VCE Maths Methods Integral Calculus Mini Test 1
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.
A function \( g: \mathbb{R} \to \mathbb{R} \) has the derivative \( g'(x) = x^3 - x \).
Given that \( g(0) = 5 \), the value of \( g(2) \) is
- A. 2
- B. 3
- C. 5
- D. 7
If \( \int_a^b f(x)\,dx = -5 \) and \( \int_a^c f(x)\,dx = 3 \), where \( a < b < c \), then \( \int_b^c 2f(x)\,dx \) is equal to
- A. -16
- B. 16
- C. -4
- D. 4
The function \( f: \mathbb{R} \to \mathbb{R} \) has average value \( k \) on the interval [0, 2] and satisfies \( f(x) = f(x + 2) \) for all \( x \in \mathbb{R} \).
The value of the definite integral \( \int_2^6 f(x)\,dx \) is
- A. 2k
- B. 3k
- C. 4k
- D. 6k
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 \( g : \mathbb{R} \setminus \{-3\} \rightarrow \mathbb{R},\ g(x) = \frac{1}{x + 3} - 2 \).
a. On the axes below, sketch the graph of \( y = g(x) \), labelling all asymptotes with their equations and axis intercepts with their coordinates. 2 marks
b. Determine the area of the region bounded by the line \( x = -2 \), the x-axis, the y-axis and the graph of \( y = g(x) \). 3 marks
The graph of \( y = x + \frac{1}{x} \) is shown over part of its domain.
Use two trapeziums of equal width to approximate the area between the curve, the x-axis and the lines \( x = 1 \) and \( x = 3 \). 2 marks
End of examination questions
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