According to the diagram of life expectation at birth on page 85 in
Leonard Hayflick's _How and Why We Age_ (taken from Dublin, Lotka and
Spiegelman, _Length of Life_, The Ronald Press, New York 1936 and the
US National Center for Health Statistics) I get something that might
be an exponential after the 1700's, but the information is a bit too
uncertain to tell for sure and it could just as well be a power of
time or something different.
However, the diagram on page 88 shows life expectation at different
ages for the United states 1900-1987, and that curve family is not
exponential, it seems to be quite linear (the data is from National
Center for Health Statistics, Statistical Bulletin, Metropolitan Life
Insurance Company 1987). The rate of increase has even decreased the
last decades, as most of the easily treated causes of death have been
fixed. The increase is around 0.25 years more of life expectacy at
birth per year for males, and somewhat more for females.