Thomas Calculus Solutions Pdf 168697 | Solutions Thomas Calculus 14th Edition

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                               Thomas Calculus Early Transcendentals 14th Edition Hass SOLUTIONS 
                               MANUAL 
                                
                                                                    CHAPTER 2   LIMITS AND CONTINUITY 
                                
                                
                                  2.1      RATES OF CHANGE AND TANGENTS TO CURVES 
                                                    f        f (3)f (2)                                                                                   f       f (1)f (1)      
                                                                                   289                                                                                                      20  
                                    1.   (a)                                                19                                                   (b)                                              1
                                                    x            32                 1                                                                      x          1(1)               2   
                                                                                                 
                                                     g        g (3) g (1)             (    )                                                              g       g (4) g ( 2)      
                                                                                     3  1                                                                                                     8 8
                                    2.   (a)                                                     2                                               (b)                                                  0
                                                     x            3  1                 2                                                                    x          4  ( 2)               6    
                                                                 3                                                                                                            
                                                                                                                                                                       h         h                     
                                                               h   h                                                                                                   
                                                                   4             4           11          4                                                               2         6       0  3         3  3
                                                    h                                                                                                       h              
                                    3.   (a)                                                                                                    (b)                                                 
                                                                  3                                                                                                                         
                                                     t                                                                                                     t                                             
                                                                    4      4                    2                                                                           2     6                     3 
                                                                                                                  
                                                    g       g () g (0)      (21)(21)                                                                   g       g () g ()      (21)(21)
                                                                                                               2
                                    4.   (a)                                                                                                    (b)                                                            0
                                                     t            0                                                                                     t           ( )                    2         
                                                                                           0                 
                                                      R(2) R(0)      
                                           R                               811      
                                                                                             31  
                                    5.                                                              1
                                                        20                   2              2   
                                                      P (2) P (1)      (81610)(145)
                                           P                                     
                                    6.                                                                    2  2  0
                                                         21                           1
                                                                         2           2                        2                       2
                                                     y       ((2h ) 5)(2 5)                                              4h h
                                                                                                  44h h 51
                                    7.   (a)                                                                                           4  h. As h  0, 4  h  4  at  P(2, 1)  the slope is 4.
                                                                                                                           
                                                     x                     h                              h                      h      
                                           (b)       y  (1)  4( x  2)  y  1  4 x  8  y   4 x  9
                                
                                                                             2            2                        2                     2
                                                     y      (7(2h ) )(72 )                                                4hh
                                                                                                  744h h 3
                                    8.   (a)                                                                                              4  h. As h  0, 4  h  4  at P(2, 3)  the slope
                                                                                                                                                                                                  
                                                     x                      h                              h                       h       
                                                    is  4. 
                                           (b)       y  3  (4)( x  2)  y  3  4 x  8  y  4 x  11
                                                                        2                        2                                 2                                     2
                                                    y       ((2h) 2(2h)3)(2 2(2)3)                             44h h 42h3(3)       2hh
                                    9.   (a)                                                                                                                               2  h. As h  0, 2  h  2  at
                                                    x                                 h                                                h                            h
                                                    P(2,  3) the slope is 2. 
                                           (b)  y  (3)  2( x  2)  y  3  2 x  4  y  2x  7.
                                                                       2                   2                            2                           2          
                                                    y        ((1h) 4(1h))(1 4(1))                     12h h 44h(3)                    h 2h 
                                                                                                                                                            h  2. As h  0, h  2  2  at P(1,  3) the
                                  10.   (a)               
                                                    x                                                                      h                    
                                                                                 h                                                                    h 
                                                    slope is 2. 
                                           (b)  y  (3)  (2)( x  1)  y  3  2 x  2  y  2x 1.
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                                                                      3  3                         2     3                     2     3                                    
                                                    y       (2h) 2              812h 4h  h 8                12h 4h h                                   2                                             2
                                  11.   (a)                                                                                            12  4h  h . As h  0, 12  4h  h   12,  at P(2, 8)
                                                                                                                                                                
                                                    x               h                          h                            h           
                                                    the slope is 12. 
                                           (b)  y  8  12( x  2)  y  8  12 x  24  y  12x 16.
                                                                          3         3                         2     3                     2     3                                    
                                                    y       2(1 h) (21 )                                                                                              2                                             2
                                                                                              213h3h h 1                  3h3h h
                                  12.   (a)                                                                                                      3  3h  h . As h  0, 3 3h  h   3,  at
                                                                                                                                                                           
                                                    x                    h                               h                            h           
                                                    P(1, 1) the slope is 3. 
                                           (b)  y  1  (3)( x  1)  y  1  3x  3  y  3x  4.
                                
                                
                                
                                
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                                                   Chapter 2  Limits and Continuity 
                                 62
                                 62 
                                                                      3                   3                              2     3                                         2     3
                                                    y        (1h) 12(1h)(1 12(1))                    13h3h h 1212h(11)                          9h3h h                                    2 
                                                                                                                                                       
                                  13.   (a)                                                                                                                                       9  3h  h .
                                                                                                                                                            
                                                    x                           h                                               h                                       
                                                                                           2                                                                          h
                                                    As h  0, 9  3h  h  9  at P(1,  11) the slope is 9. 
                                           (b)  y  (11)  (9)( x  1)  y  11  9 x  9  y  9x  2.
                                                                      3              2          3         2                            2     3                   2                  2     3                       
                                                    y       (2h) 3(2h) 4(2 3(2) 4)                             812h 6h h 1212h 3h 40     3h  h                                             2
                                  14.   (a)                                                                                                                                                 3h  h  .
                                                    x                            2  h                                                          h                                     h      
                                                    As h  0, 3h  h  0  at P(2, 0) the slope is 0. 
                                           (b)  y  0  0( x  2)  y  0.
                                                                 1       1
                                                     y                       2(2h)  
                                                               2h     2                         1            1       
                                  15.   (a)                                                                        .
                                                     x            h            2(2h)   h             2(2h)
                                                                               1              1                               1                             1  
                                                    As h  0,                             ,  at P 2,                                 the slope is               .
                                                                                                                                   
                                                                          2(2 h)         4                                     2                             4
                                                                                                                                    
                                                              1                                             
                                                                        1                                  1      1        1                 1
                                                     y                     ( x  (2))  y                         x    y                    x  1
                                           (b)                                                                                     
                                                              2          4                                 2        4         2                 4  
                                                                (4h )       4                                                                    
                                                     y                       
                                  16.   (a)                    2(4h )     24         4 h        2      1      4h2(2h)   1                 1        1  
                                                                                                                                                             .
                                                                                                       
                                                                                                                 
                                                     x               h                2h         1      h             2  h           h      2h          2h 
                                                                           1            1                                                          1 
                                                    As h  0,                       ,  at P(4,  2) the slope is   .
                                                                          2 h        2                                                               2 
                                                                         1                                  1                        1 
                                           (b)    y  (2)   ( x  4)  y  2   x  2  y   x  4 
                                                                        2                                    2                          2 
                                                                                                                                         
                                                     y        4h  4         4h 2      4h  2                    (4 h)4                  1       
                                                                                                                                                
                                                                                                                                           
                                  17.   (a)                                                                                                          .
                                                                                         h        
                                                     x              h                                4h  2      h(   4h 2)                4h 2
                                                                              1                  1           1                                                      1 
                                                    As h  0,                                            ,  at P(4, 2) the slope is   .
                                                                                                             4                                                           4
                                                                             4h 2          4  2                                                                     
                                                                   1                                  1                       1 
                                           (b)    y  2   ( x  4)  y  2   x  1  y   x  1 
                                                                   4                                    4                         4 
                                                                                                                                                                           
                                                     y        7(2h)  7(2)                     9h 3        9h 3      9h 3                   (9h)9                 1      
                                                                                                                                                                                  
                                                                                                                                                                              
                                  18.   (a)                                                                                                                                          .
                                                                                                                           h           
                                                     x                       h                           h                              9h 3     h(  9h 3)                  9h 3
                                                                            1               1        1                                                            1 
                                                    As h  0,                                              ,       at P(2, 3)  the slope is                             .
                                                                             9h 3          9 3            6                                                           6 
                                                                                                                                                   8
                                                                  1                                        1        1                 1        
                                           (b)       y  3            ( x  (2))  y  3                      x    y                  x    
                                                                   6                                         6         3                6          3
                                 19.  (a)             Q               Slope of PQ  p
                                                                                                                      t         
                                                    Q(10, 225)                        650225  42.5 m/sec 
                                                      1                                 2010 
                                                    Q (14, 375)                       650375  45.83 m/sec
                                                       2                                2014   
                                                    Q (16.5, 475)                     650475  50.00 m/sec
                                                       3                               2016.5                                 
                                                    Q (18, 550)                       650550  50.00 m/sec
                                                       4                                2018 
                                           (b)   At t  20, the sportscar was traveling approximately 50 m/sec or 180 km/h. 
                                
                                                                                                                  p 
                                 20.   (a)          Q                   Slope of PQ 
                                                                                                                 t  
                                                    Q(5, 20)                          8020  12 m/sec 
                                                      1                                105 
                                                    Q (7, 39)                         8039  13.7 m/sec
                                                       2                               107  
                                                    Q (8.5, 58)                       8058  14.7 m/sec
                                                       3                              108.5                             
                                                    Q (9.5, 72)                       8072  16 m/sec 
                                                       4                              109.5
                                           (b)   Approximately 16 m/sec
                                
                                
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Section 2.1  Rates of Change and Tangents to Curves          63
                                                                                                                                                                                                                           
                                                                                                                                                        21.   (a)
                                                                                                                                                                                                                                                                                p 
                                                                                                                                                                                                                                                              200 
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                       )      160 
                                                                                                                                                                                                                                                       0s
                                                                                                                                                                                                                                                       100    120 
                                                                                                                                                                                                                                                       (
                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                       it
                                                                                                                                                                                                                                                       f
                                                                                                                                                                                                                                                       o           80
                                                                                                                                                                                                                                                       r
                                                                                                                                                                                                                                                  P
                                                                                                                                                                                                                                                                   40 
                                                                                                                                                                                                                                                                         0                                                                    t 
                                                                                                                                                                                                                                                                                                       2010   2011   2012   2013   2014 
                                                                                                                                                                                                                                                p                                                                                                                  Ye ar 
                                                                                                                                                                                                                                                                                               17462         112
                                                                                                                                                                                                   (b)                                                                                                                                                                              56 thousand dollars per year
                                                                                                                                                                                                                                                  t                                        20142012                                                                                                   2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                p                                                      6227     
                                                                                                                                                                                                   (c)   The average rate of change from 2011 to 2012 is                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             35 thousand dollars per year.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             t      20122011
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                p    11162     49  thousand dollars per year.
                                                                                                                                                                                                                                           The average rate of change from 2012 to 2013 is t      20132012 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1                                                                                                                                                                                                                                                                               
                                                                                                                                                                                                                                           So, the rate at which profits were changing in 2012 is approximately  (35  49)  42 thousand dollars
                                                                                                                                                                                                                                           per year.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        2 
                                                                                                                                                        22.   (a)  F ( x)  ( x  2)/( x  2) 
                                                                                                                                                                                                                                                     x           1.2             1.1               1.01                1.001             1.0001                  1
                                                                                                                                                                                                                                                      F ( x)                                                         4.0                                                                                          3.4                                                                                                   3.04                                                                                                                     3.004                                                                                                          3.0004                 3
                                                                                                                                                                                                                                            F                                             4.0(3)                                                                                                                                                                                                                                                                                                                      F                                              3.4 (3) 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                              x                                                          1.21                                                        5.0;                                                                                                                                                                                                                                              x                                                           1.11                                                        4.4;
                                                                                                                                                                                                                                            F                                             3.04(3)                                                                                                                                                                                                                                                                                                                     F                                              3.004(3) 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                              x                                                          1.011                                                               4.04;                                                                                                                                                                                                                                     x                                                           1.0011                                                                      4.004;
                                                                                                                                                                                                                                            F                                              3.0004(3) 
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                              x                                                                                                                                                4.0004;
                                                                                                                                                                                                                                                                                                           1.00011 
                                                                                                                                                                                                   (b)   The rate of change of F ( x) at x  1 is 4. 
                                                                                                                                                                                                                                            g       g (2) g (1)                                                                                                                                                                                                                                                                                                                                                         g       g (1.5) g (1)     
                                                                                                                                                                                                                                                                                                                                                                                         21                                                                                                                                                                                                                                                                                                                                                                                      1.5 1
                                                                                                                                                        23.   (a)                                                                                                                                                                                                                                                                 0.414213                                                                                                                                                                                                                                                                                                                                                                                        0.449489
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                             x                                                                 21                                                                            21                                                                                                                                                                                                                         x                                                               1.51                                                                                             0.5
                                                                                                                                                                                                                                            g       g (1h) g (1)        1h 1
                                                                                                                                                                                                                                                                                                                                                                                           
                                                                                                                                                                                                                                             x                                                           (1h)1                                                                                                              h
                                                                                                                                                                                                                                                                                                           
                                                                                                                                                                                                   (b)  g ( x)                                                                                                      x
                                                                                                                                           
                                                                                                                                                                                                                                        1  h                                                                                                                                               1.1            1.01            1.001            1.0001          1.00001      1.000001 
                                                                                                                                                                                                                                                       1  h                                                                                                                                1.04880    1.004987    1.0004998    1.0000499    1.000005    1.0000005 
                                                                                                                                                                                                                                                               1  h  1 /h                                                                                                                 0.4880      0.4987        0.4998          0.499            0.5              0.5 
                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                   (c)   The rate of change of g ( x) at x  1 is 0.5. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                               1h 1                                                                       1 
                                                                                                                                                                                                   (d)   The calculator gives lim                                                                                                                                                                                                                                                                                            . 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     h           2
                                                                                                                                                                                                                                                                                                                                                                                                                        h0                                                                                                                           
                                                                                                                                                                                                                                                                                                                                                                               1                  1         1
                                                                                                                                                                                                                                                                          f (3)f (2)                                                                                                  
                                                                                                                                                                                                                                                                                                                                                                               3    2          6              1
                                                                                                                                                        24.   (a)   i)                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                             32                                                                        1                                             1                                                  6
                                                                                                                                                                                                                                                                                                                                                                                    1    1          2                                                                 T  
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                                          f(T)f (2)                                                                                                                                                                                                                        2T                                                                                    2T                                                                                                          1         
                                                                                                                                                                                                                                                                                                                                                                                 T                    2         2T                                                2T  
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      , T     2
                                                                                                                                                                                                                                           ii)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                  T 2                                                                           T 2                                              T 2                                                                        2T (T 2)                                                                            2T (2T )         2T 
                                                                                                                                                                                                                                            T                                                                                                                                                                                                                            2.1                  2.01               2.001            2.0001            2.00001        2.000001 
                                                                                                                                                                                                 (b)
                                                                                                                                                                                                                                                  f (T )                                                                                                                                                                                                                 0.476190        0.497512       0.499750      0.4999750      0.499997      0.499999 
                                                                                                                                                                                                                                           ( f (T )  f (2))/(T  2)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         
                                                                                                                                                                                                                                                                                                                                                                                                                                                                         0.2381         0.2488        0.2500        0.2500          0.2500       0.2500
                                                                                                                                                                                                                                           
                                                                                                                                                                                                   (c)   The table indicates the rate of change is 0.25 at t  2.
                                                                                                                                                                                                                                                                                             1                                                                         1
                                                                                                                                                                                                   (d)                                        lim                                                                                       
                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                         T 2  2T            4
                                                                                                                                           
                                                                                                                                                       NOTE: Answers will vary in Exercises 25 and 26.
                                                                                                                                                                                                                                                                                             s                                     150                                                                                                                                                                                               s                                       2015                                                            10                                                                                                                                                          s                                      3020   
                                                                                                                                                        25.   (a)                                                                        [0, 1]:                                                                                                                               15 mph; [1, 2.5]:                                                                                                                                                                                                                                                                      mph; [2.5, 3.5]:                                                                                                                                                                                                                           10 mph
                                                                                                                                                                                                                                                                                              t       10                                                                                                                                                                                                                               t       2.51       3                                                                                                                                                                                                                                                  t      3.52.5
                                                                                                                                           
                                                                                                                                           
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Copyright  2018 Pearson Education, Inc. 
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