Consider the family f(x) = a(1-e^-bx). Assume the parameters a an b are positive.
a). Use limits to determine if this family has any horizontal asymptotes. (Check both directions)
b). Does this family have any axis intercepts? If so, where?
c), Use calculus tools to prove that every curve in this family is increasing.
d). Use your calculus tools to determine if the curves in this family have any inflection points and identify the intervals of concavity.
e). Make a sketch of a single generic curve in this family. Include any asymptotes or points of interest.
f). How do the curves in this family change if b is fixed but a increases? Describe and sketch.
g). Calculate the slope of a general curve in this family at x = 0. What happens to the slope as b increases if a is fixed? How does this affect the graph?