How to find a tangent equation when only given a curve equation and the x axis intercept of the tangent

Question

The Equation of the curve is y = 3*sqrt(4x+1). I am told the tangent intercepts the x axis at x = -4. I need to find at what point the tangent equation will intercept the curve.

Answer 1

Hi Seb R.,

So, the tangent line has a value = the first derivative of the given function: f ' = 3*(1/2)*(4x+1)^(-1/2)*4 = 6*(4x+1)^(-1/2) [check my chain rule math!]. Now, how can you assemble everything you have? Write the slope of the tangent line to an unknown x-coordinate, on the function curve. That might look like:

slope = .delta.y / .delta.x = 3*(4x+1)^(0.5) / (x+ 4.1) Rationale: .delta.y is from the tangent point down to the x-axis; .delta.x is the total x-separation from the tangent point back to the x-axis intercept.

that must equal the first derivative at the desired tangent point = 6*(4x+1)^(-1/2)

I'm hoping you have the algebra to solve that equality, it really isn't that hard?

-- Cheers, -- Mr. d.