<p>The diagram shows part of the graph \(y = x^{\cos 2x}\), for&nbsp;\(0 \le x \le 5\), which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by \(x\) units and its vertical distance travelled is denoted by \(y\) units.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1783573222728-q8fnjduy6xq.png" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;\(\frac{\mathrm{d}y}{\mathrm{d}x} = x^{\cos 2x} \left[ \frac{\cos 2x}{x} - (2 \sin 2x) \ln x \right]\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">At the point on the graph where&nbsp;\(x = \pi\), find the rate at which the roller coaster is moving vertically when it is moving horizontally at a rate of \(8\) units per hour.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the acute angle that the tangent to the graph where&nbsp;\(x = \pi\) makes with the horizontal.<span class="inline-marks">[1]</span></div></li></ol>

The diagram shows part of the graph \(y = x^{\cos 2x}\), for \(0 \le x \le 5\), which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by \(x\) units and its vertical distance travelled is denoted by \(y\) units.

Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = x^{\cos 2x} \left[ \frac{\cos 2x}{x} - (2 \sin 2x) \ln x \right]\).

At the point on the graph where \(x = \pi\), find the rate at which the roller coaster is moving vertically when it is moving horizontally at a rate of \(8\) units per hour.

Find the acute angle that the tangent to the graph where \(x = \pi\) makes with the horizontal.

Applications of Differentiation

International Baccalaureate Higher Level Mathematics

2024 ACJC P2 Q1

2024 ACJC P2 Q1