topics_11_1_11_2

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 13

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online