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tlimit

W: %pi*(D^4-x^4)/(32*x);

t: (D-x)/2;

ex1: M/W*tlimit(ex1,x...

Calculate

tlimit

W: %pi*(D^4-x^4)/(32*x);

t: (D-x)/2;

ex1: M/W*tlimit(ex1,x...

Calculate

tlimit

Run Example
(%i1)tlimit(5*x^4+2*x^2-6,x,5),numer;
(%o1)                               3169.0
(%i2) 
Run Example
globalsolve: true;
(%o1)                                true
(%i2) realonly: true;
(%o2)                                true
(%i3) /* Standard (x,y)->
 (r,theta) coordinate translation */x(t) := r(t) * cos(theta(t));
(%o3)                     x(t) := r(t) cos(theta(t))
(%i4) y(t) := r(t) * sin(theta(t));
(%o4)                     y(t) := r(t) sin(theta(t))
(%i5) /* Constant course assumption */declare (slope,constant);
(%o5)                                done
(%i6) declare (slope,real);
(%o6)                                done
(%i7) declare (yintercept,constant);
(%o7)                                done
(%i8) declare (yintercept, real);
(%o8)                                done
(%i9) y(t)=slope * x(t) + yintercept;
(%o9)     r(t) sin(theta(t)) = slope r(t) cos(theta(t)) + yintercept
(%i10) eq1: y(t)=slope * x(t) + yintercept;
(%o10)    r(t) sin(theta(t)) = slope r(t) cos(theta(t)) + yintercept
(%i11) /* Known & constant sound velocity */declare (soundvel,constant);
(%o11)                               done
(%i12) assume(soundvel>
0);
(%o12)                          [0 < soundvel]
(%i13) /* Known and constant base frequency */declare (basefrequency, constant);
(%o13)                               done
(%i14) assume(basefrequency>
0);
(%o14)                        [0 < basefrequency]
(%i15) /* f-- observed frequency. Standard doppler equation*/declare(f, real);
(%o15)                               done
(%i16) let (diff(r(t),t), (f(t)/basefrequency-1)*soundvel);
                  d                           f(t)
(%o16)            -- (r(t)) --> soundvel (------------- - 1)
                  dt                      basefrequency
(%i17) diff(r(t),t) = (f(t)/basefrequency-1)*soundvel;
                   d                         f(t)
(%o17)             -- (r(t)) = soundvel (------------- - 1)
                   dt                    basefrequency
(%i18) /* Differentiate eq1 and solve for r(t) */eq3: letsimp(solve([diff(y(t),t) = slope * diff(x(t),t)],r(t)));
(%o18) [r(t) = 
                     slope soundvel cos(theta(t))
- -------------------------------------------------------------------
                       d                               d
  slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
                       dt                              dt
                         soundvel sin(theta(t))
 + -------------------------------------------------------------------
                        d                               d
   slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
                        dt                              dt
 + (slope soundvel f(t) cos(theta(t)))/(basefrequency slope sin(theta(t))
  d                                             d
 (-- (theta(t))) + basefrequency cos(theta(t)) (-- (theta(t))))
  dt                                            dt
 + (- (soundvel f(t) sin(theta(t)))/(basefrequency slope sin(theta(t))
  d                                             d
 (-- (theta(t))) + basefrequency cos(theta(t)) (-- (theta(t)))))]
  dt                                            dt
(%i19) /* Differentiate eq3 and solve for slope */eq4: letsimp(diff(eq3,t));
                     soundvel f(t)
(%o19) [- soundvel + ------------- = 
                     basefrequency
                                      d
              soundvel cos(theta(t)) (-- (theta(t)))
                                      dt
-------------------------------------------------------------------
                     d                               d
slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
                     dt                              dt
                                            d
              slope soundvel sin(theta(t)) (-- (theta(t)))
                                            dt
 + -------------------------------------------------------------------
                        d                               d
   slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
                        dt                              dt
                    d
 + (slope soundvel (-- (f(t))) cos(theta(t)))
                    dt
                                     d
/(basefrequency slope sin(theta(t)) (-- (theta(t)))
                                     dt
                                d
 + basefrequency cos(theta(t)) (-- (theta(t))))
                                dt
                 d
 + (- (soundvel (-- (f(t))) sin(theta(t)))
                 dt
                                     d
/(basefrequency slope sin(theta(t)) (-- (theta(t)))
                                     dt
                                d
 + basefrequency cos(theta(t)) (-- (theta(t)))))
                                dt
                                    d
 + (- (soundvel f(t) cos(theta(t)) (-- (theta(t))))
                                    dt
                                     d
/(basefrequency slope sin(theta(t)) (-- (theta(t)))
                                     dt
                                d
 + basefrequency cos(theta(t)) (-- (theta(t)))))
                                dt
                                          d
 + (- (slope soundvel f(t) sin(theta(t)) (-- (theta(t))))
                                          dt
                                     d
/(basefrequency slope sin(theta(t)) (-- (theta(t)))
                                     dt
                                d
 + basefrequency cos(theta(t)) (-- (theta(t)))))
                                dt
                                                        2
                                                       d
 + (slope soundvel cos(theta(t)) (slope sin(theta(t)) (--- (theta(t)))
                                                         2
                                                       dt
                   2
                  d                                d             2
 + cos(theta(t)) (--- (theta(t))) - sin(theta(t)) (-- (theta(t)))
                    2                              dt
                  dt
                        d             2
 + slope cos(theta(t)) (-- (theta(t))) ))
                        dt
                       d                               d              2
/(slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t))))
                       dt                              dt
                                                     2
                                                    d
 + (- (soundvel sin(theta(t)) (slope sin(theta(t)) (--- (theta(t)))
                                                      2
                                                    dt
                   2
                  d                                d             2
 + cos(theta(t)) (--- (theta(t))) - sin(theta(t)) (-- (theta(t)))
                    2                              dt
                  dt
                        d             2
 + slope cos(theta(t)) (-- (theta(t))) ))
                        dt
                       d                               d              2
/(slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))) )
                       dt                              dt
 + (- (slope soundvel f(t) cos(theta(t))
                                      2
                                     d
 (basefrequency slope sin(theta(t)) (--- (theta(t)))
                                       2
                                     dt
                                 2
                                d
 + basefrequency cos(theta(t)) (--- (theta(t)))
                                  2
                                dt
                                d             2
 - basefrequency sin(theta(t)) (-- (theta(t)))
                                dt
                                      d             2
 + basefrequency slope cos(theta(t)) (-- (theta(t))) ))
                                      dt
                                         d
/expt(basefrequency slope sin(theta(t)) (-- (theta(t)))
                                         dt
                                d
 + basefrequency cos(theta(t)) (-- (theta(t))), 2))
                                dt
 + (soundvel f(t) sin(theta(t)) (basefrequency slope sin(theta(t))
   2                                              2
  d                                              d
 (--- (theta(t))) + basefrequency cos(theta(t)) (--- (theta(t)))
    2                                              2
  dt                                             dt
                                d             2
 - basefrequency sin(theta(t)) (-- (theta(t)))
                                dt
                                      d             2
 + basefrequency slope cos(theta(t)) (-- (theta(t))) ))
                                      dt
                                         d
/expt(basefrequency slope sin(theta(t)) (-- (theta(t)))
                                         dt
                                d
 + basefrequency cos(theta(t)) (-- (theta(t))), 2)]
                                dt
(%i20) eq6: trigsimp(solve(eq4, slope));
                        2                                          2
(%o20) [slope = (sqrt((f (t) - 2 basefrequency f(t) + basefrequency )
   2
  d              2                               d           d
 (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t))) (-- (theta(t)))
    2                                            dt          dt
  dt
   2
  d                       2                                             2
 (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t) - 8 basefrequency )
    2
  dt
  d             4    d         2  d             2
 (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
  dt                 dt           dt
                                  2
 + ((2 f(t) - 2 basefrequency) sin (theta(t)) - f(t) + basefrequency)
   2
  d
 (--- (theta(t))) + (2 basefrequency - 2 f(t)) cos(theta(t)) sin(theta(t))
    2
  dt
  d             2    d              d             2             d
 (-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) sin (theta(t))) (-- (theta(t))))
  dt                 dt             dt                          dt
                                                           2
                                                          d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
                                                            2
                                                          dt
                                  2
 + ((2 f(t) - 2 basefrequency) sin (theta(t)) + 2 f(t) - 2 basefrequency)
  d             2      d
 (-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t))
  dt                   dt
  d                                  2
 (-- (theta(t)))), slope = - (sqrt((f (t) - 2 basefrequency f(t)
  dt
                     2
                2   d              2                               d
 + basefrequency ) (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t)))
                      2                                            dt
                    dt
                   2
  d               d                       2
 (-- (theta(t))) (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t)
  dt                2
                  dt
                  2   d             4    d         2  d             2
 - 8 basefrequency ) (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
                      dt                 dt           dt
                                  2
 + ((2 f(t) - 2 basefrequency) cos (theta(t)) - f(t) + basefrequency)
   2
  d
 (--- (theta(t))) + (2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t))
    2
  dt
  d             2    d              d             2             d
 (-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) cos (theta(t))) (-- (theta(t))))
  dt                 dt             dt                          dt
                                                           2
                                                          d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
                                                            2
                                                          dt
                                  2
 + ((2 basefrequency - 2 f(t)) cos (theta(t)) + 4 f(t) - 4 basefrequency)
  d             2      d                                       d
 (-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t)) (-- (theta(t))))]
  dt                   dt                                      dt
(%i21) /* Now we have a number of estimates for the slope constant. *//* We can use any statistical technique to estimate the actual constant *//* Constant speed assumption */eq7: letsimp(diff(letsimp(diff(x(t),t)),t)) = 0;
                                             2
                                            d
(%o21) (- basefrequency r(t) sin(theta(t)) (--- (theta(t)))
                                              2
                                            dt
                                     d             2
 - basefrequency r(t) cos(theta(t)) (-- (theta(t)))
                                     dt
                                  d
 - 2 soundvel f(t) sin(theta(t)) (-- (theta(t)))
                                  dt
                                           d
 + 2 basefrequency soundvel sin(theta(t)) (-- (theta(t)))
                                           dt
             d
 + soundvel (-- (f(t))) cos(theta(t)))/basefrequency = 0
             dt
(%i22) eq8: solve(eq7, r(t));
(%o22) [r(t) = - ((2 soundvel f(t) - 2 basefrequency soundvel) sin(theta(t))
  d                          d
 (-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t)))
  dt                         dt
                                2
                               d
/(basefrequency sin(theta(t)) (--- (theta(t)))
                                 2
                               dt
                                d             2
 + basefrequency cos(theta(t)) (-- (theta(t))) )]
                                dt
(%i23) eq9: subst(eq8, eq1);
(%o23) - (sin(theta(t)) ((2 soundvel f(t) - 2 basefrequency soundvel)
                d                          d
 sin(theta(t)) (-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t))))
                dt                         dt
                                2
                               d
/(basefrequency sin(theta(t)) (--- (theta(t)))
                                 2
                               dt
                                d             2
 + basefrequency cos(theta(t)) (-- (theta(t))) ) = 
                                dt
yintercept - (slope cos(theta(t)) ((2 soundvel f(t) - 2 basefrequency soundvel)
                d                          d
 sin(theta(t)) (-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t))))
                dt                         dt
                                2
                               d
/(basefrequency sin(theta(t)) (--- (theta(t)))
                                 2
                               dt
                                d             2
 + basefrequency cos(theta(t)) (-- (theta(t))) )
                                dt
(%i24) eq10: solve(eq9, yintercept);
(%o24) [yintercept = - (((2 soundvel f(t) - 2 basefrequency soundvel)
    2
 sin (theta(t)) + (2 basefrequency slope soundvel - 2 slope soundvel f(t))
                               d
 cos(theta(t)) sin(theta(t))) (-- (theta(t)))
                               dt
             d
 - soundvel (-- (f(t))) cos(theta(t)) sin(theta(t))
             dt
                   d             2
 + slope soundvel (-- (f(t))) cos (theta(t)))
                   dt
                                2
                               d
/(basefrequency sin(theta(t)) (--- (theta(t)))
                                 2
                               dt
                                d             2
 + basefrequency cos(theta(t)) (-- (theta(t))) )]
                                dt
(%i25) /* Now we have a number of estimates for the yintercept constant. *//* We can use any statistical technique to estimate the actual constant *//* This gives us r(theta) in terms of slope and y_intercept*/eq_rt: solve(eq1, r(t));
                                     yintercept
(%o25)           [r(t) = -----------------------------------]
                         sin(theta(t)) - slope cos(theta(t))
(%i26) tlimswitch: true;
(%o26)                               true
(%i27) tlimit (eq6, theta, 0);
                                   2
(%o27) [slope = limit      (sqrt((f (t) - 2 basefrequency f(t)
                theta -> 0
                     2
                2   d              2                               d
 + basefrequency ) (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t)))
                      2                                            dt
                    dt
                   2
  d               d                       2
 (-- (theta(t))) (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t)
  dt                2
                  dt
                  2   d             4    d         2  d             2
 - 8 basefrequency ) (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
                      dt                 dt           dt
                                  2
 + ((2 f(t) - 2 basefrequency) sin (theta(t)) - f(t) + basefrequency)
   2
  d
 (--- (theta(t))) + (2 basefrequency - 2 f(t)) cos(theta(t)) sin(theta(t))
    2
  dt
  d             2    d              d             2             d
 (-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) sin (theta(t))) (-- (theta(t))))
  dt                 dt             dt                          dt
                                                           2
                                                          d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
                                                            2
                                                          dt
                                  2
 + ((2 f(t) - 2 basefrequency) sin (theta(t)) + 2 f(t) - 2 basefrequency)
  d             2      d
 (-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t))
  dt                   dt
  d                                              2
 (-- (theta(t)))), slope = limit      (- (sqrt((f (t) - 2 basefrequency f(t)
  dt                       theta -> 0
                     2
                2   d              2                               d
 + basefrequency ) (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t)))
                      2                                            dt
                    dt
                   2
  d               d                       2
 (-- (theta(t))) (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t)
  dt                2
                  dt
                  2   d             4    d         2  d             2
 - 8 basefrequency ) (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
                      dt                 dt           dt
                                  2
 + ((2 f(t) - 2 basefrequency) cos (theta(t)) - f(t) + basefrequency)
   2
  d
 (--- (theta(t))) + (2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t))
    2
  dt
  d             2    d              d             2             d
 (-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) cos (theta(t))) (-- (theta(t))))
  dt                 dt             dt                          dt
                                                           2
                                                          d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
                                                            2
                                                          dt
                                  2
 + ((2 basefrequency - 2 f(t)) cos (theta(t)) + 4 f(t) - 4 basefrequency)
  d             2      d
 (-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t))
  dt                   dt
  d
 (-- (theta(t)))))]
  dt
(%i28) 
Run Example
W: %pi*(D^4-x^4)/(32*x);
                                       4    4
                                 %pi (D  - x )
(%o1)                            -------------
                                     32 x
(%i2) t: (D-x)/2;
                                     D - x
(%o2)                                -----
                                       2
(%i3) ex1: M/W*tlimit(ex1,x,D);
                                  32 ex1 x M
(%o3)                            -------------
                                       4    4
                                 %pi (D  - x )
(%i4) 

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