First Order Taylor Approximation

Description:

• 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.
• we have $f(x)$
• Find $\nabla f(z)=\left[\begin{array}{c}\frac{\partial f}{\partial z_0}(z)\\ \frac{\partial f}{\partial z_1}(z)\\ ...\end{array}\right]$
• The approximation is $\hat{f}(x)=\nabla f(z)+\nabla f(z{)}^T(x-z)$
  • which is the linear approximation at point x
  • $\nabla f(z)$ is calculated by derive, then put the value to find in

For higher term:

• $f(x)\approx f(x_0)+\nabla f(x_0{)}^T(x-x_0)+H.O.T.$
  • higher order terms