VCE Maths Methods Differential Calculus Mini Test 12
Number of marks: 10
Reading time: 2 minutes
Writing time: 15 minutes
Instructions – No Calculator
• 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.
a. Let \( y = 3x e^{2x} \).
Find \( \frac{dy}{dx} \). 1 mark
b. Find and simplify the rule of \( f'(x) \), where \( f : \mathbb{R} \rightarrow \mathbb{R},\ f(x) = \frac{\cos(x)}{e^x} \). 2 marks
Let \( g : \mathbb{R} \rightarrow \mathbb{R}, \quad g(x) = \sqrt[3]{x - k}+ m, \quad \text{where } k \in \mathbb{R} \setminus \{0\} \text{ and } m \in \mathbb{R} \).
Let the point \( P \) be the y-intercept of the graph of \( y = g(x) \).
a. Find the coordinates of \( P \), in terms of \( k \) and \( m \). 1 mark
b. Find the gradient of \( g \) at \( P \), in terms of \( k \). 2 marks
c. Given that the graph of \( y = g(x) \) passes through the origin, express \( k \) in terms of \( m \). 1 mark
d. Let the point \( Q \) be a point different from the point \( P \), such that the gradient of \( g \) at points \( P \) and \( Q \) are equal.
Given that the graph of \( y = g(x) \) passes through the origin, find the coordinates of \( Q \) in terms of \( m \). 3 marks
End of examination questions
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