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(Section 1.10: Difference Quotients)
1.10.1
SECTION 1.10: DIFFERENCE QUOTIENTS
LEARNING OBJECTIVES
• Define average rate of change (and average velocity) algebraically and graphically.
• Be able to identify, construct, and evaluate (and simplify) various forms of
difference quotients.
PART A: DISCUSSION
• This section will revisit Section 0.14 on slopes of lines, Section 1.1 on function
evaluations, and Section 1.2 on graphs.
• A difference quotient
is used to find the
slope
of a secant line
to a graph or to find
an average rate of change
(perhaps an average velocity
). Difference quotients will
form the basis for derivatives, tangent lines, and instantaneous rates of change in
Section 1.11.
PART B: SECANT LINES and AVERAGE RATE OF CHANGE
The secant line
to the graph of a function
f
on the interval
a
,
b
±
²
³
´
, where
a
<
b
,
is the line that passes through the points
a
,
fa
()
and
b
,
fb
.
The average rate of change
of
f
on
a
,
b
±
²
³
´
is equal to the
slope
of this secant line,
which is given by:
rise
run
=
±
b
±
a
.
We call this a difference quotient
, because it has the form:
difference of outputs
difference of inputs
.
• (See Footnote 1 on assumptions about
f
.)
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1.10.2
PART C: AVERAGE VELOCITY
The following development of average velocity will help explain the association
between slope and average rate of change.
Example 1 (Average Velocity)
A car is driven due north 100 miles during a twohour trip.
What is the average velocity of the car?
• Let
t
=
the time (in hours) elapsed since the beginning of the trip.
• Let
y
=
st
()
, where
s
is the position function
for the car (in miles).
s
gives the
signed
distance of the car from the starting position.
•• The position (
s
) values would be
negative
if the car were
south
of
the starting position.
• Let
s
0
=
0
, meaning that
y
=
0
corresponds to the starting position.
Therefore,
s
2
=
100
(miles).
The average velocity
on the timeinterval
a
,
b
±
²
³
´
is the
average rate of change of position with respect to time
. That is,
change in position
change in time
=
±
s
±
t
where
±
(uppercase delta) denotes “change in”
=
sb
±
sa
b
±
a
, a
difference quotient
Here, the average velocity on
0, 2
±
²
³
´
is:
s
2
±
s
0
2
±
0
=
100
±
0
2
=
50
miles
hour
or
mi
hr
or mph
²
³
´
µ
¶
·
TIP 1
:
The unit of velocity is the unit of
slope
given by:
unit of
s
unit of
t
.
(Section 1.10: Difference Quotients)
1.10.3
The average velocity is 50 mph on
0, 2
±
²
³
´
in the three scenarios below.
It is the slope of the orange secant line.
We will define instantaneous velocity
(or simply velocity
) in Section 1.11.
• Here, the velocity is constant (50 mph).
• Here, the velocity is increasing; the car is accelerating.
• Here, the car overshoots the destination and then backtracks.
WARNING 1
:
The car’s velocity is
negative
in value when it is
backtracking; this happens when the graph falls.
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.
 Fall '11
 staff
 Calculus, Algebra, Rate Of Change

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