Calculus Of One Real Variable – By Pheng Kim
Ving
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1. Same Relation As Cartesian Or Polar Equation |
{1.1} Example 4.1.
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Fig. 1.1 |
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Fig. 1.2 |
In this section we discuss the sketching of polar curves, which are curves of polar equations.
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2. Sketching By Using Cartesian Equations |
To sketch a polar curve given with its polar equation, we can
first try to find its Cartesian equation. If the Cartesian equation is
a function y = f(x), or if it's a relation f(x, y) = 0 which
is recognized (eg (x – h)2 + ( y – k)2 – r2 = 0 or (x – h)2 + ( y – k)2 =
r2 is recognized as the circle with centre (h, k) and radius
r), then we can sketch the curve.
Example 2.1
Solution
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Fig. 2.1 Curve For Example 2.1. |
EOS
Example 2.2
Solution
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Fig. 2.2 Circle With Centre (0, a/2) And Radius a/2. |
EOS
When determining the direction of the curve, we rely on the original polar equation.
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3. Properties Of Polar Curves |
There are cases where the Cartesian equation is of the form f(x, y) = 0 that isn't recognized (eg (x2 + y2 + ax)2 = a2(x2 + y2), where a is a positive constant). There are also cases
where we're asked to use the polar equation itself directly without finding
its corresponding Cartesian equation. A table of values of course is helpful.
Also there are properties of polar curves that can
help us in sketching a polar curve with more accuracy than by using a table of
values alone.
Symmetry
Symmetry About The x-Axis
If one replacement yields the same equation, we can conclude
that the curve is symmetric about the x-axis,
without having to
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Fig. 3.1 Symmetry About The x-Axis. |
bother about the other replacement.
For a curve symmetric about the x-axis,
getting its part above (or below) the x-axis is
enough to sketch the entire curve, as the
part below (or above) the x-axis is
simply the reflection about the x-axis of the
part above (or below).
Symmetry About The y-Axis
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Fig. 3.2 Symmetry About The y-Axis. |
Symmetry About The Origin
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Fig. 3.3 Symmetry About The Origin. |
Tangent
Slope Of Tangent
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Tangents At The Origin
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Fig. 3.4 Tangents At The Origin. |
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Angle From Ray To Tangent
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Fig. 3.5 Angle From Ray To Tangent. |
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4. Sketching By Using Polar Equations |
General
Procedure
{4.1} Part 1.
Solution
the slope of the tangent at the origin at (0, 0) is:
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Fig. 4.1 The Cardioid. |
EOS
The origin of the name cardioid is clear from the shape of the curve.
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Fig. 4.2 An Incorrect Sketch Of The Cardioid At The Origin. |
Solution
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Fig. 4.3 |
Direction.
EOS
Example 4.3 – The
Four-Leaved Rose
Solution
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Fig. 4.4 The Four-Leaved Rose. |
EOS
Example 4.4 – The Lemniscate
Solution
4) Points Where Tangent Is Horizontal Or Vertical.
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Fig. 4.5 The Lemniscate. |
Direction.
EOS
Example 4.5 – The
Spiral Of Archimedes
Solution
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Fig. 4.6 The Spiral Of Archimedes. |
EOS
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5. Intersections Of Polar Curves |
The Origin
{fig4.1}
Fig. 4.1
{exa4.1} Example 4.1
{fig4.3}
Fig. 4.3
{exa4.2}
Example 4.2
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Fig. 5.1 The Cardioid. |
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Fig. 5.2 |
Points Other
Than The Origin
{fig4.4}
Fig. 4.4
{exa4.3}
Example 4.3
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Fig. 5.3 A Circle And A Four-Leaved Rose. |
Finding Points
Of Intersection
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separately, where k is an arbitrary integer. |
Example 5.1
Solution
For The Origin. For the circle r = a/2, r can never be 0. So the origin isn't a point of intersection.
For Points Other Than The Origin.
EOS
Problems & Solutions |
{exa4.1} Example 4.1.
Solution
4) Points Where Tangent Is Horizontal Or Vertical
Solution
Solution
Direction.
Solution
Solution
For The Origin. For the circle r = a, r can never be 0. So the origin isn't a point of intersection.
For Points Other Than The Origin.
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