VCE Maths Methods Integral 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.
If \(\int_1^4 f(x)dx = 4\) and \(\int_2^4 f(x)dx = -2\), then \(\int_1^2 (f(x)+x)dx\) is equal to
- A. 2
- B. 6
- C. 8
- D. \(\frac{7}{2}\)
- E. \(\frac{15}{2}\)
The graph of a function \(f\), where \(f(-x) = f(x)\), is shown below.
The graph has \(x\)-intercepts at \((a, 0)\), \((b, 0)\), \((c, 0)\) and \((d, 0)\) only.
The area bound by the curve and the \(x\)-axis on the interval \([a, d]\) is
- A. \(\int_a^d f(x)dx\)
- B. \(\int_a^b f(x)dx - \int_b^c f(x)dx + \int_c^d f(x)dx\)
- C. \(2\int_a^b f(x)dx + \int_b^c f(x)dx\)
- D. \(2\int_a^b f(x)dx - 2\int_b^{b+c} f(x)dx\)
- E. \(\int_a^b f(x)dx + \int_c^b f(x)dx + \int_d^c f(x)dx\)
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 \(f: [0, 2] \to \mathbb{R}\), where \(f(x) = \frac{1}{\sqrt{2}}\sqrt{x}\).
a. Find the domain and the rule for \(f^{-1}\), the inverse function of \(f\). 2 marks
The graph of \(y = f(x)\), where \(x \in [0, 2]\), is shown on the axes below.
b. On the axes above, sketch the graph of \(f^{-1}\) over its domain. Label the endpoints and point(s) of intersection with the function \(f\), giving their coordinates. 2 marks
c. Find the total area of the two regions: one region bounded by the functions \(f\) and \(f^{-1}\), and the other region bounded by \(f\), \(f^{-1}\) and the line \(x=1\). Give your answer in the form \(\frac{a-b\sqrt{b}}{6}\), where \(a, b \in Z^+\). 4 marks
End of examination questions
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