VCE Methods Differential Calculus Application Task 7

Number of marks: 12

Reading time: 2 minutes

Writing time: 18 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.


Question 1 [2020 Exam 2 Section B Q5]

Let \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = x^3 - x\).
Let \(g_a: \mathbb{R} \to \mathbb{R}\) be the function representing the tangent to the graph of \(f\) at \(x=a\), where \(a \in \mathbb{R}\).
Let \((b, 0)\) be the \(x\)-intercept of the graph of \(g_a\).

a. Show that \(b = \frac{2a^3}{3a^2-1}\). 3 marks

b. State the values of \(a\) for which \(b\) does not exist. 1 mark

c. State the nature of the graph of \(g_a\) when \(b\) does not exist. 1 mark

d.

i. State all values of \(a\) for which \(b = 1.1\). Give your answers correct to four decimal places. 1 mark

ii. The graph of \(f\) has an \(x\)-intercept at \((1, 0)\).
State the values of \(a\) for which \(1 \le b < 1.1\). Give your answers correct to three decimal places. 1 mark

The coordinate \((b, 0)\) is the horizontal axis intercept of \(g_a\).
Let \(g_b\) be the function representing the tangent to the graph of \(f\) at \(x=b\), as shown in the graph below.

Graph of f(x) with two tangent lines, g_a and g_b.

e. Find the values of \(a\) for which the graphs of \(g_a\) and \(g_b\), where \(b\) exists, are parallel and where \(b \ne a\). 3 marks

Let \(p: \mathbb{R} \to \mathbb{R}\), \(p(x) = x^3 + wx\), where \(w \in \mathbb{R}\).

f. Show that \(p(-x) = -p(x)\) for all \(w \in \mathbb{R}\). 1 mark

A property of the graphs of \(p\) is that two distinct parallel tangents will always occur at \((t, p(t))\) and \((-t, p(-t))\) for all \(t \ne 0\).

g. Find all values of \(w\) such that a tangent to the graph of \(p\) at \((t, p(t))\), for some \(t > 0\), will have an \(x\)-intercept at \((-t, 0)\). 1 mark


End of examination questions

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