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      <text>10.2.8 (Tangents and normals)</text>
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      <text><![CDATA[<h1>Tangents and normals</h1>
<p>In the previous section we saw how we can define the derivative at a point on a curve as the derivative of the tangent to the curve at that point. Now we proceed to see how we can use the derivative to find the equation of both tangents and normals to a given curve. Firstly, we recall the following:</p>
<p>The <strong>tangent</strong> to a curve at a point A is the best approximating line to the curve at A.</p>
<p>Recall that we met tangents in weeks 1 and 6, where we examined lines intersecting quadratics and circles. In these cases the tangent <em>touches</em> the curve once - at the point A. However, note that for some functions, the tangent may touch the curve somewhere else. Also, it is possible for the tangent to pass through the curve at the point of tangency. This is called a <strong>point of inflection. </strong>An example is shown in the figure below.</p>
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<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Point%20of%20inflection.png" alt="diagram showing a point of inflection" width="452" height="231" /></p>
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<p>Suppose that we have a curve \(y = f(x)\). If A is the point with x-coordinate \(a\) then the gradient of the tangent to the curve is given by the derivative \(f'(a)\). Then the equation of the tangent is given by 
\[
y - f(a) = f'(a)(x-a) \implies y = f'(a)(x-a) +f(a). 
\]</p>]]></text>
<file name="Point of inflection.png" path="/" encoding="base64">[encoded data, 50604 characters]</file>
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