VCE Methods Integral Calculus Application Task 13
Number of marks: 11
Reading time: 2 minutes
Writing time: 16 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 \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = x^2e^{-x^2}\).
a. Find \(f'(x)\). 1 mark
b.
i. State the nature of the stationary point on the graph of \(f\) at the origin. 1 mark
ii. Find the maximum value of the function \(f\) and the values of \(x\) for which the maximum occurs. 2 marks
iii. Find the values of \(d \in \mathbb{R}\) for which \(f(x) + d\) is always negative. 1 mark
c.
i. Find the equation of the tangent to the graph of \(f\) at \(x = -1\). 1 mark
ii. Find the area enclosed by the graph of \(f\) and the tangent to the graph of \(f\) at \(x = -1\), correct to four decimal places. 2 marks
d. Let \(M(m, n)\) be a point on the graph of \(f\), where \(m \in [0, 1]\).
Find the minimum distance between \(M\) and the point \((0, e)\), and the value of \(m\) for which this occurs, correct to three decimal places. 3 marks
End of examination questions
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