VCE Maths Methods Differential Calculus Mini Test 7
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.
Part of the graph of a cubic polynomial function \(f\) and the coordinates of its stationary points are shown below.
\(f'(x) < 0\) for the interval
- A. \((0, 3)\)
- B. \((-\infty, -5) \cup (0, 3)\)
- C. \((-\infty, -3) \cup (\frac{5}{3}, \infty)\)
- D. \((-3, \frac{5}{3})\)
- E. \((-\frac{400}{27}, 36)\)
The average rate of change of the function with the rule \(f(x) = x^2 - 2x\) over the interval \([1, a]\), where \(a > 1\), is 8.
The value of \(a\) is
- A. 9
- B. 8
- C. 7
- D. 4
- E. \(1 + \sqrt{2}\)
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.
Let \(f: (\frac{1}{3}, \infty) \to \mathbb{R}\), \(f(x) = \frac{1}{3x-1}\).
a.
i. Find \(f'(x)\). 1 mark
ii. Find an antiderivative of \(f(x)\). 1 mark
b. Let \(g: \mathbb{R}\setminus\{-1\} \to \mathbb{R}\), \(g(x) = \frac{\sin(\pi x)}{x+1}\).
Evaluate \(g'(1)\). 2 marks
The graph of the relation \(y = \sqrt{1-x^2}\) is shown on the axes below. \(P\) is a point on the graph of this relation, \(A\) is the point \((-1, 0)\) and \(B\) is the point \((x, 0)\).
a. Find an expression for the length \(PB\) in terms of \(x\) only. 1 mark
b. Find the maximum area of the triangle \(ABP\). 3 marks
End of examination questions
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