180
3.2. USING DERIVATIVES TO DESCRIBE FAMILIES OF FUNCTIONS
find all values of t such that L ′′ (t) = 0 and hence construct a second derivative
sign chart. For which values of t is a function in this family concave up?
concave down?
(c) What is the value of lim
t→∞
A
1 + ce −kt ? lim
t→−∞
A
1 + ce −kt ?
(d) Find the value of L(x) at the inflection point found in (b).
(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 L and how this behavior depends on A, c, and kcritical number.
(f) Explain why it is reasonable to think that the function L(t) models the growth
of a population over time in a setting where the largest possible population the
surrounding environment can support is A.
⊳
Summary
In this section, we encountered the following important ideas:
• Given a family of functions that depends on one or more parameters, by investigating
how critical numbers and locations where the second derivative is zero depend on the
values of these parameters, we can often accurately describe the shape of the function
in terms of the parameters.
• In particular, just as we can created first and second derivative sign charts for a single
function, we often can do so for entire families of functions where critical numbers and
possible inflection points depend on arbitrary constants. These sign charts then reveal
where members of the family are increasing or decreasing, concave up or concave down,
and help us to identify relative extremes and inflection points.
critical number
Exercises
1. Consider the one-parameter family of functions given by p(x) = x 3 − ax 2 , where a > 0.
(a) Sketch a plot of a typical member of the family, using the fact that each is a
cubic polynomial with a repeated zero at x = 0 and another zero at x = a.
(b) Find all critical numbers of p.
(c) Compute p ′′ and find all values for which p ′′ (x) = 0. Hence construct a second
derivative sign chart for p.
3.2. USING DERIVATIVES TO DESCRIBE FAMILIES OF FUNCTIONS
find all values of t such that L ′′ (t) = 0 and hence construct a second derivative
sign chart. For which values of t is a function in this family concave up?
concave down?
(c) What is the value of lim
t→∞
A
1 + ce −kt ? lim
t→−∞
A
1 + ce −kt ?
(d) Find the value of L(x) at the inflection point found in (b).
(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 L and how this behavior depends on A, c, and kcritical number.
(f) Explain why it is reasonable to think that the function L(t) models the growth
of a population over time in a setting where the largest possible population the
surrounding environment can support is A.
⊳
Summary
In this section, we encountered the following important ideas:
• Given a family of functions that depends on one or more parameters, by investigating
how critical numbers and locations where the second derivative is zero depend on the
values of these parameters, we can often accurately describe the shape of the function
in terms of the parameters.
• In particular, just as we can created first and second derivative sign charts for a single
function, we often can do so for entire families of functions where critical numbers and
possible inflection points depend on arbitrary constants. These sign charts then reveal
where members of the family are increasing or decreasing, concave up or concave down,
and help us to identify relative extremes and inflection points.
critical number
Exercises
1. Consider the one-parameter family of functions given by p(x) = x 3 − ax 2 , where a > 0.
(a) Sketch a plot of a typical member of the family, using the fact that each is a
cubic polynomial with a repeated zero at x = 0 and another zero at x = a.
(b) Find all critical numbers of p.
(c) Compute p ′′ and find all values for which p ′′ (x) = 0. Hence construct a second
derivative sign chart for p.