3.2. USING DERIVATIVES TO DESCRIBE FAMILIES OF FUNCTIONS
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behavior of p if the constant a is positive? Why? What if the constant a is
negative? In each case, describe the relative extremes of p.
(c) Find p ′′ (x) and construct a second derivative sign chart for p. What does this
tell you about the concavity of p? What role does a play in determining the
concavity of p?
(d) Without using a graphing utility, sketch and label typical graphs of p(x) for the
cases where a > 0 and a < 0. Label all inflection points and local extrema.
(e) Finally, use a graphing utility to test your observations above by entering and
plotting the function p(x) = x 3 − ax for at least four different values of a. Write
several sentences to describe your overall conclusions about how the behavior
of p depends on a.
⊳
Activity 3.5.
Consider the two-parameter family of functions of the form h(x) = a(1 − e −bx ), where
a and b are positive real numbers.
(a) Find the first derivative and the critical numbers of h. 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 lim
x→∞
a(1 − e
−bx )? lim
x→−∞
a(1 − e
−bx )?
(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.
⊳
Activity 3.6.
Let L(t) =
A
1 + ce −kt , where A, c, and k are all positive real numbers.
(a) Observe that we can equivalently write L(t) = A(1 + ce −kt ) −1 . Find L ′ (t)
and explain why L has no critical numbers. Is L always increasing or always
decreasing? Why?
(b) Given the fact that
L
′′ (t) = Ack
2 e
−kt ce −kt − 1
(1 + ce −kt ) 3 ,
179
behavior of p if the constant a is positive? Why? What if the constant a is
negative? In each case, describe the relative extremes of p.
(c) Find p ′′ (x) and construct a second derivative sign chart for p. What does this
tell you about the concavity of p? What role does a play in determining the
concavity of p?
(d) Without using a graphing utility, sketch and label typical graphs of p(x) for the
cases where a > 0 and a < 0. Label all inflection points and local extrema.
(e) Finally, use a graphing utility to test your observations above by entering and
plotting the function p(x) = x 3 − ax for at least four different values of a. Write
several sentences to describe your overall conclusions about how the behavior
of p depends on a.
⊳
Activity 3.5.
Consider the two-parameter family of functions of the form h(x) = a(1 − e −bx ), where
a and b are positive real numbers.
(a) Find the first derivative and the critical numbers of h. 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 lim
x→∞
a(1 − e
−bx )? lim
x→−∞
a(1 − e
−bx )?
(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.
⊳
Activity 3.6.
Let L(t) =
A
1 + ce −kt , where A, c, and k are all positive real numbers.
(a) Observe that we can equivalently write L(t) = A(1 + ce −kt ) −1 . Find L ′ (t)
and explain why L has no critical numbers. Is L always increasing or always
decreasing? Why?
(b) Given the fact that
L
′′ (t) = Ack
2 e
−kt ce −kt − 1
(1 + ce −kt ) 3 ,