VCE Methods Integral Calculus Application Task 8

Question 1 [2021 Exam 2 Section B Q3]

Let \(q(x) = \log_e(x^2 - 1) - \log_e(1 - x)\).

a. State the maximal domain and the range of \(q\). 2 marks

b.

i. Find the equation of the tangent to the graph of \(q\) when \(x = -2\). 1 mark

ii. Find the equation of the line that is perpendicular to the graph of \(q\) when \(x = -2\) and passes through the point \((-2, 0)\). 1 mark

Let \(p(x) = e^{-2x} - 2e^{-x} + 1\).

c. Explain why \(p\) is not a one-to-one function. 1 mark

d. Find the gradient of the tangent to the graph of \(p\) at \(x = a\). 1 mark

The diagram below shows parts of the graph of \(p\) and the line \(y = x + 2\).

The line \(y = x + 2\) and the tangent to the graph of \(p\) at \(x = a\) intersect with an acute angle of \(\theta\) between them.

e. Find the value(s) of \(a\) for which \(\theta = 60^\circ\). Give your answer(s) correct to two decimal places. 3 marks

f. Find the x-coordinate of the point of intersection between the line \(y = x + 2\) and the graph of \(p\), and hence find the area bounded by \(y = x + 2\), the graph of \(p\) and the x-axis, both correct to three decimal places. 3 marks