MAT 136: Calculus I - Spring 2013
Section 1.1: How do we measure velocity?
Section 1.1: How do we measure velocity?
Goal
The goal of this section is to introduce the concepts of average velocity and instantaneous velocity.
Motivating Example
Example 1. Suppose you drive from Phoenix to Flagstaff and your average speed was 71 mph. What does
this mean?
Suppose that a at some moment in time, you look down at your speedometer and it says 53 mph (note:
the needle on speedometer may be moving). We say that your instantaneous velocity at that moment is 53
mph.
Average Rate of Change
Definition 2. The average rate of change of a function f over a closed interval [a, b] is defined via
AV[a,b] =
.
In particular, if f is a position function, then AV[a,b] yields the average velocity over the time interval [a, b].
The Picture:
If we let the distance between a and b be h, then we get an alternative formula:
AV[a,b] =
.
Both formulas are examples of a difference quotient.
Important Note 3.
1. The average rate of change of a function f over [a, b] is the same as the slope of the secant line, msec ,
joining (a, f (a)) and (b, f (b)).
2. The units on AV[a,b] are the units of f (t) per units of t. Examples include mph, meters per second,
etc.
This work is licensed under the Creative Commons Attribution-Share Alike 3.0 License.
Written by D.C. Ernst
MAT 136: Calculus I - Spring 2013
Section 1.1: How do we measure velocity?
Definition 4. Informally, we define the instantaneous velocity of a moving object at time t = a to be the
value that the average velocity approaches as we take smaller and smaller intervals of time containing t = a.
Examples
Let’s play with a few examples.
Example 5. Suppose a ball is thrown off a 100 foot tall building such that the height of the ball in feet at
time t in seconds is given by h(t) = −16t2 + 25t + 100.
(a) What is average rate of change over the first second of flight?
(b) How about over [0, 2]? Interpret the sign of your answer.
Example 6. The position in meters of a particle moving in a straight line is given for some values of time
t in seconds in the following table.
t
p(t)
0
0
.1
.5
.2
.7
.3
1.2
.4
3
Estimate the instantaneous velocity at t = .2 seconds.
Exploration Activities
Now, in groups of 2–4 (no one works alone), work on the following tasks from Active Calculus.
1. Do Activity 1.1. However, for part (a) only compute the average for the first 4 intervals. Also, for
part (c), see if you can answer the question with out a graphing calculator.
2. Do Activity 1.2, but skip part (b).
3. Do Activity 1.3.
Reminder: I expect you to be keeping the work you do for the Exploration Activities in a portfolio that I
will periodically collect.
This work is licensed under the Creative Commons Attribution-Share Alike 3.0 License.
Written by D.C. Ernst