Find equation of the gradient of the curve, dy/dx = 2x +a

[practice]

Given the equation of gradient of the curve ,dy/dx = 2x +a... and two points on the curve have given.. but I couldn't remember.. because it's a exam question. Find the equation of the curve.
Need steps to solve this type of question. Thank you.

 

[Deleted member 4993]

Given the equation of gradient of the curve ,dy/dx = 2x +a... and two points on the curve have given.. but I couldn't remember.. because it's a exam question. Find the equation of the curve.
Need steps to solve this type of question. Thank you.

The antiderivative of the given equation of the gradient is

y = x² + a*x + C ........................................(1)

This is the equation of the curve with two unknowns - namely 'a' & 'C'.

Suppose the given two points were:

(x₁, y₁) and (x₂, y₂). Then using the derived equation of the curve (1), we get,

y₁ = x₁² + a*x₁ + C ..............................(2)

y₂ = x₂² + a*x₂ + C ..............................(3)

Now solve for 'a' & 'C' - using (2) and (3)

 

[HallsofIvy]

One very confusing point that you introduced was that you introduced (and perhaps confused yourself) was that you titled thus "Find the equation of the gradient" but the in the problem is actually "Given the gradient, find the equation of the curve"!