Differentiation

Question

a winding river has been modelled by the curve y=x³ - x² -6x -2 and a road has been modelled by a straight line named AB. the road is a tangent to the river at the point A(1,-8).

find the equation of the tangent at point A and hence find the co-ordinates of B.

can some one explain how the answer is 5x + y+3 = 0 and point B is (-1,2)

Answer 1

First, find the derivative of the wind function.

 

y' = 3x² - 2x - 6

 

Plug in x=1.

 

y' = 3 - 2 - 6

y' = -5

 

This is the slope of the tangent line.  Use the point slope form of line,

 

y = -5(x - 1) - 8

y = -5x - 3

 

Move all terms to the left side of equation.

 

5x + y + 3 = 0

 

Set the line equation equal to the cubic to get your other point of intersection.

 

x³ - x² - 6x - 2 = -5x - 3

 

x³ - x² - x + 1 = 0

 

 

Factor the left side.

 

x²(x - 1) - (x - 1) = 0

 

(x - 1)(x² - 1) = 0

 

(x - 1)²(x + 1) = 0

 

x = 1         and          x = -1

 

 

Since we already have x=1 as the tangent coordinate, x=-1 is the intersecting coordinate.

 

The x-coordinate of point B is -1.  Evaluate the cubic at x=-1 to find the y-coordinate of point B.