Linearization of scalar valued functions, Real Analysis II

We do a quick overview of the first-order approximation for real-valued functions of n variables, emphasizing the connection to concepts like the gradient and tangent planes. In the special case where the codomain of f is R (so the output is a scalar), the Jacobian matrix Df(p) is a row vector made up of the partial derivatives of the function. This matrix is the transpose of the gradient vector, and the first-order approximation centered at p is written as L(x) = f(p) + ∇f(p).(x - p), where the dot product is used to emphasize the special form of the gradient for scalar-valued functions. [Playlist:    • Real Analysis II (nearly finished)  ] (MA 426 Real Analysis II, Lecture 35) Then we look at familiar examples, notably the tangent line approximation for functions of one variable and the tangent plane approximation for functions of two variables. We explore how to use these approximations to estimate the function's value at nearby points. Lastly, we discuss the generalization to functions of three variables and how the same principles apply. #AdvancedCalculus #FirstOrderApproximation #Gradient #TangentPlane #TangentLine #MultivariableCalculus #JacobianMatrix #Mathematics #MathEducation #RealAnalysis #Differentiation