VCE Maths Methods Integral Calculus Mini Test 12
Number of marks: 9
Reading time: 2 minutes
Writing time: 13 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 \(y = x \log_e(3x)\).
a. Find \(\frac{dy}{dx}\). 2 marks
b. Hence, calculate \(\int_1^2 (\log_e(3x)+1)dx\). Express your answer in the form \(\log_e(a)\), where \(a\) is a positive integer. 2 marks
Let \(f: R \setminus \{1\} \to R\), where \(f(x) = 2 + \frac{3}{x-1}\).
a. Sketch the graph of \(f\). Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation. 3 marks
b. Find the area enclosed by the graph of \(f\), the lines \(x=2\) and \(x=4\), and the x-axis. 2 marks
End of examination questions
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