Tangents to Curves Given Parametrically Let C be a curve parametrized by the functions x=x(0),y=t) defined on some interval I.We will assume thatI is an open interval and the parametrizing functions are differentiable. Since a parametrized curve can intersect itself,at a point of Cthere can be (①one tangent,.(ii）two or more tangents,.or(ii）no tangent at all.. one tange two tangents no tangen Figure 10.6.1 Main Menu cmew59时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically Let C be a curve parametrized by the functions x = x(t), y = y(t) defined on some interval I. We will assume that I is an open interval and the parametrizing functions are differentiable. Since a parametrized curve can intersect itself, at a point of C there can be (i) one tangent, (ii) two or more tangents, or (iii) no tangent at all
Tangents to Curves Given Parametrically To make sure that there is at least one tangent line at each point of C,we will make the additional assumption that (10.6.1) [x'(u)+[y'(u)]≠0. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically To make sure that there is at least one tangent line at each point of C, we will make the additional assumption that
Tangents to Curves Given Parametrically secant line (r+l6+h) y'(o) m= tangent line x'(to) (xtol,o》 Figure 10.6.2 The tangent line satisfies (10.6.3) y(to)[x -xo]-x'(to)[y -yo]=0. Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically The tangent line satisfies
Tangents to Curves Given Parametrically Example 1 Find an equation for each tangent to the curve x(0=R,y(0=1-1t∈(-00，∞) at the point(8,-1). Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically Example 1 Find an equation for each tangent to the curve x (t) = t 3 , y (t) = 1 – t at the point (8, -1). t(-, )
Tangents to Curves Given Parametrically Example 2 Find the points on the curve x(1)=3-4sin5,y(1)=4+3cos1 t∈（-oo,o∞) at which there is (i)a horizontal tangent,(ii)a vertical tangent. Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically Example 2 Find the points on the curve x ( t ) = 3 – 4 sin t, y ( t ) = 4 + 3cos t at which there is (i) a horizontal tangent, (ii) a vertical tangent. t (−, )
Tangents to Curves Given Parametrically (3,7) -1,4) (3.4) (7,41 3,1) Figure 10.6.3 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically
Tangents to Curves Given Parametrically Example 3 The curve parametrized by the functions 1-2 x0=1+2 0= 1(1-2) 1+2 1∈(-0o,0o) is called a strophoid.Find equations for the line tangent to the curve at the origin.Then find the points at which there is a horizontal tangent. Main Menu cmew59时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically Example 3 The curve parametrized by the functions is called a strophoid. Find equations for the line tangent to the curve at the origin. Then find the points at which there is a horizontal tangent
Figure 10.6.4 Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Tangents to Curves Given Parametrically Example 4 Take a>0.Find the slope of the spiral r=a at 2 Figure 10.6.5 Main Meny☐ o8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically Example 4 Take a > 0. Find the slope of the spiral r = a θ at θ = . 2 1
Tangents to Curves Given Parametrically Example 5 Find the points of the cardioid r=1-cos 0 at which the tangent line is vertical. [2] polar axis Figure 10.6.6 Main Menu cm墙d
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Tangents to Curves Given Parametrically Example 5 Find the points of the cardioid r = 1 – cos θ at which the tangent line is vertical