Overview
In Differential Calculus, we have regularly asked the question “What does the derivative of a given function tell us about the function itself?” Indeed, in many different settings, we have been given information about a function’s instantaneous rate of change, and used that information to determine characteristics of the function itself. In Chapter 4, we provide the most accurate and sophisticated answers to this big question. In particular, we will introduce the notion of the definite integral, which is an important counterpart to the derivative.
In Section 4.1, we start with a familiar question, but in reverse: where previously we were interested in an object whose position we know and sought to find its velocity, now we start with velocity, and see if we can find changes in position and/or distance travelled.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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Understand how we can compute the distance of an object moving at a constant rate over a given time period.
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Understand how we interpret the physical meaning of area under a curve that represents the velocity of a moving object.
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Recall that if we know the position, \(s(t)\), of a moving object at time \(t\), then \(s'(t)\) represents the object’s instantaneous velocity at time \(t\).
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Understand how we can compute the distance travelled by a moving object when all we know about the object is its velocity.
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Understand the difference between distance travelled and change in position (displacement).
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Realize the important role that antiderivatives play in the context of the velocity-distance problem.
To prepare for class
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
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Read subsection 4.1.1 in Active Calculus: Area under the graph of the velocity function
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Do some experimentation with the following interactive applets (by Marc Renault):
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Watch the following video (by GVSUmath) which shows how the distance travelled can be estimated using data:
After class
- Read subsection 4.1.2 in Active Calculus: Two approaches: area and antidifferentiation
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Watch the following video (by GVSUmath) where antidifferentiation is used to find the exact distance travelled, and which also mentions why we cannot always use this approach:
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Read subsection 4.1.3 in Active Calculus: When velocity is negative
