fix typo

preliminaries/index.md

@@ -1294,7 +1294,7 @@ slides: true
             <img src="00.Preliminaries.key-stage-0095anim0.png" data-images="00.Preliminaries.key-stage-0095anim0.png,00.Preliminaries.key-stage-0095anim1.png,00.Preliminaries.key-stage-0095anim2.png,00.Preliminaries.key-stage-0095anim3.png,00.Preliminaries.key-stage-0095anim4.png,00.Preliminaries.key-stage-0095anim5.png" class="slide-image" />
 
             <figcaption>
-            <p    >Here is the whole process again in a simple animation.<br></p><p     class="list-item">We start with a function with two inputs and one output. In this case the function f(<span class="red">x</span>, <span class="blue">y</span>) = <span class="red">x</span><sup>2</sup> + <span class="blue">y</span><sup>2</sup> - ¼<span class="red">x</span><span class="blue">y</span> + <span class="blue">x</span> - <span class="red">y</span> +1<br></p><p     class="list-item">We pick a point on the function, in this case (<span class="red">10</span>, <span class="blue">20</span>).<br></p><p     class="list-item">We can define two partial derivatives. One is <span class="red">the derivative of f as x varies and y is kept fixed</span>.<br></p><p     class="list-item">The other is the derivative of <span class="blue">f as y varies and x is kept fixed</span>.<br></p><p     class="list-item">Both of these are functions of one variable, so we can apply what we know from univariate calculus to work out the derivatives. At our point (<span class="red">10</span>, <span class="blue">20</span>), this gives us a tangent line touching the red function and a tangent line touching the blue function. <br></p><p     class="list-item">Since these lines cross, they lie in a shared hyperplane. That is the plane that (in most cases) just touches but does not cross f. Like the tangent line, the tangent hyperplane functions as a locally linear approximation of f: in a small enighborhood around the point (<span class="red">10</span>, <span class="blue">20</span>), it behaves as much like f as any linear function can.</p><p     class="list-item"></p>
+            <p    >Here is the whole process again in a simple animation.<br></p><p     class="list-item">We start with a function with two inputs and one output. In this case the function f(<span class="red">x</span>, <span class="blue">y</span>) = <span class="red">x</span><sup>2</sup> + <span class="blue">y</span><sup>2</sup> - ¼<span class="red">x</span><span class="blue">y</span> + <span class="blue">x</span> - <span class="red">y</span> +1<br></p><p     class="list-item">We pick a point on the function, in this case (<span class="red">10</span>, <span class="blue">20</span>).<br></p><p     class="list-item">We can define two partial derivatives. One is <span class="red">the derivative of f as x varies and y is kept fixed</span>.<br></p><p     class="list-item">The other is the derivative of <span class="blue">f as y varies and x is kept fixed</span>.<br></p><p     class="list-item">Both of these are functions of one variable, so we can apply what we know from univariate calculus to work out the derivatives. At our point (<span class="red">10</span>, <span class="blue">20</span>), this gives us a tangent line touching the red function and a tangent line touching the blue function. <br></p><p     class="list-item">Since these lines cross, they lie in a shared hyperplane. That is the plane that (in most cases) just touches but does not cross f. Like the tangent line, the tangent hyperplane functions as a locally linear approximation of f: in a small neighborhood around the point (<span class="red">10</span>, <span class="blue">20</span>), it behaves as much like f as any linear function can.</p><p     class="list-item"></p>
             </figcaption>
             <span class="hint">click image for animation</span>
        </section>