find g ( ) and b) determine the domain and range of ( ) 00, g 8 The demand function of a product can be represented b p= + 00, where p is the price (in dollars) and is the quantit (in units) (Assume 0 00 ) a) Give the revenue function b) Determine the quantit that will maimize the revenue c) What is the maimum revenue? d) What price gives maimum revenue?

2 9 For the quadratic function f( ) = + +, determine the verte, ais of smmetr, and the - and - intercepts 0 A product has a profit function given b P ( ) = a) Determine the number of units that will give maimum profit b) What is the maimum profit? Determine k if ( ) is a factor of f k ( ) = The graph of a fourth degree polnomial function is given below Write an equation for the function = f( ) - ( + )(9 7) Given: g ( ) = ( + ) a) Give the coordinates of the hole in the graph of g b) Give the domain of g c) Give the equation of all asmptotes [Be sure to indicate the tpe of asmptote] Using information on asmptotes, intercepts, end behavior, sign analsis, etc, sketch the graph of ( )( + 8+ ) f( ) = ( )( ) Given: f( ) = ( + ) ( )( ) a) Determine the zeros of f ( ) b) Determine the sign in each of the intervals c) Using the above information and plotting several crucial points, sketch the graph of f ( ) Given the polnomial function P( ) = + + a) Using the Theorem on Rational Zeros, list all the possible rational zeros of P ( ) b) Use Descartes Rules of Sign to determine the possibilities for the number of positive, negative, and comple (nonreal) roots c) Using the information obtained from parts a and b, give all roots of the equation P ( ) = 0 7 Sketch the graph of f ( ) = log Then sketch the following on separate aes a) f ( ) =log b) f ( ) = log ( ) c) f( ) = log + d) f( ) = log ( + ) e) f ( ) = log f) f ( ) = log g) f( ) = log + 8 Epress log + log log as the logarithm of a single term

3 9 Epress as the sum or difference of logs: ( ) log + 0 If log ( ) = log log ( ), find ( ) Determine the domain of lo g Determine the - and -intercepts of = ln An open bo with a square base is made from a square piece of cardboard 0 inches on a side b cutting out a square from each corner and turning up the sides a) Epress the volume V of the bo as a function of the length of the side of the square cut from each corner b) Epress the surface area S of the bo as a function of ( ) ( ) 9 9 a) Epress in simplest factored form: f( ) = + b) Determine the value(s) of where f( ) = 0 c) Determine the value(s) of where f ( ) will be undefined The recursive definition of a sequence is given b a = and a = a + Determine its fourth term The sith term of a geometric sequence is 0 and its 0th term is 0 Find the second term 7 A college graduate earns $9,000 during his first ear on the job Suppose that his salar increases % for the net five ears Find his total income for those five ears n n 8 Find the fifteenth term in the epansion of ( ) Consider the epansion of ( ) a) Which term contains b) Give that term? 0 How much should one invest now at 0% compounded monthl to have $000 in two ears? Determine the number of ears it would take to triple an investment if the interest rate is 88% compounded continuousl Set up as a linear programming problem and solve You have $0,000 with which to invest in stocks and bonds You want to invest at least $,000 and no more than $,000 in stocks Furthermore, ou don t want to invest more than $0,000 in bonds Currentl, the earnings on stocks and bonds are 8% and 7 %, respectivel a) Determine the amount that should be invested in each in order that earnings are maimized b) What are the maimum earnings?

4 If f( ) = +, use the binomial epansion to find f ( + h) f( ) h Solve, using matrices: + z = + z = z = Represent in sigma ( ) notation the following two series a) b) 9 Determine the sum of Simplif completel k = k log k + 8 Using sum formulas and properties of summation, evaluate (k k + k) (Answers to Part I) and 9 or, 0 9 ± =,000; =,000 log+ log log log k = [ ] and 9 (, ] (,) (-,-) 8-9 (,) - (,-) 0 - (,-) - - (-e,) - - [ ) ( 9,, + ) + h

5 a) a) 7 a) 8 a) + + g ( ) 00 R b) domain f : [, ) ; domain : g (, ) (, ) ; domain ( ) ( ] f g :, (, ) b) D :(,) (, + ); D :(, ) (, + ); D :(, ) (,) (, + ) f = b) domain: [ 0,0 ]; range: [ 0,0] ( ) 00 = + b) 0 units c) $000 d) $00 g g f 9 verte at (, ); ais of smmetr at = ; -intercepts at (,0) and (, 0) ; -intercept at (0,) 0 a) 0 units b) $,00 = ( + )( )( ) a) (, ) b) (, ) (,0) ( 0, + ) c) VA at = 0; oblique at = 9 crossover at (-,) - - (,) a), 0,, b) + + c) - 0 (,) - (,-) -, -, 7 = log (,) a) = log (,-) b) = log ( ) (-,) - - c) = log + (,) d) = log ( + ) (,) e) = log (,) - - f) = log g) = log + 8 log - -

6 9 log log( ) log( ) ( ) ( ( ) (, + ),0 and 0,ln ) a) V( ) = (0 ) b) S ( ) = (0 )(0 + ) 9 ( ) a) f( ) = ( + )(9 ) b) = -, 0, c) = -, -, 00 7 appro $,7 8 9 a) 7 th term b),0 0 $0970 ears let = amount to be invested in stocks = amount to be invested in bonds objective: maimize E = , 0 + 0, 000 subject to:, 000, 000 0, 000 a) $,000 in stocks; $8,000 in bonds b) $,0 8 + h+ 8h + h h (-,,) a) 7 log 8 0,90 k = k + k b) ( ) k = k k + Part II: Multiple Choice A product has a total cost function C( ) = and total revenue function R( ) = 0 The equation for its profit function P ( ) is (A) P ( ) = 0+ 0 (B) P ( ) = 0 0 (C) P ( ) = 0 (D) P ( ) = 0+ 0 (E) none of these + 0 Which of the following are asmptotes of f( ) =? I = II = III = IV = V = 0 (A) I & II onl (B) III & IV onl (C) I, II, & V (D) III, IV, & V (E) none of these

7 Which of the following functions have inverses that are not functions? I = + II = + III = + (A) I onl (B) II onl (C) III onl (D) I & II (E) II & III Which of the graphs of the following functions have smmetr with respect to the -ais? I = II = + III = + IV = (A) I onl (B) I & III (C) II onl (D) I & IV (E) none of these Which one of the following epressions is equal to 0? ln + ln ln ln (A) ( ) ( ) (B) ( ) + ( ) (C) ln ( ) ln ( ) (E) none of these (D) ln( e) ln The coefficient of the middle term in the epansion of ( ) 0 0 (A) (B) 0 ( )( ) 0 (E) none of these (D) ( ) ( ) can be found b 0 (C) ( ) ( ) (Answers to Part II) B C E D B D 7

REVIEW. cos 4. x x x on (0, x y x y. 1, if x 2

REVIEW. cos 4. x x x on (0, x y x y. 1, if x 2 Math ` Part I: Problems REVIEW Simplif (without the use of calculators). log. ln e. cos. sin (cos ). sin arccos( ). k 7. k log (sec ) 8. cos( )cos 9. ( ) 0. log (log) Solve the following equations/inequalities.