VCE Maths Methods Diff 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.
Let \( f(x) = \log_e x \), where \( x > 0 \) and \( g(x) = \sqrt{1 - x} \), where \( x < 1 \).
The domain of the derivative of \( (f \circ g)(x) \) is
- A. \( x \in \mathbb{R} \)
- B. \( x \in (-\infty, 1] \)
- C. \( x \in (-\infty, 1) \)
- D. \( x \in (0, \infty) \)
- E. \( x \in (0, 1) \)
\[ f(x) = \begin{cases} \tan\left(\frac{x}{2}\right) & 4 \leq x < 2\pi \\ \sin(ax) & 2\pi \leq x \leq 8 \end{cases} \]
The value of \( a \) for which \( f \) is continuous and smooth at \( x = 2\pi \) is- A. −2
- B. \( -\sqrt{2} \)
- C. \( -\frac{1}{2} \)
- D. \( \frac{1}{2} \)
- E. 2
Two functions, \( f \) and \( g \), are continuous and differentiable for all \( x \in \mathbb{R} \). It is given that
\( f(-2) = -7, \quad g(-2) = 8, \quad f'(-2) = 3, \quad g'(-2) = 2 \)
The gradient of the graph \( y = f(x) \times g(x) \) at \( x = -2 \) is
- A. −10
- B. −6
- C. 0
- D. 6
- E. 10
The gradient of the graph of \( y = e^{3x} \) at the point where the graph crosses the vertical axis is equal to
- A. 0
- B. \( \frac{1}{e} \)
- C. 1
- D. e
- E. 3
A function \( g \) is continuous on the domain \( x \in [a, b] \) and has the following properties:
• The average rate of change of \( g \) between \( x = a \) and \( x = b \) is positive.
• The instantaneous rate of change of \( g \) at \( x = \frac{a + b}{2} \) is negative.
Therefore, on the interval \( x \in [a, b] \), the function must be
- A. many-to-one
- B. one-to-many
- C. one-to-one
- D. strictly decreasing
- E. strictly increasing
The value of an investment, in dollars, after n months can be modelled by the function
\( f(n) = 2500 \times (1.004)^n \)
where \( n \in \{0, 1, 2, ...\} \).
The average rate of change of the value of the investment over the first 12 months is closest to
- A. $10.00 per month.
- B. $10.20 per month.
- C. $10.50 per month.
- D. $125.00 per month.
- E. $127.00 per month.
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. Let \( y = \frac{x^2 - x}{e^x} \). Find and simplify \( \frac{dy}{dx} \). 2 marks
b. Let \( f(x) = \sin(x) e^{2x} \). Find \( f'\left( \frac{\pi}{4} \right) \). 2 marks
End of examination questions
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