Guided Practice for 4.1: Determining distance traveled from velocity

Overview

We begin the next chapter by looking at a new question in calculus: Given a moving object and its velocity function, can we tell how far it’s traveled over a time interval? This is easy if the velocity is constant; just use the old formula distance equals rate times time. But what if the velocity is constantly changing? In answering this question, we’ll invent a method for approximating the distance traveled and discover a backwards connection between this question and derivatives, via the concept of the antiderivative.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• Explain the overall approach described in this section for determining the distance traveled by an object, given its velocity at time t.
• Use a rectangle sum to approximate the distance traveled by an object, given its velocity at time t.
• Explain what is meant by an antiderivative for a function.
• Given functions g(x) and f(x), determine if f(x) is an antiderivative for g(x).
• Interpret the real-world meaning of a negative velocity value.

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Find the exact value of the total distance travelled by an object, given its velocity function, if the velocity function is piecewise linear.
• Given a function g(x) that is a simple linear or polynomial function, find an antiderivative for g(x).

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 4.1. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus).

• Quick review: Determining distance traveled from velocity (3:37)
• Estimating distance traveled using a velocity graph (5:54)
• Estimating distance traveled with data (7:50)
• Finding distance traveled with antiderivatives (8:22)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.