Problem:
Numerically, find a critical point of the function

f(x) = sin(x + 1).


Visualization:
Using Microcalc:

  1. Choose Beginning Calculus from the initial menu and press .
  2. Use the arrow keys to move to the menu item Extrema of F(x), press and .
  3. At the prompt F(x) =, enter the formula
    sin(x+1)
  4. Enter the bounds at the prompts
    x_0 = 0 x_1 = 1
  5. You get the following table:
                     x                 F(x)             F'(x)            F"(x)
    --------------------------------------------------------------------------
           0.00000 000          0.84147 098            0.5403           -0.841
           0.10000 000          0.89120 736            0.4536           -0.891
           0.20000 000          0.93203 909            0.3624           -0.932
           0.30000 000          0.96355 819            0.2675           -0.964
           0.40000 000          0.98544 973            0.1700           -0.985
           0.50000 000          0.99749 499            0.0707           -0.997
           0.60000 000          0.99957 360           -0.0292           -1.000
           0.70000 000          0.99166 481           -0.1288           -0.992
           0.80000 000          0.97384 763           -0.2272           -0.974
           0.90000 000          0.94630 009           -0.3233           -0.946
           1.00000 000          0.90929 743           -0.4161           -0.909
    
    Extremum on  [0.50000 000, 0.60000 000]
    
    Press :
    
  6. From this table, we see that the extremum lies between 0.5 and 0.6.
  7. Press and, using the arrow keys, move to Subdivide and press to get a similar table of values between 0.5 and 0.6.
  8. Repeating the latter step several times, we finally get the following table:
                     x                 F(x)             F'(x)            F"(x)
    --------------------------------------------------------------------------
           0.57079 630          1.00000 000       2.6795 e-08           -1.000
           0.57079 631          1.00000 000       1.6795 e-08           -1.000
           0.57079 632          1.00000 000       6.7949 e-09           -1.000
           0.57079 633          1.00000 000      -3.2051 e-09           -1.000
           0.57079 634          1.00000 000      -1.3205 e-08           -1.000
           0.57079 635          1.00000 000      -2.3205 e-08           -1.000
           0.57079 636          1.00000 000      -3.3205 e-08           -1.000
           0.57079 637          1.00000 000      -4.3205 e-08           -1.000
           0.57079 638          1.00000 000      -5.3205 e-08           -1.000
           0.57079 639          1.00000 000      -6.3205 e-08           -1.000
           0.57079 640          1.00000 000      -7.3205 e-08           -1.000
                                                                              
                                                                              
    Local maximum on  [0.57079 632, 0.57079 633]                              
                                                                              
         F(0.57079 632) = 1.00000 000                                         
                                                                              
         F(0.57079 633) = 1.00000 000                                         
                                                                              
    
    which indicates that the extremum occurs at x = 0.5707963 correct to 7 decimal places.