Curves given Parametrically Assume a pair of functionsx=x(y=yt)is differentiable on the interior of an interval / y x(),y Figure 10.5.1 When t ranges over I,the point (x().())traces out a path in the xy-plane. We call such a path a parametrized curve and refer to t as the parameter. Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Curves given Parametrically Assume a pair of functions x = x(t), y = y(t) is differentiable on the interior of an interval I. When t ranges over I, the point ( x(t), y(t) ) traces out a path in the xy-plane. We call such a path a parametrized curve and refer to t as the parameter
Curves given Parametrically Example 1 Identify the curve parametrized by the functions x(0=t+1y(0=2t-5 t∈(0，∞) Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Curves given Parametrically Example 1 Identify the curve parametrized by the functions x (t) = t +1 y (t) = 2t -5 t (-, )
Curves given Parametrically Example 2 Identify the curve parametrized by the functions x0=21,y0)=2te[0,oo). Main Menu cmew59时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Curves given Parametrically Example 2 Identify the curve parametrized by the functions
2 Figure 10.5.2 Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Curves given Parametrically Example 3 Identify the curve parametrized by the functions x(t)=sin2t,y(t)=cost t∈[0，π] Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Curves given Parametrically Example 3 Identify the curve parametrized by the functions
x=1-y2 (Figure 10.5.3) 0.1t=0 (0,-1:1=x x()a sin2 1,y(r)=cos r te[0, Figure 10.5.3 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Curves given Parametrically Straight Lines Given that (o yo)=(1 1),the functions (10.5.1) x()=x0+1x1-xo), y)=%+10y1-o) 1∈(-00，) parametrize the line that passes through the points (xo.yo)and (x1,y1). Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Curves given Parametrically Straight Lines Given that (x0 , y0 ) = (x1 , y1 ), the functions parametrize the line that passes through the points (x0 , y0 ) and (x1 , y1 )
Curves given Parametrically Ellipses and Circles Usually we let t range from 0 to 2 and parametrize the ellipse by setting (10.5.2) x(t)=acost, y(t)=bsint t∈[0,2π] Ifb=a,we have a circle.We can parametrize the circle x2+y2=a2 by setting (10.5.3) x(t)=acost, y(t)=asint t∈[0,2π]. Main Meny c的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Curves given Parametrically Ellipses and Circles Usually we let t range from 0 to 2π and parametrize the ellipse by setting If b = a, we have a circle. We can parametrize the circle x 2 + y 2 = a 2 by setting
Curves given Parametrically Hyperbolas Take a,b>0.The functionsx(t)=a cosht.()=b sinh tsatisfy the identity LOILOI- a2 b2 Since x(t)=a cosh t>0 for all t,as t ranges over the set of real numbers,the point (x(t),y(t))traces out the right branch of the hyperbola x2 y2 a2- 61 导后=1 Figure 10.5.4 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Curves given Parametrically Hyperbolas Take a, b > 0. The functions x(t) = a cosh t, y(t) = b sinh t satisfy the identity Since x(t) = a cosh t > 0 for all t, as t ranges over the set of real numbers, the point (x(t), y(t)) traces out the right branch of the hyperbola ( ) ( ) 2 2 2 2 1 x t y t a b − = 2 2 2 2 1 x y a b − =
Curves given Parametrically Example 4 The line segment that joins the points(1,2)and(3,6)is the graph of the function y=2x, 1≤x≤3. Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Curves given Parametrically Example 4 The line segment that joins the points (1, 2) and (3, 6) is the graph of the function y = 2x, 1 ≤ x ≤ 3