Find the linearization of f(x) = √25 − x2 at a = 3

Answer 1

Remember that the point-slope form of a line is: y - y₁ = m(x - x₁). If you want to replace the function with a line approximating it, use the point (x₁, y₁) where x₁ = 3 and use the derivative at x = 3 as the slope m.

If x₁ = 3 then y₁ = √(25 - 3²) = 4

f '(x) = -x/(25 - x²)^1/2 (by the power rule and chain rule)

f '(3) = -3/(25 - 3²)^1/2 = -3/4

So the linearization is y - 4 = -3/4(x - 3)

You can make this "fancy" by giving it a function name like L(x) and moving the 4 to the other side like this:

L(x) = 4 - 3/4(x - 3) or multiplying it out and combining like-terms but it's still be same thing.