1.5. INTERPRETING, ESTIMATING, AND USING THE DERIVATIVE
43
1.5 Interpreting, estimating, and using the derivative
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• In contexts other than the position of a moving object, what does the derivative of
a function measure?
• What are the units on the derivative function f ′ , and how are they related to the
units of the original function f ?
• What is a central difference, and how can one be used to estimate the value of the
derivative at a point from given function data?
• Given the value of the derivative of a function at a point, what can we infer about
how the value of the function changes nearby?
Introduction
An interesting and powerful feature of mathematics is that it can often be thought of both
in abstract terms and in applied ones. For instance, calculus can be developed almost
entirely as an abstract collection of ideas that focus on properties of arbitrary functions. At
the same time, calculus can also be very directly connected to our experience of physical
reality by considering functions that represent meaningful processes. We have already
seen that for a position function y = s(t), say for a ball being tossed straight up in the air,
the ball’s velocity at time t is given by v(t) = s ′ (t), the derivative of the position function.
Further, recall that if s(t) is measured in feet at time t, the units on v(t) = s ′ (t) are feet per
second.
In what follows in this section, we investigate several different functions, each with
specific physical meaning, and think about how the units on the independent variable,
dependent variable, and the derivative function add to our understanding. To start, we
consider the familiar problem of a position function of a moving object.
Preview Activity 1.5. One of the longest stretches of straight (and flat) road in North
America can be found on the Great Plains in the state of North Dakota on state highway
46, which lies just south of the interstate highway I-94 and runs through the town of Gackle.
A car leaves town (at time t = 0) and heads east on highway 46; its position in miles from
Gackle at time t in minutes is given by the graph of the function in Figure 1.22. Three
important points are labeled on the graph; where the curve looks linear, assume that it is
indeed a straight line.
(a) In everyday language, describe the behavior of the car over the provided time
43
1.5 Interpreting, estimating, and using the derivative
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• In contexts other than the position of a moving object, what does the derivative of
a function measure?
• What are the units on the derivative function f ′ , and how are they related to the
units of the original function f ?
• What is a central difference, and how can one be used to estimate the value of the
derivative at a point from given function data?
• Given the value of the derivative of a function at a point, what can we infer about
how the value of the function changes nearby?
Introduction
An interesting and powerful feature of mathematics is that it can often be thought of both
in abstract terms and in applied ones. For instance, calculus can be developed almost
entirely as an abstract collection of ideas that focus on properties of arbitrary functions. At
the same time, calculus can also be very directly connected to our experience of physical
reality by considering functions that represent meaningful processes. We have already
seen that for a position function y = s(t), say for a ball being tossed straight up in the air,
the ball’s velocity at time t is given by v(t) = s ′ (t), the derivative of the position function.
Further, recall that if s(t) is measured in feet at time t, the units on v(t) = s ′ (t) are feet per
second.
In what follows in this section, we investigate several different functions, each with
specific physical meaning, and think about how the units on the independent variable,
dependent variable, and the derivative function add to our understanding. To start, we
consider the familiar problem of a position function of a moving object.
Preview Activity 1.5. One of the longest stretches of straight (and flat) road in North
America can be found on the Great Plains in the state of North Dakota on state highway
46, which lies just south of the interstate highway I-94 and runs through the town of Gackle.
A car leaves town (at time t = 0) and heads east on highway 46; its position in miles from
Gackle at time t in minutes is given by the graph of the function in Figure 1.22. Three
important points are labeled on the graph; where the curve looks linear, assume that it is
indeed a straight line.
(a) In everyday language, describe the behavior of the car over the provided time