PreCalc B Second Semester Final Exam Review PreCalc B Second Semester Final Exam Review...

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  • PreCalc B Second Semester Final Exam Review

    I.    Linear  velocity,  angular  velocity,  and   arc  length.  

    1A.θ = 40π t = 1min

    ω = θ t =

    20 2π( ) 1min

    i 1min 60sec

    = 2π 3

    rad/sec

     

    21B. 30 3

    20 cm/sec

    V r πω

    π

    ⎛ ⎞= = ⎜ ⎟⎝ ⎠ =

     

    1C. 0 cm/sec  

    2A.V = 5m 20sec

    = 1 4

    m/sec

    V = rω = 1 4

    m/sec = 2ω

    2B. ω = 1 8

    rad/sec

     

    ( )

    ( )( )2 2 2 3. 6 2 12feet

    1 1 6 2 36ft 2 2

    S r

    A r

    θ

    θ

    = = =

    = = =  

      II.    Trigonometry  

    1A. 2 13

    13 1B. − 13

    3 1C. −3

    2 1D. 13

    2  

    2. 2 2

    , 2 2

    ⎝ ⎜

    ⎠ ⎟  

    3A. 2

    2 3B. −1 3C. 2 3

    3 3D. 2

       

    4. A : none P : 3π

    D :°except− 5π 2

    + 3πk R :°    

                                 

    5. A : 2 P : 4 D :° R : −1.5,2.5[ ]                    

    6. A : none P : 2π

    D :°except 3π 2

    +πk R : (−∞,−2]∪[2,∞)  

           

     

     

     

     

     

     

     

    7. y = 19cos π 32

    x +10( )⎛ ⎝⎜

    ⎞ ⎠⎟ + 31

    8. y = 23cos π 9

    x − 2( )⎛ ⎝⎜

    ⎞ ⎠⎟ −1

     

     

     

  • 9A. cos2 x 2

    ⎛ ⎝⎜

    ⎞ ⎠⎟ − sin2 x

    2 ⎛ ⎝⎜

    ⎞ ⎠⎟

    = cos x

    9B. 2sin xcos x

    1+ 2cos2 x −1 = sin x

    cos x

    = tan x

    9C. tan(2(50)) = tan100o

    9D. sin(2(35)) = sin70o

     

     

    9E. tan2θ +1

    = sec2 x  

    9F. 2sinθ cosθ

    = sin2θ  

    10A. 2 1 3

    ⎛ ⎝⎜

    ⎞ ⎠⎟ 2 2 3

    ⎛ ⎝⎜

    ⎞ ⎠⎟ = 4 2

    9

    10B.1− 2 1 3

    ⎛ ⎝⎜

    ⎞ ⎠⎟ 2

    = 1− 2 1 9

    ⎛ ⎝⎜

    ⎞ ⎠⎟ =

    7 9

    10C. 1− 2 2

    3

    1+ 2 2 3

    = 3− 2 2 3+ 2 2

       

     

    11A. tan−1 −1( )= − π4  

    11B. arccos − 2

    2

    ⎝ ⎜

    ⎠ ⎟ =

    3π 4  

    11C. tan sin−1 5

    13 ⎛ ⎝⎜

    ⎞ ⎠⎟ = 5

    12  

    12. cos 150 2

    ⎛ ⎝⎜

    ⎞ ⎠⎟ =

    1+ − 3 2

    ⎛ ⎝⎜

    ⎞ ⎠⎟

    2

    = 2 − 3 4

    = 2 − 3 2

       

    13A.1− sin2β = 1− 2 1 3

    ⎛ ⎝⎜

    ⎞ ⎠⎟ 2

    = 7 9

    13B. 1− cosα 2

    = 1− 1 2 2

    = 1 2

    13C. 1− tan 2α

    2 tanα = 1− − 3

    1 ⎛ ⎝⎜

    ⎞ ⎠⎟

    2

    2 − 3( ) = 3 3

    13D. 1− 2 2

    3

    1+ 2 2 3

    = 3− 2 2 3+ 2 2

       

    14.    Proof  answers  may  vary    

    15A.sec2θ = 4; cosθ = ±1/ 2

    θ = 60o ,120o ,240o ,300o  

    15B.tan2θ +5tanθ + 6 = 0 = (tanθ + 3)(tanθ + 2)

    θ = 108.43o ,116.57o ,288.44o ,296.57o  

    15C.2sin2θ − cosθ −1= 0

    θ = 60o ,180o ,300o  

    16A. sin2 x − sin x − 6 = 0 sin x − 3( ) sin x + 2( ) = 0

    sin x = 3 or sin x = −2

    NoSolution

     

    16B. 2cot2 x + csc2 x − 2 = 0 2cot2 x +1+ cot2 x − 2 = 0

    3cot2 x = 1 cot2 x = 1 3

    tan2 x = 3 tan x = ± 3

    x = π 3

    , 2π 3

    , 4π 3

    , 5π 3

     

    16C. 2cos2 x − sin x −1= 0

    2 1− sin2 x( )− sin x −1= 0 2− 2sin2 x − sin x −1= 0 2sin x −1( ) sin x +1( ) = 0

    sin x = 1 2

    orsin x = 1

    x = π 6

    , 5π 6

    , 3π 2

     

    17D. sin2θ = 2 2

    θ = π 8

    , 3π 8

    , 9π 8

    ,11π 8

     

    17E. 0, π 2

    , 3π 2

    , π

    17F. tan(3θ + π 4

    ) = 1

    θ = 0,π 3

    , 2π 3

    ,π , 4π 3

    ,5π 3

     

  • 17G. sec2θ + tanθ = 0 tan2θ + tanθ +1= 0

    tanθ = −1± i 3 2

    θ = no solution

     

       

    17H. sinθ + cosθ = 1

    θ = 0,π 2

     

          III.    Polar  Graphing  

    ( )

    21. 4 25 29 5tan 111.8 2

    29,111.8

    r r

    θ θ

    = + =

    = → = −

    o

    o

     

    5 5 22. cos 5cos 4 2 5 5 2sin 5sin 4 2

    5 2 5 2, 2 2

    x r

    y r

    πθ

    πθ

    = = = −

    = = = −

    ⎛ ⎞ − −⎜ ⎟⎜ ⎟⎝ ⎠

     

    83. 3, 7 π⎛ ⎞−⎜ ⎟⎝ ⎠

     

     

    4. 2 r 2 cos2θ( )− 2 r 2 sin2θ( ) = 5r sinθ 2r 2 cos2θ − 2r 2 sin2θ = 5r sinθ

     

    5. r 2 = r − 2r sinθ

    x2 + y2 = x2 + y2 − 2y  

     

    6A.    symmetry:    x-­‐axis,  y-­‐axis , 2 π⎛ ⎞

    ⎜ ⎟⎝ ⎠  pole  

     

    6B.    symmetry:    y-­‐axis   2 π⎛ ⎞

    ⎜ ⎟⎝ ⎠  

     

    6C.    symmetry:    y-­‐axis   2 π⎛ ⎞

    ⎜ ⎟⎝ ⎠  

      6D.    symmetry:    x-­‐axis,  y-­‐axis   ,

    2 π⎛ ⎞

    ⎜ ⎟⎝ ⎠  pole  

     

        IV.  Law  of  Sines  and  Cosines  

    1A. 51o 1B. 18.59 cm

    1C. 14.45 cm

    2A. 12.64 cm 2B. 50.91o

    2C. 81.09o

    3A. 27.45 in 3B. 29.71 in

    3C. 28o

    4A. b1=51.60m b2 =5.135

    4B. B1=153.45 o B2 =2.55

    o

    4C. C1=14.55 o C2 =165.45

    o

    5A. A = 15(7)(5)(3) = 1575 = 15 7 ft2

    5B. A = 1 2

    sin38o 6( ) 4( ) = 7.39mi2

     

      V.  Conics    

    1A. x2 + 4x + 4( )− 3 y2 − 2y +1( ) = −13+ 4− 3 x + 2( )2 − 3 y −1( )2 = −12 y −1( )2

    4 −

    x + 2( )2 12

    = 1

    C : −2,1( ) V : −2,3( ) −2,−1( ) F : −2,5( ) −2,−3( ) A : y −1( ) = ± 33 x + 2( )

     

  • 1B. 4 x2 − 4x( ) + 9 y2 + 6y( ) = −61 4 x2 − 4x + 4( ) + 9 y2 + 6y + 9( ) = −61+16+81 4 x − 2( )2 + 9 y + 3( )2 = 36

    x − 2( )2 9

    + y + 3( )2

    4 = 1

    C : 2,−3( ) V : −1,−3( ) 5,−3( ) F : 2 ± 5,−3( ) Major length : 6 Minor length : 4

     

    1C. 4 x2 − 2x +1( )− y2 + 6y + 9( ) = 6+ 4− 9 4 x −1( )2 − y + 3( )2 = 1

    x −1( )2 1

    4 −

    y + 3( )2 1

    = 1

    C : 1,−3( ) V : 3

    2 ,−3

    ⎛ ⎝⎜

    ⎞ ⎠⎟

    1 2

    ,− 3 ⎛ ⎝⎜

    ⎞ ⎠⎟

    F : 1± 5 2

    ,−3 ⎛

    ⎝ ⎜

    ⎠ ⎟

    A : y + 3( ) = ±2 x −1( )

     

    2. x2 +8x +16( )− 4y = −8+16 x + 4( )2 − 4y = 8 x + 4( )2 = 4y +8 x + 4( )2 = 4 y + 2( )

    V : −4,−2( ) F : −4,−1( ) D : y = −3

     

    3. a = 5 c = 3 25− b2 = 9 b2 = 16

    x − 2( )2 25

    + y − 4( )2

    16 = 1

     

    4.V : 5,−4( ) x − h( )2 = −4a y − k( ) x −5( )2 = −8 y + 4( )

     

    ( ) ( ) ( )2 2

    5. : 4,6

    4 6 1

    5 9

    C

    x y− − + =

     

    ( ) ( )

    2 2

    2 2

    6. 4 9 5

    4 1 1

    4 5

    b b

    x y

    + = =

    − + − =

     

    ( )

    2

    2

    2

    7. 4 30 4 20 900 80 11.25

    45

    5, 0.56 ht=19.44ft

    10, 2.22 ht=17.78ft

    20, 8.89 ht=11.11ft

    x ay a a a

    x y

    x y

    x y

    x y

    = = − = − = −

    = −

    = = − →

    = = − →

    = = − →

     

      VI.    Parametrics  

    ( )

    ( ) ( ) ( ) ( ) ( )

    ( ) 4 2 cos 12

    ( ) 4 2 cos 15

    ( ) 4 2 2 cos 24 4 2 2

    2( ) 4 2 2 cos 4 2 2 15

    x t t

    or x t t

    y t t

    or y t t

    π

    π

    =

    ⎛ ⎞− − = ⎜ ⎟⎝ ⎠

    = − + −

    ⎛ ⎞− − = − + −⎜ ⎟⎝ ⎠

     

    x(t)  

      y(t)                      

  •   VII.  Sequences  and  Series  

    1A. 3, 7, 11, 15, 19; Arithmetic

    1B. −14 3

    , −13 3

    , − 4, −11 3

    ,−10 3

    ; Arithmetic

    1C. 2 3

    , 3 4

    , 4 5

    , 5 6

    , 6 7

    ; Neither

    2A. an = 32−5 n−1( ) 2B. a n = 11 3( )n−1

    3. −333 4. 3.45×1011

    5.166 = 30+ 4 n−1( ) n = 35 6A. S25 =

    25 2

    5+ 77( ) = 1025

    S5 = 5 2

    5+17( ) = 55 1025−55= 970

    6B. S23 = 3 1− 323( )

    1− 3 = 1.41×1011

    6C. 1339 280

    7. 8748

    8. 1 64

    = 2 1 2

    ⎛ ⎝⎜

    ⎞ ⎠⎟

    n−1 1

    128 = 1

    2 ⎛ ⎝⎜

    ⎞ ⎠⎟

    n−1

    ln 1 128

    ⎛ ⎝⎜

    ⎞ ⎠⎟ = n−1( ) ln 12

    ⎛ ⎝⎜