Greenwood press

140
SYMMETRY

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TANGENT
The term tangent can be used to describe a function (see Periodic Functions)
or a ratio in trigonometry applications (see Triangle Trigonometry). A geomet-
ric tangent is a segment or line that locally touches a curve or figure at one point,
but does not pass through the curve at that location. For example, 
y = x
3
− 3x
2
+ 2x − 7 has a tangent of y = 2x − 11 at the point (2,–7), as shown below. 
The slope of a tangent line represents the derivative of a function at a point.
This value is the same as the instantaneous rate of change of an object with vary-
ing rate. For example, the number of bushels, 
b, of corn removed in a field can be
modeled with the function 
b = 50 − 50e
−0.08h
, where 
h is the number of hours
past 8:00 
AM
. The rate of productivity during any hour of the day can be deter-
mined by evaluating the derivative with a specific value of 
h, which is the same
as the slope of the line tangent to the curve at that point, as shown in the figure
below. Without the derivative, the slope of the tangent line can be approximated
by finding the slope of a secant line that contains two points that are extremely
close to the point of tangency. For example, 
h = 4 at 12:00 
PM
. The production
rate at noon can be approximated by the slope of the line between 11:59:59 and
12:00:01. These times should be converted into decimals so that they can be sub-
stituted into the equation. Since there are 3,600 seconds in an hour, the difference
of 1 second from 12:00 
PM
will be measured as 1/3600, or approximately 
0.000278 hours, from 
h = 4. Using the slope formula, m =
y
2
−y
1
x
2
−x
1
, the slope of
the tangent is 
m ≈
b(4.000278)−b(3.999722)
4.000278−3.999722
≈
.001615
.000556
≈ 2.9 bushels per hour.