VCE Maths Methods Integral Calculus Mini Test 2
Number of marks: 10
Reading time: 2 minutes
Writing time: 15 minutes
Section A – Calculator Allowed
Instructions
• Answer all questions in pencil on your Multiple-Choice Answer Sheet.
• Choose the response that is correct for the question.
• A correct answer scores 1; an incorrect answer scores 0.
• Marks will not be deducted for incorrect answers.
• No marks will be given if more than one answer is completed for any question.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
If \( \frac{d}{dx} \left( x \cdot \sin(x) \right) = \sin(x) + x \cdot \cos(x) \), then \( \frac{1}{k} \int x \cos(x)\, dx \) is equal to
- A. \( k \left( x \cdot \sin(x) - \int \sin(x)\, dx \right) + c \)
- B. \( \frac{1}{k} \cdot x \cdot \sin(x) - \int \sin(x)\, dx + c \)
- C. \( \frac{1}{k} \left( x \cdot \sin(x) - \int \sin(x)\, dx \right) + c \)
- D. \( \frac{1}{k} (x \cdot \sin(x) - \sin(x)) + c \)
- E. \( \frac{1}{k} \left( \int x \cdot \sin(x)\, dx - \int \sin(x)\, dx \right) + c \)
End of Section A
Section B – No Calculator
Instructions
• 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.
Part of the graph of \( f : [-\pi, \pi] \rightarrow \mathbb{R},\ f(x) = x\sin(x) \) is shown below.
a. Use the trapezium rule with a step size of \( \frac{\pi}{3} \) to determine an approximation of the total area between the graph of \( y = f(x) \) and the x-axis over the interval \( x \in [0, \pi] \). 3 marks
b.
i. Find \( f'(x) \). 1 mark
ii. Determine the range of \( f'(x) \) over the interval \( \left[\frac{\pi}{3}, \frac{2\pi}{3} \right] \). 1 mark
iii. Hence, verify that \( f(x) \) has a stationary point for \( x \in \left[\frac{\pi}{3}, \frac{2\pi}{3} \right] \). 1 mark
c. On the set of axes below, sketch the graph of \( y = f'(x) \) on the domain \( [-\pi, \pi] \), labelling the endpoints with their coordinates.
You may use the fact that the graph of \( y = f'(x) \) has a local minimum at approximately \( (-1.1, -1.4) \) and a local maximum at approximately \( (1.1, 1.4) \). 3 marks
End of examination questions
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