MTH301 Calculus II GDB N0. 1 Discussion Spring 2014 Due Date 8th August 2014

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MTH301 Calculus II GDB N0. 1 Discussion and Solution Spring 2014 Due Date 8th August 2014


GDB Topic:
Discuss the applications of Polar coordinate System in our daily life with at least 3 examples.

Ideal Solution


For convenience, we begin with a given xy -coordinate system and then take the origin as the pole and the nonnegative x -axis as the polar axis. Given the pole O and the polar axis, the point P with polar coordinates r and θ , written as the ordered pair (r,θ , is located as follows. First find the terminal side of the angle θ , given in radians, where θ is measured counterclockwise (if θ>0 ) from the x -axis (the polar axis) as its initial side. If r≥0 , then P is on the terminal side of this angle at the distance r from the origin. If r<0 , then P lies on the terminal side of this angle at the distance r from the origin. If r<0 , then P lies on the ray opposite the terminal side at the distance ∣r∣=-r>0 from the pole. The radial coordinate r can be described as the directed distance from P the pole along the terminal side of the angle θ . Thus, if r is positive, the point P lies in the same quadrant as θ , whereas if r is negative, then P lies in the opposite quadrant. If r=0 , the angle θ does not matter; the polar coordinates (0,θ represent the origin whatever the angular coordinate θ might be. The origin, or pole, is the only point for which r=0 .

[Vuhelper]

7. Polar Coordinates

For certain functions, rectangular coordinates (those using x-axis and y-axis) are very inconvenient. In rectangular coordinates, we describe points as being a certain distance along the x-axis and a certain distance along the y-axis.


polar coordinates
But certain functions are very complicated if we use the rectangular coordinate system. Such functions may be much simpler in the polar coordinate system, which allows us to describe and graph certain functions in a very convenient way.

Polar coordinates work in much the same way that we have seen in trignometry(radians and arc length, where we used r and θ and in the polar form of complex numbers (where we also saw r and θ.

[vuassignments]

For convenience, we begin with a given xy -coordinate system and then take the origin as the pole and the nonnegative x -axis as the polar axis. Given the pole O and the polar axis, the point P with polar coordinates r and θ , written as the ordered pair (r,θ , is located as follows. First find the terminal side of the angle θ , given in radians, where θ is measured counterclockwise (if θ>0 ) from the x -axis (the polar axis) as its initial side. If r≥0 , then P is on the terminal side of this angle at the distance r from the origin. If r<0 , then P lies on the terminal side of this angle at the distance r from the origin. If r<0 , then P lies on the ray opposite the terminal side at the distance ∣r∣=-r>0 from the pole. The radial coordinate r can be described as the directed distance from P the pole along the terminal side of the angle θ . Thus, if r is positive, the point P lies in the same quadrant as θ , whereas if r is negative, then P lies in the opposite quadrant. If r=0 , the angle θ does not matter; the polar coordinates (0,θ represent the origin whatever the angular coordinate θ might be. The origin, or pole, is the only point for which r=0 .