VCE Maths Methods Diff Calculus Mini Test 3
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.
The maximum value of the function \( h: [0, 2] \to R, h(x) = (x-2)e^x \) is
- A. \(-e\)
- B. 0
- C. 1
- D. 2
- E. \(e\)
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.
The shapes of two walking tracks are shown below.
Track 1 is described by the function \( f(x) = a - x(x - 2)^2 \).
Track 2 is defined by the function \( g(x) = 12x + bx^2 \).
The unit of length is kilometres.
a. Given that \( f(0) = 12 \) and \( g(1) = 9 \), verify that \( a = 12 \) and \( b = -3 \). 1 mark
b. Verify that \( f(x) \) and \( g(x) \) both have a turning point at \( P \).
Give the coordinates of \( P \). 2 marks
c. A theme park is planned whose boundaries will form the triangle \( \triangle OAB \) where \( O \) is the origin, \( A \) is at \( (k, 0) \) and \( B \) is at \( (k, g(k)) \), as shown below, where \( k \in (0, 4) \).
Find the maximum possible area of the theme park, in km\(^2\). 3 marks
End of examination questions
VCE is a registered trademark of the VCAA. The VCAA does not endorse or make any warranties regarding this study resource. Past VCE exams and related content can be accessed directly at www.vcaa.vic.edu.au
