Question

The gradient at any point on the curve y=f(x) is given by the equation dy/dx=1/✓x-2.The curve passes through the point (2,3).Find the equation of this curve.​

Answer 1

y = -1.71x + 6.42

Explanation:

The gradient at any point on the curve is given by the expression;

`(dy)/(dx) = (1)/(√(x ) - 2) \\`

At the point (2,3), the gradient of the curve will be:

`(dy)/(dx)|(x = 2) = (1)/(√( 2)-2 ) \\\\(dy)/(dx)|(x = 2) = m = - 1.71`

The equation of this curve at this point can be given by:

`y - y_1 = m(x-x_1)`

Where
`x_1 =2, y_1 = 3`

Substituting these values into the given equation:

`y - 3 = -1.71(x-2)\\y - 3 = -1.71x + 3.42\\y = -1.71x + 3.42 + 3\\y = -1.71x + 6.42`