first order taylor approximation

• a linear function that approximates a differentiable function near a given point by using its value and gradient at that point.
• useful for simplifying complex functions and estimating their behavior around a point of interest.
• used in various applications, such as numerical methods, error analysis, and machine learning algorithms.
• obtained by adding the product of the gradient and the difference between the input variable and the given point to the value of the function at that point.
• $f(x)\approx f(x_0)+\nabla f(x_0{)}^T(x-x_0)+H.O.T.$
  • higher order terms