Chapter 1
Understanding the Derivative
1.1 How do we measure velocity?
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How is the average velocity of a moving object connected to the values of its
position function?
• How do we interpret the average velocity of an object geometrically with regard to
the graph of its position function?
• How is the notion of instantaneous velocity connected to average velocity?
Introduction
Calculus can be viewed broadly as the study of change. A natural and important question
to ask about any changing quantity is “how fast is the quantity changing?” It turns out that
in order to make the answer to this question precise, substantial mathematics is required.
We begin with a familiar problem: a ball being tossed straight up in the air from an
initial height. From this elementary scenario, we will ask questions about how the ball
is moving. These questions will lead us to begin investigating ideas that will be central
throughout our study of differential calculus and that have wide-ranging consequences.
In a great deal of our thinking about calculus, we will be well-served by remembering
this first example and asking ourselves how the various (sometimes abstract) ideas we are
considering are related to the simple act of tossing a ball straight up in the air.
1
Understanding the Derivative
1.1 How do we measure velocity?
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How is the average velocity of a moving object connected to the values of its
position function?
• How do we interpret the average velocity of an object geometrically with regard to
the graph of its position function?
• How is the notion of instantaneous velocity connected to average velocity?
Introduction
Calculus can be viewed broadly as the study of change. A natural and important question
to ask about any changing quantity is “how fast is the quantity changing?” It turns out that
in order to make the answer to this question precise, substantial mathematics is required.
We begin with a familiar problem: a ball being tossed straight up in the air from an
initial height. From this elementary scenario, we will ask questions about how the ball
is moving. These questions will lead us to begin investigating ideas that will be central
throughout our study of differential calculus and that have wide-ranging consequences.
In a great deal of our thinking about calculus, we will be well-served by remembering
this first example and asking ourselves how the various (sometimes abstract) ideas we are
considering are related to the simple act of tossing a ball straight up in the air.
1