Transcript Tangent Project

Going Off on a Tangent
Objectives:
1. To find the slope of a line tangent to a
circle at a given point
Slope
Anything that isn’t completely vertical has a
slope. This is a value used to describe its
incline or decline.
The Slope Game
Summarize your findings about slope in the
table below:
m>0
m<0
m=0
m = undef
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As the absolute value of the slope of a line
the line gets steeper.
increases, --?--.
Slope of a Line
The slope of a line (or
segment) through
P1 and P2 with
coordinates (x1,y1)
and (x2,y2) where
x1x2 is
ryse
ryse
Example 1
1. Describe the line containing the points
(6, -9) and (-3, -9).
2. Describe the line containing the points
(8, 2) and (8, -5).
3. Find the slope of the line with points at
(-1, 5) and (3, 3).
Slope as Rate of Change
Slope is often referred to as rate of change.
Why is the rate of change for any given
line always constant?
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Slope as Rate of Change
Slope is often referred to as rate of change.
Would the rate of change be constant for
other graphs, like circles or parabolas?
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Slope as Rate of Change
For these graphs, the rate of change is
different at every point along the curve.
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Slope as Rate of Change
It was the search for this rate of change that
eventually lead to the discovery of
differential calculus.
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Slope as Rate of Change
Calculus tells us that the rate of change at
any given point on a graph is equal to the
slope of the tangent line at that point.
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Tangent
Tangent
A line is a tangent if
and only if it
intersects a circle
in one point.
Tangent Line Theorem
In a plane, a line is
tangent to a circle if
and only if the line
is perpendicular to a
radius of the circle
at its endpoint on
the circle.
Example 2
Find the slope of
the tangent line
at (4, 1) to a
circle with
center at (-1, 2).
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(-1, 2)
2
(4, 1)
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Tangent Line Problem
Finding the slope of a
tangent line to a circle
is fairly easy, even
though you only have
one point on the line.
You simply find the
slope of the radius, and
then take the negative
reciprocal.
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Tangent Line Problem
But what about other
curves? For
example, shown is
the graph of
y = 6 – x2. How
would we find the
slopeof the tangent
line at, say, (1, 5)?
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Tangent Line Problem
The problem is that a
parabola, or most
other curves, do not
have a radius that is
perpendicular to the
tangent line at any
given point, and we
only have one point
on the line.
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Tangent Line Problem
It was the resolution
of this problem, by
Fermat, Newton,
and Leibniz that led
to the discovery of
differential calculus.
It begins with
another line, called
the secant line.
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Secant
Secant
A line is a secant if
and only if it
intersects a circle
in two points.
A Series of Secants
Watch how a series
of secants can get
closer and closer to
the tangent line.
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A Series of Secants
Watch how a series
of secants can get
closer and closer to
the tangent line.
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A Series of Secants
Watch how a series
of secants can get
closer and closer to
the tangent line.
10
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6
4
2
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A Series of Secants
Watch how a series
of secants can get
closer and closer to
the tangent line.
10
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6
4
2
-5
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A Series of Secants
Watch how a series
of secants can get
closer and closer to
the tangent line.
10
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Example 3
The graph of y = 6 – x2 is
shown. Confirm that the
points (1, 5) and (2, 2)
lie on the parabola; that
is, they are solutions to
the equation. Now find
the slope of the secant
line through those two
points.
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Example 4
Now find the slope of the
secant line through the
points (1, 5) and (0, 6).
What conclusion can you
draw about the slope of
the tangent line at
point (1, 5).
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Example 5
Finally, find the slope of
the tangent line at the
point (1, 5).
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Assignment
• Going Off on a
Tangent Project