<p>Let&nbsp;\(f(x) = A\mathrm{e}^{kx} + 3\). Part of the graph of \(f\) is shown below.</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/1777898934808-h4sg0zq7rni.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The \(y\)-intercept is at \( (0,13)\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(A = 10\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;\(f(15) = 3.49\) (correct to \(3\) significant figures), find the value of \(k\).<span class="inline-marks">[3]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Using your value of \(k\), find \(f'(x)\).</div></li><li><div class="list-item-paragraph">Hence, explain why \(f\) is a decreasing function.</div></li><li><div class="list-item-paragraph">Write down the equation of the horizontal asymptote of the graph \(f\).<span class="inline-marks">[5]</span></div></li></ol></li></ol>
<p>Let&nbsp;\(g(x) = -x^2 +12x -24\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the area enclosed by the graph of \(f\) and \(g\).<span class="inline-marks">[5]</span></div></li></ol>

Let \(f(x) = A\mathrm{e}^{kx} + 3\). Part of the graph of \(f\) is shown below.

Given that \(f(15) = 3.49\) (correct to \(3\) significant figures), find the value of \(k\).

Find the area enclosed by the graph of \(f\) and \(g\).

Derivatives of Exponential, Logarithmic and Trigonometric Functions

International Baccalaureate Standard Level Mathematics