MAC 0 Review for Final Eam The eam will consists of problems similar to the ones below. When preparing, focus on understanding and general procedures (when available) rather than specific question. Answers for the problems can be found at the end of this packet starting on Page.. Factor completel each polnomial over the real number sstem. + + 8 c) 8 d) + - e) 75 f) 7 - + 7 g) 8 + 5 h) 7( - 6 + 9) + 5(-) i) (-) (+) 7-5(-) (+) 6. Find the domain of the following functions. Epress our answer in interval notation. f 6 ) 9 ( f() c) f() d) g() e) f ( ) 6. Find the following values for the function f(). Simplif our answer f(0) ; f(-) ; c) f( + h) ; d) f() ; e) f(- + h) ; f) f( + h) f(). Find the difference quotient f() f() = - + 8 f (h) f () h of: 5. Given two functions f ( ) and g ( ). Find the following functions and their domains
f f + g f g c) f g d) g 6. Which of the following graphs represents a function. Eplain. 7. Use the graph of the function f given below to answer parts - l) Find f(0) and f(-). Is f(-) positive or negative? c) What is the domain of f? d) What is the range of f? e) What are the -intercepts? f) What is the -intercept? g) Find all values of for which f() = -6. h) List the interval(s) on which f is increasing. i) List the interval(s) on which f is decreasing. j) List the interval(s) on which f() >0 k) List the interval(s) on which f() < 0. l) Determine whether f is even, odd or neither 8. Determine algebraicall whether the function g() = function notation. is even, odd, or neither. Be thorough and precise. Use 9. Using transformation, graph f() = ( + ) 5.. Use a sequence of graphs. Plot accuratel at least points for the basic function and show how the change. 0. Graph the piecewise functions
f() if if if if f ( ) if if c) f() =, if ; f() =, if < < ; f() =, if >. The graph of a function f is given below. Use the graph of f as the first step toward graphing each of the following functions h() = f(-) + g() = f(-) c) p() = f ( ) d) r(t) = f(). Given the functions f() and g() 5, find and simplif g f and g f. Determine their domains.. The function f is one-to-one. Find the inverse function and check our answer. Determine the domain and range for the function and its inverse. f() f() 5. Use snthetic division to divide 6 + + + 5 b +.
5. List the potential rational zeros of the polnomial function p() = 6 + + + 5 6. Find all real zeros of the given polnomial function. Epress the function in a factored form. P() = + + 7 + 8 + 7. Find the bounds on the real zeros of p() = 6 + + + 5 8. Show that ( ) is a factor of the polnomial p() = - -9 + 87 90. 9. Consider the polnomial.f() = 5 + 9 6 What is the maimum number of zeros that f can have? List the possible rational zeros of f c) Use the Remainder Theorem to find all the zeros of f and use the snthetic division to aid in factoring f completel. 0. Function p() = + + + is given List real zeros and multiplicit Determine whether the graph crosses or touches the -ais at the -intercept. c) Determine the end behavior. d) Sketch the graph of p(). Let p() = - (-)(+) ( +5) List real zeros and their multiplicit Determine whether the graph crosses or touches the -ais at each -intercept. c) Determine the end behavior of p. d) Sketch the graph of p(). Sketch the graph of the given rational function b finding the domain, -intercept(s), -intercept, asmptotes (vertical, horizontal oblique), checking for smmetr, finding the points where the graph crosses the horizontal/oblique asmptote and constructing the sign chart to determine where the graph is above/below the -ais. f(). f() = +. Find the vertical, horizontal or oblique asmptotes, if an, of the rational function f() 5 f() 5 c) f ( ). Solve the inequalit; write the answer in the interval notation
+ + c) ( ) 9 0 0 d) 5 5. Let f ( ) and g ( ). Find ( f g)() ( g f )() c) ( f f )() d) ( g g)(0) 5. Find functions f and g so that f g H, where H ( ). 6.Determine whether the following functions are one-to-one. Eplain 7. Use the graph of the given one-to-one function to sketch the graph of the inverse function. 8. Change an eponential epression to an equivalent epression involving a logarithm N. 5 e 8 9. Change each logarithmic epression to an equivalent epression involving an eponent log b ln = 0. Find the eact value of each logarithm without using a calculator log / log c) ln e. d) 5 7 log 5
. Use transformation to graph f() = f() = - -+ +. Plot accuratel at least points on the graph of a basic function and use them to perform the transformations. Determine domain, range and the equation of horizontal asmptote. Find the domain of the given functions f ( ) log 7 f ( ) log 7 9. Given f() = - ln(+) Find the domain of f Use transformations to graph f c) Use the graph to find the range and an asmptotes of f d) Find the inverse function f - e) Graph the inverse function. Let f () = + log ( +). Use transformations to graph function f. Use the graph to find the range of f and an asmptotes 5. ( Rewrite the epression as a single logarithm and simplif 8log log log ( Epress as a single logarithm. Assume all variables are positive and b logb t logb s logb v logb u 5 6. Rewrite the epression as a sum/difference of logarithms. Epress powers as factors. 7 6 ln ( )( ) ( ) ln 7. Use properties of logarithms to find the eact value of each epression log 6 9 log 6 log 5 c) log 6 log 6 8. Solve the equations 5 log log ( ) log ( ) c) ln(+)- ln = d) 5( ) 8
e) 0 f) 6 g) log (+) + log (+) = h) 5e i) log ( + + 7) = e e e k) log 8 = 5 l) e j) 9. Use the Change-of-the base formula and a calculator to approimate log 7 5. 0.Find an equation of the parabola with given properties. Graph the equation. Verte at (0, 0); ais of smmetr the -ais; containing the point (, 9) Focus at (-, 0); directri the line = c) Focus at (-,-); directri the line =. Find the verte, focus, directri, the end points of the latus rectum segment of the parabola given b the equation + = - - 0. Graph the parabola.. Find the equation of the ellipse with given properties. For each ellipse find the center, foci, and vertices. Find the lengths of major and minor aes. Focus at (-6,0); vertices (±7, 0). Foci at (,) and (-,); verte at (-,) c) Center at (,-); verte at (7,-); focus at (,-). Find the center, foci, and vertices of the ellipse + 9 + 8-8 =. Graph the ellipse.. Find the equation of the hperbola with given properties. For each hperbola find the center, vertices, foci, equations of transverse and conjugate ais, and the equations of the asmptotes. vertices at (±,0); focus at (5,0) Foci at (0,±); asmptote the line = - c) Focus at (-,0); vertices at (-,) and (-,) 5. Graph the conic + 5 + - 50-9 = 0 6. Identif the graph of a given equation as a circle, ellipse, parabola, hperbola, not a conic. + = 6 5 + 7-6 = 0 c) + - + = 0 d) + 7 + = 0 e) + 5 + = 0 7. Solve the following sstem of equations. If there are no solutions, sa so. If there are infinitel man solutions, describe the solution set. 5
c) 8 d) 8 8. Solve the linear sstem b elimination z z 7z 0 9. Evaluate the determinants 0 6 50. Solve the sstem using Cramer s Rule. If Cramer s Rule is not applicable, sa so. 5 6 b ) 0 6 c) 5 6 z z z d) 0 5 z z 5. Graph each equation of the sstem and solve it to find points of intersection 0 6 5. Write down the first five terms of the sequence
n { an} n n n { d n } ( ) n c) a, an n an 5. Write the formula for the n th term of a sequence {a n}given b -,,-,, -5, 6,,,,, 5 5. Write out each sum n k k0 n k k 0 c) ( ) j5 j ( j ) 55. Epress each sum using summation notation. Use as the lower limit of summation and i for the inde of summation. + 6 + 9 + + 9 + 9 7 79 56. Find the sum 0 k k 5 (k 7) k 50 c) (k ) k 57. Show that the sequence {a n }= {n +7}, n > is arithmetic. Find the first term and the common difference 58. Find the 80 th term of the arithmetic sequence,, 5,... 59. Find the n th -term of the arithmetic sequence if a, d ; d is the common difference 5 th term is - and the th term is 0 (You must use a sstem of equations to solve this problem). 60. Consider the finite arithmetic sequence: 7, 78, 8, 88,, 558 Write the n th term b finding the first term and the common difference. Write the sum of all its terms in sigma form. c) Find the value of the sum b using an appropriate formula.
6. Find the sum of the first 8 terms of the arithmetic sequence, 8,, 8,, 6. Find so that +, + and 5 + are consecutive terms of an arithmetic sequence. n 6. Show that the sequence { an} n is geometric. Find the first term and the common ratio. 6. Find the fifth and the n-th term of the geometric sequence for which a, r / 65. Find the 0-th term of the geometric sequence -,, -,. 66. Find the common ratio and the formula for the n-th term of the geometric sequence given a 9 786, r a 5 /8, a8 56/ 87 6 8 67. Consider the geometric sequence:,,,...,, 5 5 5 Write the formula for the n-th term. Write the sum of the first n terms of this sequence using sigma notation. c) Find the formula for the sum of the first n terms of this sequence. d) Find the sum of the first 0 terms of this sequence 68. Determine whether each geometric series converges or diverges. Eplain. If a series converges, find its sum. k a ) 0(.06) k b ) 000 ( ) c) d) 8+ + 8 + 7 +. k k 8 69. Evaluate without using a calculator 5! 70. Use the Binomial Theorem to epand given power of a binomial ( + ) 7 ( ) 5 7. Find the coefficient of in the epansion of ( + ) 7 0 7. Find the 7-th term of ( + ) 9. 7. Prove b mathematical induction that for all natural numbers n : n n k n( n ) ( n ) k( k ) n n (n ) k c) n(n +) < (n +)
d) is a factor of n + n e) + 8 + +... + (5n - ) = n(5n + )
Answers. (+)(+) ( +)(-)(+) c) ( )( + )( + ) d) ( + )( + )( ) e) ( + 5)( 5)( + 5) f) ( + )( )( 7) g) ( + 5)( 0 + 5) h) (7 6)( ) i) ( ) ( + ) 6 ( + 9). Domain: (, ) (, /) ( /, ) (0,] c) All real numbers d) [, ) (, ) (, ) e) (, ] (, ). f(0) = 0 f( ) =. + +h c) f( + h) = +h+h + d) f() = + e) f( + h) = +h f) h h 5h +0h+5 ( )(+h ) ( h + ) 5. Function, domain c) d) h 6h+0 + +, [, 0) (0, ) +, [, 0) (0, ) + +, [, 0) (0, ), (, 0) (0, ) 6. B and D represent functions because the pass the vertical line test 7. Based on the graph: f(0) = and f( ) = 6 Negative c) (, ] d) [ 8, )
e) =,, and 5 f) = g) =, h) (, ) and (0.5, ) i) (, ) and (, 0.5) j) (, 5) and (, ] k) ( 5, ) and (, ) l) Neither 8. Odd 9. f() = ( + ) 5 0. Piecewise functions:
c). (Graph) Shift right, up Reflect about -ais c) Compress verticall b a factor of and shift down b units d) Compress horizontall b a factor of. 5 f g() = 5+ g f() = 5+5. Inverse: f () = + Domain: (, ) (, ), Range: (, ) (, ) f () = ( + ) + 5 Domain: All real numbers, Range: All real numbers. 5 8 + 9 8 + 77 5 + + 5. Potential rational zeroes: p = ±, ±, ±, ±5, ± 5, ± 5 q 6. Real zeroes:,, factored: ( + )( + )( + ) 7. Between 6 and 6 8. ( ) ( + 0) 9. (the number of zeros cannot eceed the degree) ±, ±, ±, ± 5 ; ± 5 ; ± 5 c) Zeroes:, /5; factored: (5 + )( )( + ) 0. Zeroes:, multiplicit Crosses at c) = d)
. =0 multiplicit ; = multiplicit, =- multiplicit Crosses at =0,, and touches at = - c) = - 8 d). Graph :. Find asmptotes: Vertical = ±, horizontal = No vertical or horizontal asmptotes. Oblique asmptote = +
c) Vertical =, horizontal =. [-, -] U [, ] (/, ) c) (, ) [0, ] (, 5) d) (, ) [, ) 5. Given f() and g(): (fog)() = 8 5 (gof)() = c) (fof)() = d) (gog)(0) = 5. f() = ; g() = + - 6. No fails horizontal line test Yes passes horizontal line test 7. Sketch the graph of inverse: 8. log. 5 N log e 8 = or ln8 = 9. b = e = 0. c) d).. Domain: all real numbers; Range: < (asmptote at =)
. Domain: (, ) (0, ) b ) (, ) (, ). Domain: (, ) c) Range: all real numbers; asmptote = d) f () = e e)
Range: all real numbers, vertical asmptote: = 5. log [( ) ] t log v / b s /5 u 6. ln7 + ln + ln( 6) ln( + ) ln( ) ln( ) ln ln( ) 7. 5 c) (using change of base formul 8. log 5 0. log 50 c) /(e ) d) ln8 5 ln e) = 0 (use substitution) f) = and = - ( solutions) g) = h) = ln 5.6 i) = and = j) = 6 and = k) = l) = ln 5 ln 5 9.. 87 ln 7 0.
8 8 c) ( ) 8. Equation: ( + 6) = ( ), verte: (, 6), focus: ( 5, 6), directri line : = 7/. 9 + = (+) 9 c) ( ) 5. + ( ) 5 + (+) = =
. 5. Center: (,), Foci ( 5, ), ( + 5, ), Vertices: (, ) and (,) = 6 9 = ( ) ( ) c) 8 6. Not a conic Hperbola c) Parabola d) Ellipse e) Ellipse (in fact a circle) 7. =, = No solution c) Infinitel man solutions {(,) = -+} d) (, ), (, ), (, ), (, ) 7 8. Infinitel man solutions (,, z), z, is 5 5 5 5 9. Determinants 6 50. (,-) Cramer s Rule is not applicable because the determinant is 0 c) (,,-) d) Cramer s Rule is not applicable because the determinant is 0. 5. an real number
(, ), (, ), (-,-), and (-, -) (-5, -/) 5., 9, 9, 8, 5,, 5, 7, 5 9 c),, 0,, 7 5. ( ) n n n (n+) 5. + + + + 0 55. n + + + + n c) ( ) 5 ((5) + ) + ( ) 6 ((6) + ) + ( ) 7 ((7) + ) + ( ) 8 ((8) + ) + ( ) 9 ((9) + ) + ( ) 0 ((0) + ) k k + k= n (k) k+ k= 7 c) ( ) k+ k= 56. 80 k
770 c) 76850 57. a = 0, common difference d = 58. (n ); 80 th term: 6 59. a n = + (n ) a n = 8 + (n ) 60. an = 68 + 5n 98 (68 + 5k) k= 98 c) k= (68 + 5k) = 099 8 6. ( + 5(k )) = k= 6. = 5/ 6. a =, common ratio r = ; sequence:,, 8 9, 6 7, 8, 6. 5 = 8, n = ( )( )n 65. 0 = ( )( ) 9 = 5 66. Common ratio r = ; a n = ()( ) n 67. Common ratio r =, a n = ( )n a n = () ( 5 ) n n () ( n 5 ) k= c) S n = ()( ( 5 ) ) 5 0 d) 5 0 5 68. Diverges Converges, sum = 600 c) Converges, sum =.6 d) Diverges 69. 0 0 70. n
7. 8960 6 7. 55 6