1103

• Run Multiple Choice Generator at h:\shares\physics
• Run Core Problems by giving the program 174 as your 'course'--enter this when asked for your 3-digit course number.  
• Do the 'quiz' on Exponential Functions
• If you miss a problem be sure to click on Comment and explain what you do and do not understand about the question and the given solution

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

• The t = 1 point lies higher than the t = 0 point, because when we multiply A by b we get a number bigger than A (remember b > 1)
• the horizontal asymptote will be the negative t axis, because as we move along the negative t axis we get bigger and bigger negative powers of t, so b^t will be a number with a bigger and bigger denominator, and
• the function will be increasing.

If 0 < b < 1 then

• the t = 1 point lies lower than the t = 0 point,
• the horizontal asymptote will be the positive t axis, and
• the function will be decreasing.
  
   

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.