topics_11_1_11_2

# topics_11_1_11_2 - Topics 11.1 and 11.2 Problems and...

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Topics 11.1 and 11.2, Problems and Solutions 11.1 CURVES DEFINED BY PARAMETRIC EQUATIONS Suppose that a particle moves along a curve C given below Because C fails the vertical test there is no equation of the form y = f ( x ) describing the curve C (the trace of the particle) analytically. Analogously, because the horizontal test is not satisﬁed the curve C has no equation of the form x = g ( y ). However, the x - coordinate and the y -coordinate are functions of the time t and we can determine x and y in terms of t : x = f ( t ) , y = g ( t ) , a t b. Deﬁnition. Suppose that x and y are given as functions of a third variable t called a parameter by x = f ( t ) , y = g ( t ) , a t b parametric equations . Each value of t determines a point ( x,y ) that we plot on the coordinate plane. As t varies from a to b , the point ( x,y ) = ( f ( t ) ,g ( t )) varies also and traces a curve C that we call a parametric curve . 1

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The parameter t does not necessarily represents time and other than t letters (notations) can be used to denote the parameter, for example ( θ,v,s,w etc.). However, in many applications the parameter t denotes the time and we can interpret the point ( x,y ) = ( f ( t ) ,g ( t )) on the coordinate plane as a position of a particle at the time-moment t . The initial point of the curve C is ( f ( a ) ,g ( a )) and the terminal point of the curve C is ( f ( b ) ,g ( b )) because a t b and a is the initial value of t and b is the terminal value of t . Problem 9/ page 662. (a) Sketch the curve by using its parametric equations x = t, y = 1 - t to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to ﬁnd a cartesian equation of the curve. Solution. (a) First , t 0 because t is well deﬁned only for t 0. Hence, 0
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topics_11_1_11_2 - Topics 11.1 and 11.2 Problems and...

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