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1 FINAL EXAM REVIEW 0 PRECALCULUS Name DIRECTIONS: PLEASE COMPLET E ON A SEPARATE SHEET OF PAPER. USE THE ANSWER KEY PROVIDED TO CORRECT YOUR WORK. THIS WILL BE COLLECTED!!! State the domain of the rational function. [ENRICHED] ) f() = - + ) ( - 8)( + ) ) f() = - ) Give the equation of the specified asmptote(s). [ENRICHED] - ) Vertical asmptote(s): f() = - ) ) Horizontal asmptote: h() = ) ) Oblique asmptote: f() = ) Evaluate the limit based on the graph of f shown. [ENRICHED] ) lim f => ʺapproaching from the leftʺ - ) - -

2 7) lim f => ʺapproaching positive infinitʺ 7) - - List the - and -intercepts, and graph the function. [ENRICHED] 8) f() = + 8) List the - and -intercepts, and graph the function. 9) f() = - - 9)

3 ) f() = - + ) Find an equation for the rational function graph. [ENRICHED] ) ) Solve the equation algebraicall. [ENRICHED] ) + = ) ) = ) ) - - = 0 - ) Solve the polnomial inequalit. Write the solution in interval notation. ) + - > 0 ) ) ) 7) [HONORS] 7)

4 Solve the rational inequalit. Epress the solution in interval notation. 9 8) - > - 8) 9) + + < 7 9) 0) - + < 0 [HONORS] 0) Graph the eponential function using transformations where appropriate. State the asmptote. ) f() = - ) ) f() = + )

5 ) f() = - - ) ) f() = - - ) Graph the eponential function. Label at least one point and the asmptote. ) f() = 0. - )

6 ) f() = e- ) Graph the function and analze it for domain, range, continuit, intercepts, increasing or decreasing behavior, smmetr, boundedness, etrema, asmptotes, end behavior, and concavit. 7) f() = 7) State whether the function is an eponential growth function or eponential deca function, and describe its end behavior using limits. 8) f() = e 8) Graph the function. Give the domain and range. State the asmptote. 9) f() = log 9)

7 Graph the function. Give the domain and range. 0) f() = log ( - ) 0) ) f() = log/ (-) ) Determine the function which corresponds to the given graph. ) ) The indicated point is (, -). A) = ln + B) = ln( + ) C) = ln( - ) D) = ln - 7

8 ) ) The asmptote is = -. A) = ln + B) = ln - C) = ln( - ) D) = ln( + ) Convert to logarithmic form. ) -7 = ) Convert to eponential form. ) log = - ) 9 8 Find the value of the epression. ) log 8 ) 7) log7 7 7) 8) log (-) 8) Use the change of base formula to approimate the logarithm to four decimal places. 9) log ) Write the epression using onl the indicated logarithms. 0) log using natural logarithms 0) Rewrite the epression as a single logarithm. ) (loga q - loga r) + loga p ) Rewrite the epression as a sum or difference or multiple of logarithms. 7 0 ) log9 ( sr ) ) Solve the equation. Round to the nearest hundreths place when necessar. ) (8 - ) = ) 8

9 ) 9 + = - ) ) ( - ) = ) ) + = ) 7) 00et = 90 7) 8) e = + 8) 9) e - = - e 9) 0) log = 0) ) ln = - ) ) = log 7 ) ) ln( - ) = ln - ln ( - ) ) ) log ( - 8) + log ( - 8) = ) Find the future value. ) $08 for ears at 7% compounded quarterl ) ) $9 for ears at 7% compounded continuousl ) Solve the problem. 7) Find the required annual interest rate, to the nearest tenth of a percent, for $ to grow to $9 if interest is compounded semiannuall for 8 ears. Assume eactl weeks per ear. 7) 8) The deca of mg of an isotope is given b A(t)= e-0.0t, where t is time in ears. Find the amount left after 88 ears. 8) Find the function value using coordinates of points on the unit circle. 9) cos π 9) 0) sin -π 0) 9

10 ) cos 9π ) List the quadrants in which the function has the given sign. ) cosecant is negative ) State in which quadrant the terminal side of the given angle lies. ) -7.8 ) ) 98 0ʹʹʹ ) Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. ) -7 ) ) 8π 7 ) Convert the angle to decimal degrees and round to the nearest hundredth of a degree. 7) 9 7) Convert the angle to degrees, minutes, and seconds. 8) 7.7 8) Convert from degrees to radians. 9) 88 9) Convert the radian measure to degree measure. Use the value of π found on a calculator and round answers to two decimal places. 70) - 9 π 70) 7) 7 7) Use the arc length formula and the given information to find the indicated quantit. 7) s =. ft, θ = π rad; find r 7) 7) r = 9 ft, θ = ; find s 7) Find the area of the sector of a circle. 7) Radius =.0 cm; θ = π radians 7)

11 Solve for. 7) 7) Solve the problem. 7) From a distance of feet from the base of a building, the angle of elevation to the top of the building is. Estimate the height of the building to the nearest foot. 7) 77) A police helicopter is monitoring the speed of two cars on a straight road. The helicopter is at an altitude of 00 feet directl above the road. At one instant, the angle of elevation from the first car to the helicopter is, and the angle of elevation from the second car to the helicopter is. How far apart are the two cars to the nearest foot? 77) Assume that θ is an acute angle in a right triangle satisfing the given conditions. Evaluate the indicated trigonometric function. 78) sin θ = ; cot θ 78) Suppose that θ is in standard position and the given point is on the terminal side of θ. Give the eact value of the indicated trig function for θ. 79) (, 8); find cos θ. 79) 80) (, - ); find sec θ. 80) Find the requested trigonometric function values of the angle θ shown in standard position. 8) Find sec θ, csc θ, and cot θ 8) 8) sin π 8) 8) cos (-00 ) 8) 8) tan (- π ) 8) 8) cos π 8)

12 8) cot -π 8) A) - B) C) D) - 87) tan ( ) 87) 88) cos π 88) 89) sin (-π) 89) Evaluate without using a calculator. 90) sec β, if sin β = -, with β in quadrant III 90) Find the value of the unique real number θ between 0 and π that satisfies the given conditions. 9) cos θ = - and sin θ < 0 9) 9) tan θ = - and cos θ > 0 9) Solve for in the given interval using reference triangles in the proper quadrant. 9) cos = -, π π 9) 9) cos =, π π [Honors] 9) Find the indicated value. 9) Find the phase shift for = sin (7θ - π). 9) 9) Find the vertical shift for = - + sin (θ + π ). 9)

13 Specif the period and the amplitude of the function. 97) =. cos 97) Select the functions that have identical graphs. 98) (i) = cos (ii) = - sin ( - π) 98) (iii) = cos - π Graph the function over a one-period interval. 99) = -cos + π 99) ) = + sin( - π) 0) 8 7

14 ) = + sin ( - π) ) Graph the function. ) = -tan ) ) = sec ) Find the eact value. ) cos- ) ) sin- 0. ) ) tan- (-) )

15 7) cot- undefined 7) Give the eact value of the following without using a calculator. 8) sin (arctan ) 8) 9) csc (sin- /0) 9) A) B) C) D) ) cos ( arcsin ) ) A) B) C) D) Find an algebraic epression equivalent to the given epression. ) cos (arccot u) ) ) sin (tan- u ) ) State whether the given measurements determine zero, one, or two triangles. ) A =, a =, b = 0 ) ) A = 9, a =, b = ) Solve the triangle. ) C =, b =, c = ) ) A =, a = 7, b = 8 ) 7) C =, a = 9, c = 7) Solve. 8) A gu wire to a tower makes a 70 angle with level ground. At a point ft farther from the tower than the wire but on the same side as the base of the wire, the angle of elevation to the top of the tower is 8. Find the length of the wire (to the nearest foot). 8) 9) A triangular field has sides of. m and. m, and the angle between them measures.9. Find the area of the field. 9)

16 Solve the triangle. ) ) ) a =, b =, c = 9 ) Decide whether a triangle can be formed with the given side lengths. If so, use Heronʹs formula to find the area of the triangle. ) a = 0 ) b = c = 88 ) A parallelogram has sides of length. cm and 0. cm. If the longer diagonal has a length of 8.9 cm, what is the angle opposite this diagonal? Give our answer to the nearest tenth of a degree. ) Use basic identities to simplif the epression. csc θ cot θ ) sec θ ) ) cos θ + csc θ sin θ ) sinθ ) cos θ - cos θ sinθ ) 7) sinθ + tanθ + cosθ 7) Simplif the epression. 8) sin(- ) csc 8) 9) sec (- ) cos (- ) 9) ) tan π - tan ) ) ( sin + cos ) - ( csc - cot ) )

17 ) sinθ + sin θ - + cosθ ) ) cos - sin + - sin cos ) Prove the identit. ) (sin )(tan cos - cot cos ) = - cos ) ) + sec sin = sec ) ) + csc sec = cos + cot ) 7) sin - cos + sin = csc 7) + cos 8) cot csc - = + sin sin 8) 9) sec - tan = sec + tan 9) ) tan t - cot t + + cot t = tan t + - csc t tan t tan t - ) Find all solutions of the equation. Let ʹnʹ represent an integer. ) sin t = ) ) sint = ) ) sint - cost - sin t + = 0 ) ) tan t = sin t ) ) cost - = 0 ) Solve, finding all solutions in [0, π). ) cos tan + tan = 0 ) 7) sin = 7) 8) cos = - cos 8) 9) csc = 0 9) 7

18 Write out the first five terms of the sequence. [HONORS] ) an = n - n ) ) an = (-)n - n + n - ) Evaluate the summation. [HONORS] ) (i - ) ) i= ) (-)k + (k - 8) ) k= Write the summation. Do not evaluate. [HONORS] ) (i + ) ) i= Solve the problem. [HONORS] ) Eloise contracts to work for das, receiving $0.0 the first da, $0.0 the second da, $0.8 the third da, and so on, with each daʹs pa triple that of the previous da. How much will she earn on the last da of the contract? ) Find the desired value. [HONORS] ) 0, -, -, -,... ) Find the common difference. 7) Find the eplicit formula for the arithmetic sequence. a =, d = 7) 8) -9, -, -9, -, -9,... Find an. 8) For the arithmetic sequence, write the indicated number of terms. [HONORS] 9) a =, d = -, n = 9) Find a of arithmetic sequence from given information. [HONORS] ) a=7, a0 = ) Find the sum of the first n terms of the given arithmetic series. [HONORS] ) a = -, d = - (n = 9) ) ), 7, 8,... (n = 8) ) 8

19 ) a =, a = (n = 7) A). B) C) D) ) Find the requested sum of the arithmetic sequence. [HONORS] ) (-i + ) ) i= Solve the problem. [HONORS] ) An auditorium has rows with seats in the first row, in the second row, in the third row, and so forth. How man seats are in the auditorium? ) Write out the term of the geometric sequence that satisfies the given conditions. [HONORS] ) a =, r = -, n = ) 7) a = 0, a = 80, n=7 7) Find an for the given geometric sequence. [HONORS] 8) a = 00, r = 8) 9), -, 9, -,... 9) 08 Find the first term and the common ratio for the geometric sequence. [HONORS] 70) a =, a8 = 98,0 70) Use the formula for Sn to find the sum for the geometric sequence. [HONORS] 7),,, 8, 7) 7) (-)i 7) i= Find the common ratio r for the given infinite geometric sequence. [HONORS] 7), 8,, 8,,... 7) Evaluate the series, if it converges. [HONORS] 7) ) 7) i - 7) i= 9

20 7) i - 7) i= Solve the problem. [HONORS] 77) Assuming two parents, four grandparents, etc., what is the total number of ancestors a person has going back 7 generations? 77) Evaluate the epression. [HONORS] 78) C 7 78) 79) 7!!! 79) Write the binomial epansion for the given epression. [HONORS] 80) ( + ) 80) 8) ( - ) 8) Write the indicated term of the binomial epansion. [HONORS] 8) ( + ); rd term 8) 8) ( - ); th term 8) 0

21 Answer Ke Testname: FINAL EXAM REVIEW 0 ) (-, ) ) (-, - ) (-, ) (, ) ) =, = - ) = 8 ) = - ) - 7) 0 8) -intercept: -, 0, no -intercepts ; ) -intercept: (0, 0), -intercept: (0, 0) ; ) -intercepts: (-, 0) and (, 0), -intercept: 0, ) f() = ) = ) No solution ( + )( - ) ( + )( - )

22 Answer Ke Testname: FINAL EXAM REVIEW 0 ) ) -,, ) (-, ] 7) [-, -] [, 9] 8) (, ) 9) (-, -) (-, ) 0) -, 0 0, ) = ) = ) =

23 Answer Ke Testname: FINAL EXAM REVIEW 0 ) = ) = ) =

24 Answer Ke Testname: FINAL EXAM REVIEW 0 7) The domain is all real numbers, and the range is (0, ). The function is continuous and increasing on its domain. The -intercept is (0, ). There is no -intercept. The function is neither even nor odd. It is bounded below but not above. There are no local etrema. The horizontal asmptote is = 0. The end behavior is described b lim f() = 0 and - lim f() =. The graph is concave up. 8) Eponential growth function; lim f() = 0; lim f() = - 9) domain: (0, ); range: (-, ); = ) domain: (, ); range: (-, ); =

25 Answer Ke Testname: FINAL EXAM REVIEW 0 ) domain: (-, ); range: (-, ); = ) D ) D ) log = -7 ) 9- = 8 ) 0 7) 8) Undefined 9).8 0) ln ln ) loga qp r ) 7 log log9 s - log9 r ) {} ) - ) {.9} log (/) ) log (/) 7).0 8) -.8 9).79,.79 0) {} ) = e ) 7 ) / ) ) $.9 ) $0.7 7) 7.%

26 Answer Ke Testname: FINAL EXAM REVIEW 0 8) 9) 0) - ) ) III, IV ) IV ) IV ) ; -7 ) π 7 ; - π 7 7). 8) 7 9) 8π 70) -0 7) ). π ft 7) 0 π ft 7) 87.8 cm 7) 0. 7) 8 feet 77) 79 feet 78) 79) 80) 8) sec θ = - /, csc θ =, cot θ = - 8) 8) 8) 8) 8) B 87) Undefined 88) 0 89) 0 90) -

27 Answer Ke Testname: FINAL EXAM REVIEW 0 9) π 9) π 9) 7π 9) 9π 9) 7 π 9) - 97) Amplitude =.; period = π 98) ii and iii 99) ) 8 7 7

28 Answer Ke Testname: FINAL EXAM REVIEW 0 ) ) ) ) π ) π ) -π 7) 0 8) 9) C ) D 8

29 Answer Ke Testname: FINAL EXAM REVIEW 0 ) u u + u + ) u u + u + ) Two ) Zero ) B =., A = 0., a. ) B = 7.7, C =., c =.; B =., C = 8.7, c =.8 7) No triangle is formed. 8) 8 ft 9),7 m ) B =.7, C = 0., a =. ) A., B 99., C.8 ),8.9 ) 9. ) cotθ ) cscθ ) cosθ 7) secθ 8) - 9) ) ) 0 ) sin θ ) sec sin cos ) (sin )(tan cos + cot cos ) = sin cos (sin + cos ) - cos - cos = - cos. ) + sec sin = + sin cos = + tan = sec. ) + csc sec 7) = cos + sin = cos (sin + ) sin = - cos sin cos sin sin = sin - cos = sin + (cos - cos )- cos = + cos sin = cos + cot. sin - cos + sin sin [( + cos )+( - cos )] sin = = + cos ( - cos )( + cos ) - cos = sin = csc. sin cot 8) csc - = csc - (csc - )(csc +) = = csc + = + sin csc - csc - sin. 9) sec - tan = (sec + tan )(sec - tan ) = (sec + tan )() = sec + tan. tan t ) - cot t + + cot t = tan t + - cot t = tan t csc t = tan t + - csc t. tan t tan t ( - cot t) tan t - tan t - ) t = π/ + nπ, π/ + nπ ) t = π/ + nπ ) t = π/ + nπ, π/ + nπ, π/ + nπ ) t = nπ, π/ + nπ, π/ + nπ ) No solution 9

30 Answer Ke Testname: FINAL EXAM REVIEW 0 ) = 0, π, π, 7π 7) = π, π, π, 7π, 7π, π, π, 9π 8) = π 8, 9π 8, 7π 8, π 8 9) No solution ) 0,,,, 0 ), -, 0.8, -0.7, 0.7 ) ) -8 ) ) $,88. ) - 7) an = n + 8) an = -9 + (n - ) 9),, 8,, ) ) -89 ) ) B ) - ) 0 ) -0 7) 0,80 8) 8()n- 9) * - 7 n - 70) a =, r = 7) 7) - 7) 7) 7) 0 7) Does not converge. 77) 78) 7 79) 80) ) ) 00 0

31 Answer Ke Testname: FINAL EXAM REVIEW 0 8) -,,07

9) A) f-1(x) = 8 - x B) f-1(x) = x - 8 C)f-1(x) = x + 8 D) f-1(x) = x 8

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