matumich26 said:Homework Statement
Let f be a differentiable function from an open set
[TEX]U\subseteqR^{n}[/TEX] into R. If
[TEX]x,y\inU[/TEX] and the segment [TEX]S={(1-t)x+yt : t\in[0,1]}[/TEX] is contained in U, show that
[TEX]f(y)-f(x)=(Df)_{\xi}(y-x)[/TEX] for some [TEX]\xi\inS[/TEX].
The Attempt at a Solution
The only direction I have is that I need to use the chain rule and the mean value theorem. Beyond that I do not know how to proceed.
The total derivative is a mathematical concept used to represent the rate of change of a function with respect to all of its variables. It takes into account the changes in all of the variables, rather than just one, and can be thought of as the slope of the tangent plane to a graph of a function in multiple dimensions.
A partial derivative measures the rate of change of a function with respect to one variable, while holding all other variables constant. The total derivative, on the other hand, takes into account the changes in all variables and can be thought of as the sum of all partial derivatives.
The total derivative can be used to find the equation of a tangent line to a curve at a specific point. The line segment S is then a representation of this tangent line, with the total derivative being the slope of this line.
The total derivative can be calculated using the gradient vector, which is a vector of partial derivatives. The total derivative is then the dot product of the gradient vector and a vector representing the changes in all variables.
The concept of the total derivative is used in many fields, including physics, engineering, economics, and computer science. It is particularly useful in optimization problems, where the goal is to find the maximum or minimum value of a function. It is also used in multivariable calculus and differential equations.