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. Check when necessar.. 0. 9 7.. log ( ) log (7 ).. 7. sin cos 0 on 0, 8. tan sec tan on (0, ) 9. cos 0 on 0, 0. coscsc csc (all solutions). sin cos 0. tan ( ) (solve for in terms of ).. 0 7 Graph (#-0 on separate aes)..,if, if 7. e 8. ln 9. sin( ) 0. 8 0. The adjacent figure shows a square of side surmounted b an equilateral triangle. Epress the area A of the entire enclosed region in terms of.. Determine the domain of 9 f( ).. If f ( h) f( ) f( ) 7, find. h. If f( ) and g( ), find a) f g and b) determine the domains of f g f g,, and.
. If ( ) 00, where 0 0, g a) find g ( ) and b) determine the domain and range of g.. The graph of a fourth degree polnomial function is given below. Write an equation for the function. - 7. Give all asmptotes of the following rational functions: 0 7 a) f( ) b) g ( ) 8. Using information on asmptotes, intercepts, end behavior, sign analsis, etc., sketch the graph of ( )( 8) f( ). ( )( ) 9. 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 ( ). 0. Using information derived from the Theorem on Rational Zeros, Descartes Rule of Signs, and upper/lower bounds, find the zeros of P ( ).. 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. Epress log log log as the logarithm of a single term.. If log ( ) log log ( ), find.. Determine the domain of f ( ) log.. Verif (working on one side): csc cos tan cot. Give the eact values: a) sin b) sec c) tan d) cos
7. If cot and sin 0, find cos. 8. 8 If cos and, find tan. 7 9. If, epress sin in terms of cos. 0. If 7tan and, epress 9 in terms of a trig function of.. Determine the critical values of not a critical value. ( sin )( cos ) f( ) sin in the interval 0,. Eplain wh is. Find if sin and cos.. Find sin( ) if tan, where is an acute angle, and cos, where.. Find cos if cos and sin 0.. Find cos if cos and. 7. Verif (working on one side): sin cos sin cos 7. Verif (working on one side): cos 8cos 8cos 8. In ABC, A, a, b. Find all of the remaining parts. 9. From the top of a 0-foot tower, the angle of depression of a small building is 0. Determine the distance from the building to the foot of the tower. 0. Town B is miles from town A at a bearing of S W. Town C is miles from town A at a bearing of S 7 E. Compute the distance (to the nearest mile) from town B to town C.. Give the eact values: sec tan ( ) a) b) cossin c) sin ( ) sin ( ) d) e) cos tan tan cos ( ) sin ( )
. Prove b mathematical induction:. Prove b mathematical induction: nn ( )( n ) () () ()... nn ( ) for all natural numbers n n n 9 is a factor of for all natural numbers n.. Epress 0 7 in sigma notation. 9. Give the equation of the left half of the circle 0 8 8.. Given the conic section 8 0. a) Name it. b) Give the pertinent information for its graph (verte, focus, directri). c) Sketch its graph. 7. Give the equation of the ellipse with center at ( 8, ), one verte at (, ), and one focus at (, ). 8. Give the equation of the asmptotes of the hperbola with equation given b ( ) ( ). 9 9. Write an equation of the hperbola whose vertices are (,) and (,8) and foci are at (,0) and (,0). 70. Epand and simplif: k k log k 7. Determine the twent-seventh term of the arithmetic sequence, 0,,... 7. The recursive definition of a sequence is given b a and a a. Determine its fourth term. n n 7. Find the fifteenth term in the epansion of ( ). 8 7. An auditorium has 0 rows of seats. The last row has 0 seats, and each row has fewer seats than the row behind it. How man seats are in the auditorium? 7. Epress in simplest factored form: ( ) ( ) ( ) ()... 8. sin 9.. (Answers to Part I)... 7. 8 00 00 0.. and log log. 9... 7. and log log
8. 9. 7 7,,,,, 9 9 9 9 9 9 0. k, k, where k{integers}. no solution. ( tan ). (, 0), ( 8, ).,,.. (,) 7. - - - (-,-) (-,-e+) (,-) 9. 0. 8. - - (,) (e,). A ( ) 9,,. h 7 ;, f g Df g,,. g 00... 7. a) ; ; 0 ( )( )( ) 8. crossover at (,) (-,) b) 0; 9. a), 0,, b) - 0 - - c) (,) - 0.,, i (,-). log (,) a) log (,-) b) log ( ) (-,) - -
. c) log (,) d) log ( ) (,) e) log (,) - -. f) log g) log. log - -..,,. a) b) c) 7. 8. 0 d) 9. sin cos 0. cos 7.., ; is not in the domain of f... 0.. sin cos sin cos cos sin sin cos sin cos sin( ) sin cos sin sin cos sin cos sin cos 7. cos cos( ) cos ( cos )... 8. ABC : B 9, C, c ; 9. 0 0. Appro. miles AB ' C ': B ', C ', c '. a) b) c) d) e) 9 7 9
()( ). i) P : () kk ( )( k ) ii) Assume Pk : () ()... k( k) ( k)( k)( k) Show () ()... ( k)( k) kk ( )( k ) () ()... kk ( ) ( k)( k) ( k)( k ) ( )( ) ( )( ) = kk k k k ( k)( k)( k) = P k+ is true. nn ( )( n ) Therefore, () () ()... nn ( ) b the principle of mathematical induction.. ) : 9 is a factor of. k i P k ii) Assume 9 is a factor of. k k Show: 9 is a factor of k k k k k k k k k k k k (9 ) 9 9 ( ) k k k Since 9 is a factor of 9 and 9 is a factor of ( ) b hpothesis, k 9 is a factor of 9 ( k k ) and hence 9 is a factor of k k. Pk is true. n n Therefore, 9 is a factor of b the principle of mathematical induction.. k k k. 9 ( ) 7. ( 8) ( ) 0. a) parabola b) verte at (,); focus at (,); directri at 7 8. ( ) 9. ( ) ( ) 70. log 9 00 7. 0 7 7. 7. 7. 80 7. ( ) 7
Part II: Multiple Choice. A polnomial function Phas ( ) the propert that P(7) P(8). What conclusion can be made about the closed interval 7,8? (A) The interval contains no real zeros. (B) The interval contains at least one real zero. (C) The interval contains eactl one real zero. (D) The interval ma or ma not contain one real zero. (E) None of these. Which of the following functions are not one-to-one functions? I. II. tan III. IV. ln( ) (A) I & IV (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. csc III. IV. (A) I onl (B) I & III (C) II onl (D) I & IV 0. The coefficient of the middle term in the epansion of ( ) can be found b 0 (A) (B) 0 ( )( ) (C) 0 ( ) ( ) (D) 0 ( ) ( ) nn ( )(n). Consider a proof b mathematical induction of the proposition... n. Assuming the proposition is true for the first k terms, what must be the sum of the first ( k ) terms of the series if the statement is to be proven true? (k)( k) ( k)( k)(k) kk ( )(k) (A) (B) (C) ( k)( k)(k) (D). The adjacent figure shows a portion of the graph of (A) cos (B) cos (C) sin (D) sin - 8
7. The amplitude and period of the graph of sin( ) is (A) ; (B) ; (C) ; (D) ; 8. In ABC, if 0,, and a, then b (A) (B) (C) (D) 9. In the triangle shown, which of the following about is not true? I. z II. tan III. zcos IV. zsin X (A) I and III (B) II and IV (C) III onl z (D) IV onl Y Z 0. The area of the triangle shown is (A) 00sin0 (B) 00sin 9 0 9 0 (C) 00sin 0' (D) 00sin 9 0'. Which of the graphs below represent the function sin( )? (A) (B) (C) (D) (E) (Answers to Part II). B. E. D. D. B. C 7. A 8. A 9. D 0. B. E 9