From 680c747687c10c979c3befa890df835ed77c626d Mon Sep 17 00:00:00 2001 From: Jari Kasandiredjo Date: Wed, 7 Feb 2024 14:40:44 +0100 Subject: [PATCH] fix typo --- preliminaries/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/preliminaries/index.md b/preliminaries/index.md index b11f9ce..01b5ca6 100644 --- a/preliminaries/index.md +++ b/preliminaries/index.md @@ -1280,7 +1280,7 @@ slides: true
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Here's a visualization of what it means to take the partial derivative as a point x=0, y=5, with respect to x.

We let the function vary with x, keeping y fixed. This gives us a kind of "slice" through the surface f. The result is that we have a one-dimensional function again, for which we can take the derivative. We do this by only caring about the way f changes if we change x. In other words, by treating y as a constant.

Then we do the same for y, treating x as a constant.

For both curves, the derivatives give us the slope of the tangent line. We've shown these as dotted lines here.

These lines cross the same point, so together, they lie in a shared plane. In higeher dimensions, the tangent lines of all partial derivatives lie in a shared hyperplane. This is the tangent hyperplane. The hyperplane that just touches the surface of f.

How do we describe the tangent hyperplane? Note what the tangent slopes indicate: how much the plane moves up if we take a step of 1 along the x axis and how much the plane moves up if we take a step along the y axis. These are exactly the roles of the constants in the function xw1 + yw2 + b. Or, in vector notation the elements of the vector w in the function xTw + b.

This tells us that if we take all of our partial derivatives and stick them in a vector w, the function that will describe our tangent hyperplane is xTw + b (for some value of b, which we don't usually care about).

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Here's a visualization of what it means to take the partial derivative as a point x=0, y=5, with respect to x.

We let the function vary with x, keeping y fixed. This gives us a kind of "slice" through the surface f. The result is that we have a one-dimensional function again, for which we can take the derivative. We do this by only caring about the way f changes if we change x. In other words, by treating y as a constant.

Then we do the same for y, treating x as a constant.

For both curves, the derivatives give us the slope of the tangent line. We've shown these as dotted lines here.

These lines cross the same point, so together, they lie in a shared plane. In higher dimensions, the tangent lines of all partial derivatives lie in a shared hyperplane. This is the tangent hyperplane. The hyperplane that just touches the surface of f.

How do we describe the tangent hyperplane? Note what the tangent slopes indicate: how much the plane moves up if we take a step of 1 along the x axis and how much the plane moves up if we take a step along the y axis. These are exactly the roles of the constants in the function xw1 + yw2 + b. Or, in vector notation the elements of the vector w in the function xTw + b.

This tells us that if we take all of our partial derivatives and stick them in a vector w, the function that will describe our tangent hyperplane is xTw + b (for some value of b, which we don't usually care about).

click image for animation