[微積分]考古題

[steve1012 (steve)]

九六期末第三題的B


B) Let f'(x) = kf(x) for all x in some interval.
 Prove that f(x) = Ce^kx  ,
 where C is an arbitrary constant.

兩個證明方法

 (M1)   set f(x)=ce^kx
      => f'(x)=kce^kx=k*f(x)

 (M2)   這題其實是Seperable ODE
       set y=f'(x)
       we can rewrite the equation as follow

       dy
      ---- = k y
       dx

       seperate x y

       dy
      ----  = k dx
       y

intergrate both sides

     ln y = kx  + C

    =>  y=  e^(kx+C)  = e^kx  * e^C = C' e^kx

      (((C'=e^c)))


然後cross section perpendicular to  the x-axis就是截面積跟x軸垂直
繞著X軸旋轉的意思


以上


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