VCE Methods Integral Calculus Application Task 19

Question 1 [2017 Exam 2 Section B Q1]

Let \(f: \mathbb{R} \to \mathbb{R}, f(x) = x^3 - 5x\). Part of the graph of \(f\) is shown below.

a. Find the coordinates of the turning points. 2 marks

b. \(A(-1, f(-1))\) and \(B(1, f(1))\) are two points on the graph of \(f\).

i. Find the equation of the straight line through \(A\) and \(B\). 2 marks

ii. Find the distance \(AB\). 1 mark

c. Let \(g: \mathbb{R} \to \mathbb{R}, g(x) = x^3 - kx, k \in \mathbb{R}^+\).
Let \(C(-1, g(-1))\) and \(D(1, g(1))\) be two points on the graph of \(g\).

i. Find the distance \(CD\) in terms of \(k\). 2 marks

ii. Find the values of \(k\) such that the distance \(CD\) is equal to \(k + 1\). 1 mark

d. The diagram below shows part of the graphs of \(g\) and \(y = x\). These graphs intersect at the points with the coordinates \((0, 0)\) and \((a, a)\).

i. Find the value of \(a\) in terms of \(k\). 1 mark

ii. Find the area of the shaded region in terms of \(k\). 2 marks