VCE Maths Methods Diff Calculus Mini Test 8
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.
The function \(f: R \to R, f(x) = x^3 + ax^2 + bx\) has a local maximum at \(x = -1\) and a local minimum at \(x = 3\).
The values of \(a\) and \(b\) are respectively
- A. -2 and -3
- B. 2 and 1
- C. 3 and -9
- D. -3 and -9
- E. -6 and -15
A rectangle \(ABCD\) has vertices \(A(0, 0)\), \(B(u, 0)\), \(C(u, v)\) and \(D(0, v)\), where \((u, v)\) lies on the graph of \(y = -x^3 + 8\), as shown below.
The maximum area of the rectangle is
- A. \(\sqrt[3]{2}\)
- B. \(6\sqrt[3]{2}\)
- C. 16
- D. 8
- E. \(3\sqrt[3]{2}\)
Part of the graph \(y = f(x)\) of the polynomial function \(f\) is shown below.
\(f'(x) < 0\) for
- A. \(x \in (-2, 0) \cup (\frac{1}{3}, \infty)\)
- B. \(x \in (-9, \frac{100}{27})\)
- C. \(x \in (-\infty, -2) \cup (\frac{1}{3}, \infty)\)
- D. \(x \in (-2, \frac{1}{3})\)
- E. \(x \in (-\infty, -2] \cup (1, \infty)\)
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.
a. If \( y = (-3x^3 + x^2 - 64)^3 \), find \( \frac{dy}{dx} \). 1 mark
b. Let \( f(x) = \frac{e^x}{\cos(x)} \).
Evaluate \( f'(\pi) \). 2 marks
a. Let \(f: (-2, \infty) \to R, f(x) = \frac{x}{x+2}\). Differentiate \(f\) with respect to \(x\). 2 marks
b. Let \(g(x) = (2 - x^3)^3\). Evaluate \(g'(1)\). 2 marks
End of examination questions
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