Activity 3.4.3.
Consider the two-parameter family of functions of the form \(h(x) = a(1-e^{-bx})\text{,}\) where \(a\) and \(b\) are positive real numbers.
(a)
Find the first derivative and the critical numbers of \(h\text{.}\) Use these to construct a first derivative sign chart and determine for which values of \(x\) the function \(h\) is increasing and decreasing.
(b)
Find the second derivative and build a second derivative sign chart. For which values of \(x\) is a function in this family concave up? concave down?
(c)
What is the value of \(\displaystyle \lim_{x \to \infty} a(1-e^{-bx})\text{?}\) \(\displaystyle \lim_{x \to -\infty} a(1-e^{-bx})\text{?}\)
(d)
How does changing the value of \(b\) affect the shape of the curve?
(e)
Without using a graphing utility, sketch the graph of a typical member of this family. Write several sentences to describe the overall behavior of a typical function \(h\) and how this behavior depends on \(a\) and \(b\text{.}\)
