VCE Maths Methods Integral Calculus Mini Test 10

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.


Question 1 [2018 Exam 2 Section A Q16]

Jamie approximates the area between the \(x\)-axis and the graph of \(y = 2\cos(2x) + 3\), over the interval \(\left[0, \frac{\pi}{2}\right]\) using the three rectangles shown below.

Graph of y=2cos(2x)+3 with three rectangles approximating the area underneath.

Jamie's approximation as a fraction of the exact area is

  • A. \(\frac{5}{9}\)
  • B. \(\frac{7}{9}\)
  • C. \(\frac{9}{11}\)
  • D. \(\frac{11}{18}\)
  • E. \(\frac{7}{3}\)
Correct Answer: B
Click here for full solution
Question 2 [2016 Exam 2 Section A Q9]

Given that \(\frac{d(xe^{kx})}{dx}) = (kx+1)e^{kx}\), then \(\int xe^{kx} dx\) is equal to

  • A. \(\frac{xe^{kx}}{kx+1} + c\)
  • B. \((\frac{kx+1}{k})e^{kx} + c\)
  • C. \(\frac{1}{k}\int e^{kx} dx\)
  • D. \(\frac{1}{k}(xe^{kx} - \int e^{kx} dx) + c\)
  • E. \(\frac{1}{k^2}(xe^{kx} - e^{kx}) + c\)
Correct Answer: D
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Question 3 [2021 Exam 2 Section A Q14]

A value of \(k\) for which the average value of \( y = \cos\left(kx - \frac{\pi}{2}\right) \) over the interval [0, π] is equal to the average value of \( y= \sin(x)\) over the same interval is

  • A. \( \frac{1}{6} \)
  • B. \( \frac{1}{5} \)
  • C. \( \frac{1}{4} \)
  • D. \( \frac{1}{3} \)
  • E. \( \frac{1}{2} \)
Correct Answer: E
Click here for full solution

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.


Question 1 [2018 Exam 1 Q9]

Consider a part of the graph of \(y = x\sin(x)\), as shown below.

Graph of y = x*sin(x)

a.

i. Given that \( \int (x\sin(x))dx = \sin(x) - x\cos(x) + c \), evaluate \( \int_{n\pi}^{(n+1)\pi} (x\sin(x))dx \) when \(n\) is a positive even integer or 0. Give your answer in simplest form. 2 marks

ii. Given that \( \int (x\sin(x))dx = \sin(x) - x\cos(x) + c \), evaluate \( \int_{n\pi}^{(n+1)\pi} (x\sin(x))dx \) when \(n\) is a positive odd integer. Give your answer in simplest form. 1 mark

b. Find the equation of the tangent to \( y = x\sin(x) \) at the point \( \left(-\frac{5\pi}{2}, \frac{5\pi}{2}\right) \). 2 marks

c. No longer in the curriculum

d. Let \( f: [0, 3\pi] \rightarrow \mathbb{R}, f(x) = (3\pi - x)\sin(x) \) and \( g: [0, 3\pi] \rightarrow \mathbb{R}, g(x) = (x - 3\pi)\sin(x) \).
The line \(l_1\) is the tangent to the graph of \(f\) at the point \( \left(\frac{\pi}{2}, \frac{5\pi}{2}\right) \) and the line \(l_2\) is the tangent to the graph of \(g\) at \( \left(\frac{\pi}{2}, -\frac{5\pi}{2}\right) \), as shown in the diagram below.

Graphs of f(x) and g(x) with tangents

Find the total area of the shaded regions shown in the diagram above. 2 marks


End of examination questions

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