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.

Fig. 1.1 

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} – r^{2} = 0 or (x – h)^{2} + ( y – k)^{2} =
r^{2} is recognized as the circle with centre (h, k) and radius
r), then we can sketch the curve.
Example 2.1
Solution

Fig. 2.1 Curve For Example 2.1. 
EOS
Example 2.2
Solution

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 (x^{2} + y^{2} + ax)^{2} = a^{2}(x^{2} + y^{2}), 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 xAxis
If one replacement yields the same equation, we can conclude
that the curve is symmetric about the xaxis,
without having to

Fig. 3.1 Symmetry About The xAxis. 
bother about the other replacement.
For a curve symmetric about the xaxis,
getting its part above (or below) the xaxis is
enough to sketch the entire curve, as the
part below (or above) the xaxis is
simply the reflection about the xaxis of the
part above (or below).
Symmetry About The yAxis

Fig. 3.2 Symmetry About The yAxis. 
Symmetry About The Origin

Fig. 3.3 Symmetry About The Origin. 
Tangent
Slope Of Tangent

Tangents At The Origin

Fig. 3.4 Tangents At The Origin. 

Angle From Ray To Tangent

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:

Fig. 4.1 The Cardioid. 
EOS
The origin of the name cardioid is clear from the shape of the curve.

Fig. 4.2 An Incorrect Sketch Of The Cardioid At The Origin. 
Solution

Fig. 4.3 
Direction.
EOS
Example 4.3 – The
FourLeaved Rose
Solution

Fig. 4.4 The FourLeaved Rose. 
EOS
Example 4.4 – The Lemniscate
Solution
4) Points Where Tangent Is Horizontal Or Vertical.

Fig. 4.5 The Lemniscate. 
Direction.
EOS
Example 4.5 – The
Spiral Of Archimedes
Solution

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

Fig. 5.1 The Cardioid. 

Fig. 5.2 
Points Other
Than The Origin
^{{fig4.4}}
Fig. 4.4
^{{exa4.3}}
Example 4.3

Fig. 5.3 A Circle And A FourLeaved Rose. 
Finding Points
Of Intersection
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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