VCE Methods Integral Calculus Application Task 21

Question 1 [2021 Exam 2 Section B Q5]

Part of the graph of \(f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = \sin(\frac{x}{2}) + \cos(2x)\) is shown below.

a. State the period of \(f\). 1 mark

b. State the minimum value of \(f\), correct to three decimal places. 1 mark

c. Find the smallest positive value of \(h\) for which \(f(h - x) = f(x)\). 1 mark

Consider the set of functions of the form \(g_a: \mathbb{R} \rightarrow \mathbb{R}, g_a(x) = \sin(\frac{x}{a}) + \cos(ax)\), where \(a\) is a positive integer.

d. State the value of \(a\) such that \(g_a(x) = f(x)\) for all \(x\). 1 mark

e.

i. Find an antiderivative of \(g_a\) in terms of \(a\). 1 mark

ii. Use a definite integral to show that the area bounded by \(g_a\) and the x-axis over the interval \([0, 2a\pi]\) is equal above and below the x-axis for all values of \(a\). 3 marks

f. Explain why the maximum value of \(g_a\) cannot be greater than 2 for all values of \(a\) and why the minimum value of \(g_a\) cannot be less than –2 for all values of \(a\). 1 mark

g. Find the greatest possible minimum value of \(g_a\). 1 mark