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3.2. USING DERIVATIVES TO DESCRIBE FAMILIES OF FUNCTIONS
3.2 Using derivatives to describe families of functions
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
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?
Introduction
Mathematicians are often interested in making general observations, say by describing
patterns that hold in a large number of cases. For example, think about the Pythagorean
Theorem: it doesn’t tell us something about a single right triangle, but rather a fact
about every right triangle, thus providing key information about every member of the right
triangle family. In the next part of our studies, we would like to use calculus to help
us 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 in various settings where only small changes to
certain constants occur. These constants are called parameters.
We 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, period
2π
b , phase shift c, and vertical shift d. We understand from experience with
trigonometric functions that a affects the size of the oscillation, b the rapidity of oscillation,
and c where the oscillation starts, as shown in Figure 3.13, while d affects the vertical
positioning of the graph.
In addition, there are several basic situations that we already understand completely.
For instance, every function of the form y = mx + b is a line with slope m and y-intercept
(0, b). Note that the form y = mx + b allows us to consider every possible line by using two
parameters (except for vertical lines which are of the form x = a). Further, we understand
that the value of m affects the line’s steepness and whether the line rises or falls from left
to right, while the value of b situates the line vertically on the coordinate axes.
For other less familiar families of functions, we would like to use calculus to understand
and classify 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
3.2. USING DERIVATIVES TO DESCRIBE FAMILIES OF FUNCTIONS
3.2 Using derivatives to describe families of functions
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
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?
Introduction
Mathematicians are often interested in making general observations, say by describing
patterns that hold in a large number of cases. For example, think about the Pythagorean
Theorem: it doesn’t tell us something about a single right triangle, but rather a fact
about every right triangle, thus providing key information about every member of the right
triangle family. In the next part of our studies, we would like to use calculus to help
us 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 in various settings where only small changes to
certain constants occur. These constants are called parameters.
We 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, period
2π
b , phase shift c, and vertical shift d. We understand from experience with
trigonometric functions that a affects the size of the oscillation, b the rapidity of oscillation,
and c where the oscillation starts, as shown in Figure 3.13, while d affects the vertical
positioning of the graph.
In addition, there are several basic situations that we already understand completely.
For instance, every function of the form y = mx + b is a line with slope m and y-intercept
(0, b). Note that the form y = mx + b allows us to consider every possible line by using two
parameters (except for vertical lines which are of the form x = a). Further, we understand
that the value of m affects the line’s steepness and whether the line rises or falls from left
to right, while the value of b situates the line vertically on the coordinate axes.
For other less familiar families of functions, we would like to use calculus to understand
and classify 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