# A6_Sketching_2a_Blank - MATH10121 A6 Revision Revision A6.1...

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A6. 1 Cartesian and Polar Coordinates The precise location of a point in a plane can be identified using 2 axes at right angles ( π 2 or 90 ) intersecting at an “origin” x θ first axis r second axis y ( x , y ) in Cartesian ( r , θ ) in polar coordinates If the distance of the point from the axes is x in the direction of the more clockwise axis y in the direction of the other axis then the point is at ( x, y ) in “Cartesian” coordinates If the line from the origin to the point: has length r is rotated an angle θ from the most clockwise axis then the point is at ( r, θ ) in “polar” coordinates Note: conventionally, angles are measured anti-clockwise From simple geometry r = x 2 + y 2 x = r cos θ y = r sin θ dold A6. 2 Graphs and Curve-Sketching Using axes, algebraic relationships between two real variables can be drawn as a ‘graph’ in Cartesian form Example 1. sketch the graph of x 5 + y 2 = 1 a straight line (0 , 2) and (5 , 0) on line x y x 5 + y 2 = 1 5 2 Example 2. sketch s = 2 in the ( t, s ) plane and also sketch t = 3 constant s , constant t t s 3 2 s = 2 t = 3 Example 3. sketch xy = b 2 ( b, b ) and ( b, b ) on curve curve cannot cross the axes What is represented in the case b = 0 ? for b = 0 , xy = 0 so x = 0 ( y -axis) or y = 0 ( x -axis) x y xy = b 2 ( b , b ) ( b , b ) ( b , b ) ( b , b ) How would you sketch xy = b 2 ? ( b, b ) and ( b, b ) on curve Revision Revision MATH10121 A6
A6. 3 Sketching Parametric Relationships Sometimes two variables are each given in terms of a third, tracing out a curve as the third variable changes t = 1 Example 1. Sketch ( u, v ) with u = t 3 t , v = 1 t 2 for t [ 1 , 1] t ( u, v ) 1 (0 , 0) 1 2 ( 3 8 , 3 4 ) 0 (0 , 1) 1 2
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