VCE Maths Methods Integral Calculus Mini Test 13
Number of marks: 7
Reading time: 2 minutes
Writing time: 10 minutes
Instructions – No Calculator
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Let \( f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^2e^{kx} \), where \(k\) is a positive real constant.
a. Show that \( f'(x) = xe^{kx}(kx + 2) \). 1 mark
b. Find the value of \(k\) for which the graphs of \( y = f(x) \) and \( y = f'(x) \) have exactly one point of intersection. 2 marks
Let \( g(x) = -\frac{2xe^{kx}}{k} \). The diagram below shows sections of the graphs of \(f\) and \(g\) for \(x \ge 0\).
Let \(A\) be the area of the region bounded by the curves \(y = f(x)\), \(y = g(x)\) and the line \(x = 2\).
c. Write down a definite integral that gives the value of \(A\). 1 mark
d. Using your result from part a., or otherwise, find the value of \(k\) such that \( A = \frac{16}{k} \). 3 marks
End of examination questions
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