A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>Let&nbsp;\(f(x) = A\mathrm{e}^{kx} + 3\). Part of the graph of \(f\) is shown below.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="262" style="width:300px;height:262px;max-width:100%;display:block;margin:0 auto;"></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 \(f'(x)\).<span class="inline-marks">[5]</span></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 \(f\).</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 graphs of \(f\) and \(g\).<span class="inline-marks">[6]</span></div></li></ol>

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

Applications of Exponential and Logarithmic Functions

Increasing and Decreasing Functions

Asymptotes and Discontinuities

Derivatives of Exponential Functions

Area between Two Curves

Exponential and Logarithmic Functions

Derivatives of Exponential, Logarithmic and Trigonometric Functions

Integration

Techniques of Differentiation

Graphing Techniques

Applications of Differentiation

<p>Find the area bounded by the curves \(y = x^{2} - 3x - 2\) and \(y = -2x^{2} - 2x + 8\).</p>

Find the area bounded by the curves \(y = x^{2} - 3x - 2\) and \(y = -2x^{2} - 2x + 8\).

<p>The figure shows part of the graphs of the curves \(y = \cos x\) and \(y = \sin 2x\). Find the area of the region bounded by the curves.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767948384807-idh5axy0fsg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDgzODQ4MDctaWRoNWF4eTBmc2cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTQ0NCwiZXhwIjoxNzkyNzMxNzQ0fQ.p8voyQWYvS9EPW-ZNUuCcS4q8O6fSgIf5EmSsq2XZa8" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure shows part of the graphs of the curves \(y = \cos x\) and \(y = \sin 2x\). Find the area of the region bounded by the curves.

<p>The figure shows the two curves \(y = x^2 - 2x - 2\) and \(y = -x^2 + 4x - 2\). Find the area of the region bounded by the curves.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767948270984-948f64b1w9l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDgyNzA5ODQtOTQ4ZjY0YjF3OWwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTQ0NCwiZXhwIjoxNzkyNzMxNzQ0fQ.2idtPaAUnv6_BNP6UodIxV6J-gRxyjUnykNC1oB3ex4" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure shows the two curves \(y = x^2 - 2x - 2\) and \(y = -x^2 + 4x - 2\). Find the area of the region bounded by the curves.

<p>The figure shows the two curves \(y = -x^2 + 8x - 8\) and \(y = -\frac{(x-7)^2}{7} - 1\). Find the area of the region bounded by the curves.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767948175334-sey1w2zmu6n.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDgxNzUzMzQtc2V5MXcyem11Nm4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTQ0NCwiZXhwIjoxNzkyNzMxNzQ0fQ.Yr8n2GoWGaq0ZvWGbLdQ8apVN_T7EfnHHzmavkKQ3cA" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure shows the two curves \(y = -x^2 + 8x - 8\) and \(y = -\frac{(x-7)^2}{7} - 1\). Find the area of the region bounded by the curves.

Math Question Bank