VCE Maths Methods Diff Calculus Mini Test 11
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 tangent to the graph of \(y = \log_e(2x)\) has a gradient of 2.
This tangent will cross the y-axis at
- A. \(0\)
- B. \(-0.5\)
- C. \(-1\)
- D. \(-1 - \log_e(2)\)
- E. \(-2\log_e(2)\)
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.
Consider the functions \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = 3+2x-x^2\) and \(g: \mathbb{R} \to \mathbb{R}\), \(g(x) = e^x\).
a. State the rule of \(g(f(x))\). 1 mark
b. Find the values of \(x\) for which the derivative of \(g(f(x))\) is negative. 2 marks
c. State the rule of \(f(g(x))\). 1 mark
d. Solve \(f(g(x)) = 0\). 2 marks
e. Find the coordinates of the stationary point of the graph of \(f(g(x))\). 2 marks
f. State the number of solutions to \(g(f(x)) + f(g(x)) = 0\). 1 mark
End of examination questions
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