VCE Methods Integral Calculus Application Task 7
Number of marks: 9
Reading time: 2 minutes
Writing time: 13 minutes
Section B – Calculator Allowed
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 \( g(x) = f(\sin(2x)) \), where the function \( f \) is differentiable.
The following table gives values for \( f(x) \) and \( f'(x) \):
| \( x \) | \( \frac{1}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{3}}{2} \) |
|---|---|---|---|
| \( f(x) \) | \( -2 \) | \( 5 \) | \( 3 \) |
| \( f'(x) \) | \( 7 \) | \( 0 \) | \( \frac{1}{9} \) |
a. Find the value of \( g\left( \frac{\pi}{6} \right) \). 1 mark
The derivative of \( g \) with respect to \( x \) is given by \( g'(x) = 2 \cdot \cos(2x) \cdot f'(\sin(2x)) \).
b. Show that \( g'\left( \frac{\pi}{6} \right) = \frac{1}{9} \). 1 mark
c. Find the equation of the tangent to \( g \) at \( x = \frac{\pi}{6} \). 2 marks
d. Find the average value of the derivative function \( g'(x) \) between \( x = \frac{\pi}{8} \) and \( x = \frac{\pi}{6} \). 2 marks
e. Find four solutions to the equation \( g'(x) = 0 \) for the interval \( x \in [0, \pi] \). 3 marks
End of examination questions
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