/sci/ - Science & Math

Can someone help me intuitively understand this "Mean Value Proposition"?
Its statement is as follows:

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Let \textbf{x} be a point in \mathbb{R}^n and let \mathit{r} be a positive numer. Suppose that the function \mathit{f}: B_r(\textbf{x}) \to \mathbb{R} has first-order partial derivatives. Then if the point \textbf{x} + \textbf{h} belongs to B_r(\textbf{x}), there are points \textbf{z}_1, \textbf{z}_2, ... , \textbf{z}_n in B_r(\textbf{x}) such that:

\mathit{f}(\textbf{x} + \textbf{h}) - \mathit{f}(\textbf{x}) = \sum_{i=1}^{n} h_i \frac{\partial f}{\partial x_i}(\textbf{z}_i)

and

||\textbf{x} - \textbf{z}_i|| < ||\textbf{h}|| for each index \mathit{i} with 1 <= \mathit{i} <= n

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I grok what it means but I can't really visualize it not reason through it.
Thank you!