Answers to First Semester Review Problems

(Please e-mail me if you think an answer is wrong.)

1.  x = 36  (Remember to check all answers since both sides were squared)

2.  x = -3,1 (Watch out for zero in the denominator)

3.  x = -2,3  (Solve two equations)

4.  x < 1, x > 5  (Squiggly Line)

5.  (a)  -1   (b)  2   (c) 
     (d) 
                                                closed circle on (0,2)
                                                open circle on (0,1)
     (Be sure to look at correct interval for answers)

6.  x < -3, x > 3  (Squiggly Line -- Radicand must be positive)

7.    (Denominator cannot be zero)

8.  -2 < x < -1, x > 1 (Radicand must be positive and denominator cannot be zero -- use squiggly line)

9.  Use the squiggly line idea to graph it.  Then check your graph in your calculator in a [-5, 5] by [-25, 10] window.

10.   (a)  -1 < x < 0, x > 1  (Graph it in the calculator and look at it) (Just look at the first 4 or 5 lines in this link)

        (b)  x < -1, 0 < x < 1

        (c)  even  (Check f(-x))

        (d)  (-1,-3), (1,-3) (Use 2nd CALC Min in your calculator)

        (e)  (0,0)

11.  (Graph using reflections, translations, and dilations)

       (a)   (b)   (c) 

       (d)   (e)   (f) 

       (g)   (h) 
12.   (a)  x2 + 2x + 4  (Combine the two functions by adding them)

        (b) 

        (c)  4x2 + 12x + 10

        (d)  2x2 + 5

        (e)   (Interchange x and y and solve for y)

        (f)  x - 3
                2

        (g) 

13.  t = 0,2  (Factor)

14.  y = ax (x + 2) (x + 1)(x - 1) (x - 2) where a is any constant.

15.  2x2 - 4x + 3

16.  -1/2, 1, 2, 4  (Use synthetic division and the rational root theorem)

17.  (Check in your calculator in a [-2,10] by [-10, 10] window in dot mode)
       Vertical Asymptote:  x = 4
       Horizontal Asymptote:  y = 1
       x-intercept:  (2,0)
       y-intercept:  (0, 1/2)
       Does not hit its horizontal asymptote
       Plot one more point to the right of x = 4

18.  (Check in your calculator in a [-5,5] by [-5, 5] window in dot mode)
       Vertical Asymptote:  x = 2, x = -2
       Horizontal Asymptote:  y = 1
       x-intercepts:  (1,0), (-1,0)
       y-intercept:  (0,1/4)
       Does not hits its horizontal asymptote
       Even Function / Symmetric to the y-axis
       Plot one more point on the right of x = 2
       Use symmetry to plot the left side

19.  (Check in your calculator in a Zoom 6 window in dot mode.  Trace to x = 6, 7, 8, 9 to see what happens there.  It crosses the horizontal asymptote and comes back toward it.)
       No vertical asymptotes
       Horizontal Asymptote:  y = 3
       Hits its horizontal asymptote at x = 8
       x-intercepts:  ((-1 ± \/61)/6, 0) @ 7/6 and -3/2  (Use quadratic formula)
       y-intercepts:  (0,-5)

20.  (Check in your calculator in a [-5,5] by [-5, 5] window in dot mode)
       Vertical Asymptotes:  x = 1, x = -1
       Slant Asymptote:  y = x
       Hits its slant asymptote at x = 0.
       x-intercept:  (0,0)
       y-intercept:  (0,0)
       Symmetric to origin / Odd Function
       Plot a point between x = 0 and x = 1; plot a point for x > 1
       Use symmetry to graph the left side

21.  (This reduces to y = x - 3.  The graph is the line y = x - 3 with a hole where x = -3  (-3,6).  Graph it in your calculator in a[-15.8, 15.8] by [-7.6, 7.6] window to check it.)

22.  (Check it by graphing it in the calculator.  You should be able to do it by hand.)

23.  2, -1  ± i \/3  (Factor, then use the quadratic formula)

24.  (x + 1)2 + (y - 5)2 = 25

25  2 - i, -1   (Use the complex conjugates theorem and division)

26.  odd  (Test f (-x))

27.  y = 2x - 6  (Use slope-intercept form)

28.  x = -1 and 3

29.  y = 6(x + 1)2 - 2

30.  (9,21) and (-1,1)  (Set the two equations equal to each other and solve for x)

31.  3

32.  none.


33. 

34. 

35.  10

36.  2.5 x 2.5

37.  point discontinuity

38.   y = -0.26x + 124,   r = -0.229   There is a weak negative correlation, so a predicted value for age 48 will not be reliable.

39.  25 + 9i

40.     (Use rules of exponents)
       
41.  (a)  x = 4  (Solving equations)

       (b)  x = 3/2

       (c)  x = -1

       (d)  x = log (4/5) @ -.0969

       (e)  ln .75  @ -.2877

42.  (a)  (Graph in the calculator in a zoom 6 window remembering the graph really continues on in the negative x direction)

       (b)  turns the (a) graph upside down

       (c)  brings the (a) graph down 4 units

       (d)  moves the (a) graph 2 units to the right

       (e)  keep the right side; erase the left side; reflect the right over to the left

43.  (a)  (Graph in the calculator in a zoom 6 window using the change of base formula and graphing y = log x / log 2 remembering the graph really continues on in the negative y direction)

       (b)   (Graph in the calculator in a zoom 6 window remembering the graph really continues on in the negative y direction)

       (c)  Moves the (b) graph 1 unit to the right

       (d)  Turns the negative y-values into positive y-values on the (b) graph.  Just reflect the bottom part over the x-axis and keep the rest.

       (e)  Keep the right side of the (b) graph; erase the left side; reflect the right over to the left

       (f)  Turns the (b) graph upside down and then reflects it over the y-axis

44.  log 5   or    ln 5  @  1.46497  (Use change of base formula)
       log 3          ln 3

45.  2 log x - 2 log y - 3 log z  (Use laws of logarithms)

46.      (Use laws of logarithms)
         
47.  Graph in your calculator.  It is a parabola with vertex (1.5,6.25), zeros at x = 4, -1, facing down, shaded above the curve.

48.  Graph in your calculator.  It is a V with vertex at (1,0) and it is shaded above the curve.

49.   x > 6 or x < -12

50.  Increases for x < -5/3  and x > 2;  Decreases for -5/3 < x < 2

51.  vertical:  x = -5; x = -1;  slant:  y = 2x - 4 (divide)

52.  (a)  x = 5  (Solving equations)

       (b)  x = e3.8 + 1 @ 45.7012

       (c)  x = ln 10 @ 2.3026

       (d)  x = log 570 @ 2.7559

       (e)  x = log 7 @ 1.7712
                    log 3

53.  x = 5.0618, .000000000038  (Solve in your calculator: F2 Solve or F5 zero)

54.  (a)  ln  x + 1   =   ln x2
                  x - 2
             x + 1   =    x2
             x - 2
             x + 1 = x3 - 2x2
             0 = x3 - 2x2 - x - 1
             x = 2.5468   (Solve in calculator with F2 solve or F5 zero)
             (This is a good answer since the domain of the original is x > 2)

       (b) ln [x(x - 2)] = 1
             x2 - 2x = e1
             x2 - 2x - e = 0
               or   x @ 2.9283, -.9283 (Can use the quadratic formula)
             (Throw away the negative answer because the domain of the original is x > 2)

       (c)  2 ln x = 7
             ln x = 7/2
             x = e7/2   @    33.1155
             Alternate Solution:  ln x2 = 7
                                           x2 = e7
                                           x =   =  ± 33.1155
                                   (Throw away - 33.1155 because the domain of the original is x > 0)
                                          x = 33.1155

55.    (Take the i's out first)


56.  -1 + 7i  (Multiply by the conjugate of the denominator over itself)
           10

57.    (Definition of e)

58.  x < -1, x > 1   (Use the squiggly line and the fact that the x2 - 1 > 0)

59.  x > 19.5801 (ln both sides)

60.  y = (x - 2.5)2  - 5.25

61.  y = (7/3) xz2     y = 175
62.   (a) 

        (b)  x < 1/3, x > 1

63.  (a)  y = x + 1 -     4     or   y = x2 - x - 6   is one answer.
                                x - 2                   x - 2
           (Can start with  (x + 2) (x + a)  =  x + 1 +   R       )
                                               x - 2                            x - 2
           (The left side includes the zero and the vertical asymptote while the right side includes the slant asymptote and the vertical asymptote)

       (b)  y = 3(x - 1)  is one answer.
                     x + 2

64.  (a)  7p/3, -5p/3 (There are others; just add ± 2p as many times as you like to these)

       (b)  472o, -248o(Others may be obtained by adding ± 360o as many times as you like to these)

65.  120o  (Substitute 180o for p    or  multiply by 180/p)

66.  13p/9  (Multiply by p / 180)

67.  Study your chart for special angles  OR  your special right triangles  OR  your trig circle  OR  simply memorize them

68. 
       ( x = -3 \/5 using the Pythagorean Theorem and the fact that A is in quadrant II)

69.  (a)  b = 2 \/3  (Pythagorean Theorem)
             A = 30o  (Use a trig function)
             B = 60o  (Subtract 90 - A)

       (b)  B = 60o (Subtract 90 - A)
             c = 50 \/3  @  86.6025  (Use a trig function)
             a = 25 \/3 @  43.3013   (Use the Pythagorean Theorem)

       (c)  B = 45o (Subtract 90 - A)
             a = 20   (Since the triangle is isosceles)
             c = 20 \/2  (45o- 45o right triangle)

70.  (Using the Pythagorean Theorem, r = \/101)
      

71.  (Using the Pythagorean Theorem and the fact that A is in quadrant III, x = - \/3)
       cos A = - \/3 / 2 ;  tan A = 1/ \/3 ;  sec A = -2 / \/3 ;  csc A = -2 ;  cot A = \/3

72.  (a)  1   (b)  -1/2   (c)  - \/2 / 2   (d)  2    (e)  -2    (f)  -1
       (These should be done from memory)

73.  (a)  .6428      (b)  -1.2799 (Type 1/tan 142 in degree mode)   (c)  2.5593 (Type 1/cos 67 in degree mode)

       (d)  -1.7434  (Type 1/sin 215 in degree mode)

74.  (Check these by putting them in  your calculator in radian mode and graphing them in the zoom 7 window.  You should be able to graph them by hand.)

       (a)  Amplitude:  2     Period:  2p/3      Horizontal Shift:  right p/4      Vertical Shift:  up 1

       (b)  Amplitude:  1     Period:  2p         Horizontal Shift:  right p/3      Vertical Shift:  0      Graph is upside down

       (c)  y-intercept: 2      Period:  2p         Horizontal Shift:  0                 Vertical Shift:  0
             (Graph the cosine first; then draw asymptotes where the cosine = 0; then draw U's at high and low points)

       (d)  (p/4, 0) is a point on the graph     Period:  p      Horizontal Shift:  0     Vertical Shift:  down 1
 
75.  303° 13´ 12´´

76.  114.436°

77.         |
              |  +    + -+--+
              |
              |                .

78.  7.54 m/s

79.  y =          500                      (Substitute the points in and solve simultaneously)

             1 + 3.148e.275t

80.  t = 32.600  (Let y = 100 and solve for t)

81.  (a)   y = 5.2449(1.5524)x  

       (b)  y = 5.2449e0.4398x  

       (c)  about 1.58 hrs

82.  (a)  75.5225  (Type 2nd cos (1/4) in degree mode.  The mode is a matter of choice or instruction here)

       (b)  5.7392   (Type 2nd sin (1/10) in degree mode)

       (c)  80.5377  (Type 2nd tan (6) in degree mode)

       (d)  85.2198  (Type 2nd cos (1/12) in degree mode)

83.  (a)  Check your graph by graphing tan-1x in radian mode in a zoom 7 window)

        (b)  Move (a) graph down 3 units

        (c)  Double the y-values in the graph of (a)

        (d)  Everything below the x-axis in the (a) graph gets reflected above the x-axis

        (e)  Keep the right side of the (a) graph; erase the left side; reflect the right over to the left

        (f)  Move (a) graph 3 units to the right

       (Check all of these in your calculator.  You should be able to do them by hand)

84.  3/5  (Inverse Trig Functions)

85.  8/5

86.    for  0 < x < 2

87.  

88.  120o  or  2p/3

89.  -13/5

90.


91. 

92.  16 ft

93. 

94.  3/5

95.  4/3

96.  a)  18.5o    b)  58.5o     c)  24 hours   

       d)   y = 18.5 sin[(p/12)(t + 6)] + 58.5  OR  y = 18.5 cos  (pt /12 ) + 58.5

       e)  53.7o   within 2 degrees