VCE Maths Methods Diff Calculus Mini Test 5
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 \(f(x) = e^{g(x^2)}\), where \(g\) is a differentiable function, then \(f'(x)\) is equal to
- A. \(2xe^{g(x^2)}\)
- B. \(2xg(x^2)e^{g(x^2)}\)
- C. \(2xg'(x^2)e^{g(x^2)}\)
- D. \(2xg'(2x)e^{g(x^2)}\)
- E. \(2xg'(x^2)e^{g(2x)}\)
A right-angled triangle, \(OBC\), is formed using the horizontal axis and the point \(C(m, 9-m^2)\), where \(m \in (0, 3)\), on the parabola \(y = 9 - x^2\), as shown below.
The maximum area of the triangle \(OBC\) is
- A. \(\frac{\sqrt{3}}{3}\)
- B. \(\frac{2\sqrt{3}}{3}\)
- C. \(\sqrt{3}\)
- D. \(3\sqrt{3}\)
- E. \(9\sqrt{3}\)
Let \(f(x) = -\log_e(x+2)\).
A tangent to the graph of \(f\) has a vertical axis intercept at \((0, c)\).
The maximum value of \(c\) is
- A. -1
- B. \(-1 + \log_e(2)\)
- C. \(-\log_e(2)\)
- D. \(-1 - \log_e(2)\)
- E. \(\log_e(2)\)
Let \(f: R\setminus\{4\} \to R, f(x) = \frac{a}{x-4}\), where \(a > 0\).
The average rate of change of \(f\) from \(x=6\) to \(x=8\) is
- A. \(a\log_e(2)\)
- B. \(\frac{a}{2}\log_e(2)\)
- C. \(2a\)
- D. \(-\frac{a}{4}\)
- E. \(-\frac{a}{8}\)
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. Differentiate \( y = 2e^{-3x} \) with respect to \(x\). 1 mark
b. Evaluate \( f'(4) \), where \( f(x) = x\sqrt{2x+1} \). 2 marks
a. Let \(y = x^2\sin(x)\).
Find \(\frac{dy}{dx}\). 1 mark
b. Evaluate \(f'(1)\), where \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = e^{x^2 - x + 3}\). 2 marks
End of examination questions
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