Chapter 2.2
In this section we consider the difference between
• average rate of change, and
• instantaneous rate of change. But first we need to consider a new definition.
A line which touches a curved graph at a single point and which is sloped in the same way as the graph at that point is defined as a tangent line.
The point at which a
tangent line touches the graph is the point of
tangency.
There is exactly one tangent line for every point of tangency.
That is, they will intersect the
graph at a
second point near the point of
tangency.
P3
slope too small slope too large
The solid line, T, is a tangent. All
other lines which intersect the graph at the point of tangency will be
A tangent line will lie completely on one side of a graph as shown below …..
….. unless the point of tangency is an inflection point. Then the tangent will cut the curve at the inflection point.
time (hrs) distance (miles)
0 0
.28 22
.65 42
.90 53
1.05 59
1.42 74
1.75 92
The average rate of change
for 1st 1.05 hrs, slope of secant
(1.05, 59) (0, 0)
average velocity over 1st 1.05 hrs =
for entire trip, slope of secant
(1.75, 92)
(0, 0)
average velocity over entire trip =
What is the velocity at exactly 0.28 hours into the trip?
It would be the slope of a secant line connecting the ordered pair (0.28, 22) to a point very close to it.
(.28, 22)
P1=(.80, 49) P2=(.50, 34.5)
P3=(.35, 26.5)
Consider the following sequence of points, P1, P2, and P3, each closer to (0.28, 22):
The slopes of the secants connecting the points, P1, P2, and P3, to (0.28, 22) provide the average velocities
As the sequence of points, P1, P2, and P3, more closely approach (0.28, 22), the more accurately does the slope of the secants connecting the points to (0.28, 22) approach the velocity at exactly 0.28 hours into the road trip.
(.28, 22)
P1 P2
P3
S1 S2 S3
As the sequence of points, P1, P2, and P3, more closely approach (0.28, 22), the secant lines connecting the points more nearly resemble the tangent line at (0.28, 22).
(.28, 22)
S1 T
(.28, 22)
S2 T
(.28, 22)
S3 T
(.28, 22)
S1
average velocity between (0.28, 22) and P1:
P1=(.80, 49)
(.28, 22)
S2
average velocity between (0.28, 22) and P2:
P2=(.50, 34.5)
(.28, 22)
S3
average velocity between (0.28, 22) and P3:
P3=(.35, 26.5)
(.28, 22)
P1 P2
P3
Average velocity
between (0.28, 22) and P1 = 51.9 miles/hr
P2 = 56.8 miles/hr P3 = 64.3 miles/hr Summary
We can graphically estimate the slope of the tangent line by drawing the tangent line at the point (0.28, 22) and then calculating the slope by
using any two ordered pairs on the tangent line.
The slope of the tangent line gives the instantaneous velocity at exactly 0.28 hours since the beginning of the road trip.
(.28, 22)
T (.8, 58)
(0, 3)
instantaneous rate of change = slope of tangent = instantaneous velocity =
(.28, 22)
P1 P2
P3
Average velocity
between (0.28, 22) and P1 = 51.9 mph
P2 = 56.8 mph P3 = 64.3 mph
Instantaneous velocity at (0.28, 22)
= 68.8 mph Summary
• The slope of a secant provides the
average rate of change of
distance with respect to time over an interval of time (average
velocity).
• The slope of the tangent provides
the instantaneous rate of change
of distance with respect to time at a point in time (instantaneous
In general, for any function, f(x):
• The slope of a secant provides the
average rate of change of the
output over an interval of the input variable (rate of change over an
interval).
• The slope of the tangent line
provides the instantaneous rate of
change of the output at a specified input variable point (rate of change at a point).
The instantaneous rate of change of a function may be abbreviated to rate of change of the function.
Problem 1:
The daily profit associated with a food item is shown in the following graph.
units produced daily daily profit ($Th)
Initially profit increases with number of units of food produced.
units produced daily daily profit ($Th)
Eventually, however, supply exceeds demand and some of the units are not sold, decreasing profit.
units produced daily daily profit ($Th)
On the graph a tangent line is drawn at an input value of 60 units and another point on
the tangent line, (70, 28), has been estimated.
units produced daily daily profit ($Th)
Use the two ordered pairs to estimate the rate of change of daily profit when 60 units have been produced.
units produced daily daily profit ($Th)
When 60 units have been produced, the rate of change of daily profit is decreasing. That is:
• Producing less than 60 units generates a larger profit.
• Producing more than 60 units generates a smaller profit.
• When 60 units are produced, profit is decreasing at ���� �� �h���� �� 60�����= (28−29.5) $ h�
(70−60) �����
���� �� �h���� �� 60 �����=−0.15 $ h�
�����
units produced daily daily profit ($Th)
Problem 1 Solved (cont):
The graph also suggests profit will decrease even more severely when more than 60 units are produced.
