Testing in j2se 1.5.0 shows that the vertical scrollbar is at the bottom of its track after each text addition for both JTextArea and TextArea (you did not specify Swing or AWT components). The horizontal scrollbar is all the way to the left in JScrollPane and the ScrollPane scrollbar seems to stay wherever it was placed before the text addition.
It isn't clear from your post what behavior you want.
If you want the scrollbars to be positioned at the beginning of the document you can use setCaretPoistion(0) for each component.
Also, Swing has the JComponent method scrollRectToVisible(Rectangle aRect) (in the JTextArea api) that you can call or override.
For example, say if you want to scroll to the position of the text just added you could save the position of the end of the document before adding the new text,
append the text, use the JTextComponent method modelToView(int pos) to get a Rectangle that you can work with to pass to scrollRectToView. You can use getParent to access the JViewport for size information and use this to accurately specify your Rectangle for scrolling.
A more drastic approach is to get a reference to one or both scrollbars and to programatically set them according to your need.