The Geometric Meaning of Differentials by masterwu

steemiteducation · @masterwu · Jan 9 '18 (edited)

$15.96

The Geometric Meaning of Differentials

Over my last 3 posts, we've learned to construct a tangent line <em>L</em>(<em>x</em>) on any given point on a differentiable curve. The picture below shows the geometric meaning of a differential.

<center>![w8.png](https://steemitimages.com/DQmUDeFu81XHDc5X3DBNYd8PPTusnvDbNMrtNk3tqrK25q1/w8.png)<br/>Figure 1.<br/><em>Created with: www.desmos.com/calculator</em></center>

Now to explain what it means. Suppose we have the function...

<center>![w1.png](https://steemitimages.com/DQmTVVWrGej1QPBC3WuempdfVJH1xJmuZ9RAsh9AMo3PSpi/w1.png)</center>

We note that <em>y</em> is the <em>dependent variable</em> and <em>x</em> is the <em>independent variable</em> which can be any real number.

If <em>f</em>(<em>x</em>) is a differentiable function, then the differential <em>dy</em> can be written in terms of the differential <em>dx</em> as...

<center>![w2.png](https://steemitimages.com/DQmUtZ1Vmht9K2XViTLrfaiqpGE7FweSWg5Vh2cD5r2QmL9/w2.png)</center>

Thus <em>dy</em> is also a dependent variable, since it depends on the values of <em>x</em> and <em>dx</em>. What the graph shows above is that <em>dy</em> measures the rise or fall of the tangent line <em>L</em>(<em>x</em>) at a given point along the curve.

However, what we can see is in the picture above, given the distance of <em>dx</em>, there is quite a large discrepancy between the rise of the tangent line <em>dy</em> and the rise of the curve <em>f</em>(<em>x</em>).

A more detailed picture of this is shown below, where Δ<em>y</em> is the rise of the curve.

<center>![w14.png](https://steemitimages.com/DQmPMKaoAKUTZ48fP99Jv4RY1xC9SdsT9CDta6hQRTcH7Yo/w14.png)<br/>Figure 2.<br/><em>Created with: www.desmos.com/calculator</em></center>

Now, from Figure 2, the coordinates of the point <em>Q</em> are:

<center>![w15.png](https://steemitimages.com/DQmXR6megTgG3CHHuqXmPd39sPv41JBFvW2zLJAzNpbRTUL/w15.png)</center>

The coordinates of the point <em>P</em> are simply:

<center>![w16.png](https://steemitimages.com/DQmbvy5M7GsJ2CcL9gq44qdKVKdCHCv6rBJfJixRd9qe44M/w16.png)</center>

Thus the vertical distance between <em>Q</em> and <em>P</em> then is the rise of the curve Δ<em>y</em>...

<center>![w17.png](https://steemitimages.com/DQmVvJS1dQTzuEhfxhshsY1vzjapurEFAZmBKSekgqym3KV/w17.png)</center>

As you can see from the picture, the difference between Δ<em>y</em> and <em>dy</em> become smaller and smaller as Δ<em>x</em> or <em>dx</em> decreases.

In the next tutorial, we will compare Δ<em>y</em> and <em>dy</em> for a defined function.

<hr />

Here's a list of posts created so far on the subject of Linear Approximations and Differentials:

<ol>
<li><a href="https://steemit.com/steemiteducation/@masterwu/linear-approximations-part-1-interpolating-between-empirical-data">Linear Approximations Part 1 - Interpolating between Empirical Data</a></li>
<li><a href="https://steemit.com/steemiteducation/@masterwu/linear-approximations-part-2-estimating-values-of-f-x-x-2-near-x-2">Linear Approximations Part 2 - Estimating values of f(<em>x</em>) = √(<em>x</em>+2) near <em>x</em> = 2</a></li>
<li><a href="https://steemit.com/steemiteducation/@masterwu/linear-approximations-part-3-why-sin-x-x-near-0">Linear Approximations Part 3 - why sin(<em>x</em>) ≅ <em>x</em> near 0</a></li>
<li>The Geometric Meaning of Differentials</li>
</ol>

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