Tangent Line

[JH_33]

A line tangent to y=x^2 + 1 at x=a, a > 0, intersects the x-axis at point P.

1. Write an expression for the area of the triangle formed by the tangent line, the x-axis, and the line x=a.

2. For what value of a is the area of the triangle a minimum?


I am stumped on how to do this problem, or really even where to begin. Any help would be greatly appreciated.

 

[stapel]

To find the tangent line: The tangent line is the line through the point with that point's slope. And "slope at a point" is "derivative". So take the derivative first. This will give you dy/dx = y'(x). "Evaluate" this at x = a.

Also plug x = a into the original function. This will give you (x, y) = (a, y(a)). (No, this may not be pretty.) With this "point" and the "slope", find the line equation using the old point-slope formula: y - y₁ = m(x - x₁).

1) This is actually easier, I think. Looking at the graph of y, you know that the tangent line for a > 0 has to be increasing, so the base must have width a - P. (Draw the situation, if you're not sure.) The height has to be y(a) = a² + 1. Since the area is fully determined by the height and the width, I think this is all you need.

2) I think this is where we see the need for the tangent line's actual equation: You need to find that x-intercept, so you can plug a value in for "P". Then the "area" formula will be in terms only of "a", and you can minimize.

Eliz.