Section3.2Using derivatives to describe families of functions

Motivating Questions

Given a family of functions that depends on one or more parameters, how does the shape of the graph of a typical function in the family depend on the value of the parameters?

How can we construct first and second derivative sign charts of functions that depend on one or more parameters while allowing those parameters to remain arbitrary constants?

Mathematicians are often interested in making general observations, say by describing patterns that hold in a large number of cases. Think about the Pythagorean Theorem: it doesn’t tell us something about a single right triangle, but rather a fact about every right triangle. In the next part of our studies, we use calculus to make general observations about families of functions that depend on one or more parameters. People who use applied mathematics, such as engineers and economists, often encounter the same types of functions where only small changes to certain constants occur. These constants are called parameters.

You are already familiar with certain families of functions. For example, \(f(t) = a \sin(b(t-c)) + d\) is a stretched and shifted version of the sine function with amplitude \(a\text{,}\) period \(\frac{2\pi}{b}\text{,}\) phase shift \(c\text{,}\) and vertical shift \(d\text{.}\) We know that \(a\) affects the size of the oscillation, \(b\) the rapidity of oscillation, and \(c\) where the oscillation starts, as shown in Figure 3.2.1, while \(d\) affects the vertical positioning of the graph.

As another example, every function of the form \(y = mx + b\) is a line with slope \(m\) and \(y\)-intercept \((0,b)\text{.}\) The value of \(m\) affects the line’s steepness, and the value of \(b\) situates the line vertically on the coordinate axes. These two parameters describe all possible non-vertical lines.

For other less familiar families of functions, we can use calculus to discover where key behavior occurs: where members of the family are increasing or decreasing, concave up or concave down, where relative extremes occur, and more, all in terms of the parameters involved. To get started, we revisit a common collection of functions to see how calculus confirms things we already know.

Preview Activity3.2.1.

Let \(a\text{,}\)\(h\text{,}\) and \(k\) be arbitrary real numbers with \(a\neq 0\text{,}\) and let \(f\) be the function given by the rule \(f(x)=a(x-h)^2+k\text{.}\)

Which of the following statements are true about \(f\text{?}\) Select all that apply.

If \(a>0\text{,}\) then \(f\) has a global minimum.

An extreme value occurs at the point \((h,k)\text{.}\)

The maximum value of \(f(x)\) is \(h\text{.}\)

The graph of \(y=f(x)\) is a parabola.

The graph of \(y=f(x)\) is a line.

If \(a>0\text{,}\) then \(f\) has a global maximum.

None of the above

Next we use some calculus to develop familiar ideas from a different perspective. To start, treat \(a\text{,}\)\(h\text{,}\) and \(k\) as constants and compute \(f'(x)\text{.}\)

\(f'(x) =\)

Find a critical value of \(f\text{.}\) (This will depend on at least one of \(a\text{,}\)\(h\text{,}\) and \(k\text{.}\))

Critical value =

Assume that \(a\lt 0\text{.}\) Make a derivative sign chart for \(f\text{.}\) Based on this information, classify the critical value of \(f\) as a maximum or minimum."

A. maximum

B. neither

C. minimum

Subsection3.2.1Describing families of functions in terms of parameters

Our goal is to describe the key characteristics of the overall behavior of each member of a family of functions in terms of its parameters. By finding the first and second derivatives and constructing sign charts (each of which may depend on one or more of the parameters), we can often make broad conclusions about how each member of the family will appear.

Example3.2.2.

Consider the two-parameter family of functions given by \(g(x) = axe^{-bx}\text{,}\) where \(a\) and \(b\) are positive real numbers. Fully describe the behavior of a typical member of the family in terms of \(a\) and \(b\text{,}\) including the location of all critical numbers, where \(g\) is increasing, decreasing, concave up, and concave down, and the long term behavior of \(g\text{.}\)

Solution.

We begin by computing \(g'(x)\text{.}\) By the product rule,

Since we are given that \(a \ne 0\) and we know that \(e^{-bx} \ne 0\) for all values of \(x\text{,}\) the only way this equation can hold is when \(-bx + 1 = 0\text{.}\) Solving for \(x\text{,}\) we find \(x = \frac{1}{b}\text{,}\) and this is therefore the only critical number of \(g\text{.}\)

We construct the first derivative sign chart for \(g\) that is shown in Figure 3.2.3.

Because the factor \(ae^{-bx}\) is always positive, the sign of \(g'\) depends on the linear factor \((1-bx)\text{,}\) which is positive for \(x \lt \frac{1}{b}\) and negative for \(x \gt \frac{1}{b}\text{.}\) Hence we can not only conclude that \(g\) is always increasing for \(x \lt \frac{1}{b}\) and decreasing for \(x \gt \frac{1}{b}\text{,}\) but also that \(g\) has a global maximum at \((\frac{1}{b}, g(\frac{1}{b}))\) and no local minimum.

We turn next to analyzing the concavity of \(g\text{.}\) With \(g'(x) = -abxe^{-bx} + ae^{-bx}\text{,}\) we differentiate to find that

We observe that \(abe^{-bx}\) is always positive, and thus the sign of \(g''\) depends on the sign of \((bx-2)\text{,}\) which is zero when \(x = \frac{2}{b}\text{.}\) Since \(b\) is positive, the value of \((bx-2)\) is negative for \(x \lt \frac{2}{b}\) and positive for \(x \gt \frac{2}{b}\text{.}\) The sign chart for \(g''\) is shown in Figure 3.2.4. Thus, \(g\) is concave down for all \(x \lt \frac{2}{b}\) and concave up for all \(x \gt \frac{2}{b}\text{.}\)

Finally, we analyze the long term behavior of \(g\) by considering two limits. First, we note that

This limit has indeterminate form \(\frac{\infty}{\infty}\text{,}\) so we apply L’Hôpital’s Rule and find that \(\lim_{x \to \infty} g(x) = 0\text{.}\) In the other direction,

because \(ax \to -\infty\) and \(e^{-bx} \to \infty\) as \(x \to -\infty\text{.}\) Hence, as we move left on its graph, \(g\) decreases without bound, while as we move to the right, \(g(x) \to 0\text{.}\)

All of this information now helps us produce the graph of a typical member of this family of functions without using a graphing utility (and without choosing particular values for \(a\) and \(b\)), as shown in Figure 3.2.5.

Note that the value of \(b\) controls the horizontal location of the global maximum and the inflection point, as neither depends on \(a\text{.}\) The value of \(a\) affects the vertical stretch of the graph. For example, the global maximum occurs at the point \((\frac{1}{b}, g(\frac{1}{b})) = (\frac{1}{b}, \frac{a}{b}e^{-1})\text{,}\) so the larger the value of \(a\text{,}\) the greater the value of the global maximum.

The work we’ve completed in Example 3.2.2 can often be replicated for other families of functions that depend on parameters. Normally we are most interested in determining all critical numbers, a first derivative sign chart, a second derivative sign chart, and the limit of the function as \(x \to \infty\text{.}\) Throughout, we prefer to work with the parameters as arbitrary constants. In addition, we can experiment with some particular values of the parameters present to reduce the algebraic complexity of our work. The following activities offer several key examples where we see that the values of the parameters substantially affect the behavior of individual functions within a given family.

Activity3.2.2.

Consider the family of functions defined by \(p(x) = x^3 - ax\text{,}\) where \(a \ne 0\) is an arbitrary constant.

Find \(p'(x)\) and determine the critical numbers of \(p\text{.}\) How many critical numbers does \(p\) have?

Construct a first derivative sign chart for \(p\text{.}\) What can you say about the overall 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 \(p\text{.}\)

Find \(p''(x)\) and construct a second derivative sign chart for \(p\text{.}\) What does this tell you about the concavity of \(p\text{?}\) What role does \(a\) play in determining the concavity of \(p\text{?}\)

Without using a graphing utility, sketch and label typical graphs of \(p(x)\) for the cases where \(a\gt 0\) and \(a \lt 0\text{.}\) Label all inflection points and local extrema.

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 \(a\text{.}\) Write several sentences to describe your overall conclusions about how the behavior of \(p\) depends on \(a\text{.}\)

Activity3.2.3.

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.

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.

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?

What is the value of \(\lim_{x \to \infty} a(1-e^{-bx})\text{?}\)\(\lim_{x \to -\infty} a(1-e^{-bx})\text{?}\)

How does changing the value of \(b\) affect the shape of the curve?

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 \(b\text{.}\)

Activity3.2.4.

Let \(L(t) = \frac{A}{1+ce^{-kt}}\text{,}\) where \(A\text{,}\)\(c\text{,}\) and \(k\) are all positive real numbers.

Observe that we can equivalently write \(L(t) = A(1+ce^{-kt})^{-1}\text{.}\) Find \(L'(t)\) and explain why \(L\) has no critical numbers. Is \(L\) always increasing or always decreasing? Why?

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?

What is the value of \(\lim_{t \to \infty} \frac{A}{1+ce^{-kt}}\text{?}\)\(\lim_{t \to -\infty} \frac{A}{1+ce^{-kt}}\text{?}\)

Find the value of \(L(x)\) at the inflection point found in (b).

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 \(A\text{,}\)\(c\text{,}\) and \(k\) number.

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 \(A\text{.}\)

Subsection3.2.2Summary

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.

Exercises3.2.3Exercises

1.

A function with parameters \(a\) and \(b\) is given. Describe the critical points and possible points of inflection of \(f\) in terms of \(a\) and \(b\text{.}\) Assume \(a, b > 0\text{.}\)

A function with parameters \(a\) and \(b\) is given. Describe the critical points and possible points of inflection of \(f\) in terms of \(a\) and \(b\text{.}\)

\begin{equation*}
f(x) = (x - a)(x - b)
\end{equation*}

Critical point:

Possible point of inflection: (enter DNE if none exist)

3.

Let \(p(x)=x^2(a - x)\text{,}\) where \(a\) is constant and \(a>0\text{.}\)

Find the local maxima and minima of \(p\text{.}\)

(Enter your maxima and minima as comma-separated xvalue,classification pairs. For example, if you found that\(x = -2\)was a local minimum and\(x = 3\)was a local maximum, you should enter(-2,min), (3,max). If there were no maximum, you must drop the parentheses and enter-2,min.)

maxima and minima:

What effect does increasing the value of \(a\) have on the \(x\)-position of the maximum(s) you found? (Enter left, none or right if it moves left, has no effect, or moves right.)

What effect does increasing the value of \(a\) have on the \(x\)-position of the minimum(s) you found? (Enter left, none or right if it moves left, has no effect, or moves right.)

What effect does increasing the value of \(a\) have on the \(y\)-coordinate of the maximum(s) you found? (Enter up, none or down if it moves up, has no effect, or moves down.)

What effect does increasing the value of \(a\) have on the \(y\)-coordinate of the minimum(s) you found? (Enter up, none or down if it moves up, has no effect, or moves down.)

4.

Consider \(f (x) = a x + x \ln(b x)\) for \(a>0\text{,}\)\(b>1\text{,}\) and \(x>1\text{.}\)

Find \(f'(x)\text{:}\)\(f'(x) =\)

Based on your expression for \(f'(x)\text{,}\) is \(f(x)\) increasing or decreasing? (Enter increasing or decreasing.)

(Be sure that you can see why this is true for all values \(x > 1\text{.}\))

Find \(f''(x)\text{:}\)\(f''(x) =\)

Based on your expression for \(f''(x)\text{,}\) is \(f(x)\) concave up or concave down? (Enter up or down.)

(Be sure that you can see why this is true for all values \(x > 1\text{.}\))

5.

Find constants \(a\) and \(b\) in the function \(f(x)=a x e^{b x}\) such that \(f(\frac{1}{9})=1\) and the function has a local maximum at \(x=\frac{1}{9}\text{.}\)

\(a =\)

\(b =\)

6.

Consider the one-parameter family of functions given by \(p(x) = x^3-ax^2\text{,}\) where \(a \gt 0\text{.}\)

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 \(x = a\text{.}\)

Find all critical numbers of \(p\text{.}\)

Compute \(p''\) and find all values for which \(p''(x) = 0\text{.}\) Hence construct a second derivative sign chart for \(p\text{.}\)

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?

7.

Let \(q(x) = \frac{e^{-x}}{x-c}\) be a one-parameter family of functions where \(c \gt 0\text{.}\)

Explain why \(q\) has a vertical asymptote at \(x = c\text{.}\)

Determine \(\lim_{x \to \infty} q(x)\) and \(\lim_{x \to -\infty} q(x)\text{.}\)

Compute \(q'(x)\) and find all critical numbers of \(q\text{.}\)

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 \(q\text{.}\)

Sketch a typical member of this family of functions with important behaviors clearly labeled.

8.

Let \(E(x) = e^{-\frac{(x-m)^2}{2s^2}}\text{,}\) where \(m\) is any real number and \(s\) is a positive real number.

Compute \(E'(x)\) and hence find all critical numbers of \(E\text{.}\)

Construct a first derivative sign chart for \(E\) and classify each critical number of the function as a local minimum, local maximum, or neither.

It can be shown that \(E''(x)\) is given by the formula

Find all values of \(x\) for which \(E''(x) = 0\text{.}\)

Determine \(\lim_{x \to \infty} E(x)\) and \(\lim_{x \to -\infty} E(x)\text{.}\)

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 \(s\text{.}\)