VCE Methods Integral Calculus Application Task 1

Question 1 [2024 Exam 2 Section B Q5]

The graph below shows the compositions \( g \circ f \) and \( f \circ g \), where \( f(x) = \sin(x) \) and \( g(x) = \sin(2x) \).

a.

i. The graph of \( y = (g \circ f)(x) \) has a local maximum whose \( x \)-value lies in the interval \( [0, \frac{\pi}{2}] \).
Find the coordinates of this local maximum, correct to one decimal place. 1 mark

ii. State the range of \( g \circ f \) where \( x \in [0, 2\pi] \). 1 mark

b.

i. Find the derivative of \( f \circ g \). 1 mark

ii. Show that the equation \( \cos(\sin(2x)) = 0 \) has no real solutions. 2 marks

iii. Find the \( x \)-values of the stationary points of \( f \circ g \) where \( x \in [0, 2\pi] \). 1 mark

iv. Find the range of \( f \circ g \) where \( x \in [0, 2\pi] \). 1 mark

c.

i. Write a single definite integral that gives the area bounded by the graphs of \( y = (f \circ g)(x) \) and \( y = (g \circ f)(x) \) in the interval \( [0, 2\pi] \). 1 mark

ii. Hence, state the area bounded by the graphs of \( y = (f \circ g)(x) \) and \( y = (g \circ f)(x) \) in the interval \( [0, 2\pi] \), correct to two decimal places. 1 mark

d. Let \( f_1 : (0, 2\pi) \rightarrow \mathbb{R}, \, f_1(x) = \sin(x) \).
Find all values of \( x \) in the interval \( (0, 2\pi) \) for which the composition \( f_1 \circ g \) is defined. 2 marks