06Tangents and Normals - slope 12 3 6 3 4 12 3 6 3 4 12 3 4...

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T ANGENTS AND N ORMALS We already know that the gradient (slope) of a curve at any point is equal to the gradient of the tangent to the curve at that point The normal is the line that is perpendicular to the tangent of a curve at a point. The gradient of the normal will therefore be the negative reciprocal of the gradient of the tangent. Ex . Find the equations of the tangent line and the normal line to the curve 2 2 + = - x xe y at the point where the curve crosses the y-axis. Pt. ) 2 , 0 ( ) 2 1 ( 2 2 2 2 x e xe e y x x x - = - = - - - 1 ) 0 1 ( ) 0 ( 0 = - = e y Gradient of the tangent = 1 Gradient of the normal = -1 Tangent line 2 ) 0 ( 1 2 + = - = - x y x y Normal line 2 ) 0 ( 1 2 + - = - - = - x y x y Ex . Find the equation of the normal to the curve 2 ) ( + = x x f at the point where 3 = y . 7 2 9 2 3 = + = + = x x x pt. (7, 3) ( 29 2 1 2 2 1 ) ( ' - + = x x f 6 1 2 7 2 1 ) 7 ( ' = + = f Gradient of the tangent = 6 1 Gradient of the normal = -6 Normal line 45 6 ) 7 ( 6 3 + - = - - = - x y x y Ex . Show that if the line c mx y + = is a tangent to the curve 12 3 4 2 2 = + y x , then . 4 3 2 2 + = m c If c mx y + = is tangent, then the point of contact must be given by an equation with a repeated root (because the equation is an ellipse and the point can’t be an endpoint (vertical
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Unformatted text preview: slope)). 12 3 ) ( 6 ) 3 4 ( 12 3 6 3 4 12 ) ( 3 4 2 2 2 2 2 2 2 2 2 =-+ + + =-+ + + = + + c x mc x m c mxc x m x c mx x The discriminant must be 0, therefore ac b 4 2 = , therefore ) 12 3 )( 3 4 ( 4 36 2 2 2 2-+ = c m c m 4 3 192 144 48 144 36 192 48 36 2 2 2 2 2 2 2 2 2 2 + = + =-+-= m c m c m c m c c m Assign: Exercise 18.2 # 2dfg, 6, 7, 8, 10, 12, 14, 15, and questions below. 1) Find the coordinates of the points of intersection of the parabolas x y = 2 and 2 y x = . What are the equations of the tangents to the curves at these points? 2) Show that the equation of the tangent to the rectangular hyperbola 2 c xy = at the point ) , ( k h may be written 2 2 =-+ c yh xk . Find the equation of the tangent which passes through the point ) , ( c ....
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This note was uploaded on 07/12/2011 for the course MATH 241 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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06Tangents and Normals - slope 12 3 6 3 4 12 3 6 3 4 12 3 4...

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