<p>Let&nbsp;\(f(x) = 3 \sin{x} + 4\cos{x}\), for&nbsp;\(-2\pi \leq x \leq 2\pi\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch the graph of \(f\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Write down</div><ol class="i-list"><li><div class="list-item-paragraph">the amplitude;</div></li><li><div class="list-item-paragraph">the period;</div></li><li><div class="list-item-paragraph">the \(x\)-intercept that lies between \(-\frac{\pi}{2}\) and \(0\).<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">Hence write \(f(x)\) in the form \(p \sin{(qx + r)}\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Write down one value of \(x\) such that&nbsp;\(f'(x) = 0\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Write down the two values of \(k\) for which the equation \(f(x) = k\) has exactly two solutions.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Let&nbsp;\(g(x) = \ln{(x +1)}\), for&nbsp;\(0 \leq x \leq \pi\). There is a value of \(x\), between \(0\) and \(1\), for which the gradient of \(f\) is equal to the gradient of \(g\). Find this value of \(x\).<span class="inline-marks">[5]</span></div></li></ol>

Let \(f(x) = 3 \sin{x} + 4\cos{x}\), for \(-2\pi \leq x \leq 2\pi\).

Hence write \(f(x)\) in the form \(p \sin{(qx + r)}\).

Write down one value of \(x\) such that \(f'(x) = 0\).

Write down the two values of \(k\) for which the equation \(f(x) = k\) has exactly two solutions.

Let \(g(x) = \ln{(x +1)}\), for \(0 \leq x \leq \pi\). There is a value of \(x\), between \(0\) and \(1\), for which the gradient of \(f\) is equal to the gradient of \(g\). Find this value of \(x\).

Trigonometric Functions and Graphs

International Baccalaureate Standard Level Mathematics