Question

Find the equation of the tangent line, in a slope intercept form, to the curve?

Answer 1

The equation of the tangent is `y = 23/6x +3`

Explanation:

We use implicit differentiation to find the slope of the tangent line.

`2y(dy/dx) - (7y + 7x(dy/dx)) + 3x^2 - 2 = 0`

`2y(dy/dx) - 7y - 7x(dy/dx) + 3x^2 - 2 = 0`

`2y(dy/dx) - 7x(dy/dx) = 2 - 3x^2 + 7y`

`dy/dx = (2 - 3x^2 + 7y)/(2y - 7x)`

The slope of the tangent will be given by evaluating our point within the derivative.

`dy/dx|_(0 , 3) = (2 - 3(0)^2 + 7(3))/(2(3) - 7(0))`

`dy/dx|_(0,3) = 23/6`

Then the equation of the tangent is

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

`y - 3 = 23/6(x - 0)`

`y = 23/6x + 3`

Here is a graphical verification:

Hopefully this helps!