9.6 Gradients of Tangents, Equations of Tangents and Normals

9.6 Gradients of Tangents, Equations of Tangents and Normals


If A(x1, y1) is a point on a line y = f(x), the gradient of the line (for a straight line) or the gradient of the tangent of the line (for a curve) is the value of dy dx when x = x1.

(A) Gradient of tangent at A(x1, y1):



(B) Equation of tangent:



(C) Gradient of normal at A(x1, y1):




(D) Equation of normal :  




Example 1 (Find the Equation of Tangent)
Given that y= 4 ( 3x1 ) 2 . Find the equation of the tangent at the point (1, 1).

Solution:
y= 4 ( 3x1 ) 2 =4 ( 3x1 ) 2 dy dx =2.4 ( 3x1 ) 3 .3 dy dx = 24 ( 3x1 ) 3 At point ( 1, 1 ),  dy dx = 24 [ 3( 1 )1 ] 3 = 24 8 =3 Equation of tangent at point ( 1, 1 ) is, y1=3( x1 ) y1=3x+3 y=3x+4


Example 2 (Find the Equation of Normal)
Find the gradient of the curve y= 7 3x+4 at the point (-1, 7). Hence, find the equation of the normal to the curve at this point.

Solution:
y= 7 3x+4 =7 ( 3x+4 ) 1 dy dx =7 ( 3x+4 ) 2 .3 dy dx = 21 ( 3x+4 ) 2 At point ( 1, 7 ),  dy dx = 21 [ 3( 1 )+4 ] 2 =21 Gradient of the normal = 1 21 Equation of the normal is y y 1 =m( x x 1 ) y7= 1 21 ( x( 1 ) ) 21y147=x+1 21yx148=0