<p>The following diagram shows part of the graph of the function \(f(x) = 2x^2\)</p>
<div class="graph-block" data-graph-config="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" data-width="400" data-height="350" style="width:400px;height:350px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The line \(T\) is the tangent to the graph of \(f\) at \(x = 1\).</p>
<ol class="abc-list"><li data-list-item-id="item-1777893622771-gli135sff" data-list-item-path="item-1777893622771-gli135sff"><div class="list-item-paragraph">Show that the equation of \(T\) is \(y= 4x -2\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1777893623294-k434vm72f" data-list-item-path="item-1777893623294-k434vm72f"><div class="list-item-paragraph">Find the \(x\)-intercept of \(T\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1777893623734-a0g6ub5pw" data-list-item-path="item-1777893623734-a0g6ub5pw"><div class="list-item-paragraph"><p>The shaded region \(R\) is enclosed by the graph of \(f\), the line \(T\), and the \(x\)-axis.</p></div><ol class="i-list"><li data-list-item-id="subitem-1777893713669-mcerunzpm" data-list-item-path="item-1777893623734-a0g6ub5pw/subitem-1777893713669-mcerunzpm"><div class="list-item-paragraph">Write down an expression for the area of \(R\).</div></li><li data-list-item-id="subitem-1777893716301-asjtghmcb" data-list-item-path="item-1777893623734-a0g6ub5pw/subitem-1777893716301-asjtghmcb"><div class="list-item-paragraph">Find the area of \(R\).<span class="inline-marks">[9]</span></div></li></ol></li></ol>

The following diagram shows part of the graph of the function \(f(x) = 2x^2\)

The line \(T\) is the tangent to the graph of \(f\) at \(x = 1\).

Show that the equation of \(T\) is \(y= 4x -2\).

<p>The shaded region \(R\) is enclosed by the graph of \(f\), the line \(T\), and the \(x\)-axis.</p>

Applications of Differential Calculus

International Baccalaureate Standard Level Mathematics