3.2. USING DERIVATIVES TO DESCRIBE FAMILIES OF FUNCTIONS
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(d) Describe how the location of the critical numbers and the inflection point of p
change as a changes. That is, if the value of a is increased, what happens to
the critical numbers and inflection point?
2. Let q(x) =
e −x
x − c
be a one-parameter family of functions where c > 0.
(a) Explain why q has a vertical asymptote at x = c.
(b) Determine lim
x→∞
q(x) and lim
x→−∞
q(x).
(c) Compute q ′ (x) and find all critical numbers of q.
(d) Construct a first derivative sign chart for q and determine whether each critical
number leads to a local minimum, local maximum, or neither for the function
q.
(e) Sketch a typical member of this family of functions with important behaviors
clearly labeled.
3. Let E(x) = e
−
(x−m) 2
2s 2
, where m is any real number and s is a positive real number.
(a) Compute E ′ (x) and hence find all critical numbers of E.
(b) Construct a first derivative sign chart for E and classify each critical number
of the function as a local minimum, local maximum, or neither.
(c) It can be shown that E ′′ (x) is given by the formula
E
′′ (x) = e
−
(x−m) 2
2s 2
(x − m) 2 − s 2
s 4
.
Find all values of x for which E ′′ (x) = 0.
(d) Determine lim
x→∞
E(x) and lim
x→−∞
E(x).
(e) Construct a labeled graph of a typical function E that clearly shows how
important points on the graph of y = E(x) depend on m and s.