1103
Basic idea about numbers: If I multiply a positive number by another number and get a bigger number then the number I multiplied by is greater than 1.
The basic points of the exponential function y = A b^t are (0, A) and ( 1, A * b).
If b > 1 then
If 0 < b < 1 then
From before (see notes from last time): To find the
doubling time from the graph of y vs. t draw two horizontal lines which
intersect the graph, one horizontal line twice as far from the t axis as the
other. Project the points where the two lines intersect the graph down to the t
axis. The difference between the two values of t is the doubling time, and will
be the same wherever you draw the lines.
Interest at rate r with interest compounded annually gives us growth rate r and
growth factor b = 1+ r.
The exponential function corrsponding to growth rate r is Q(t) = Q0 ( 1 + r ) ^
t.
If the growth rate is - r the the growh factor is 1 + growth rate = 1 + (-r ) =
1 - r and the exponential function is Q0 ( 1 - r ) ^ t.
The exponential function Q(t) = Q0 * b ^ t doubles between t = 0 and the clock
time t when Q(t) = 2 Q0.
An equation for the doubling time is 2 Q0 = Q0 * b^t, which simplifies to b^t =
2.