VCE Methods Differential Calculus Application Task 3

Question 1 [2024 Exam 2 Section B Q3]

The points shown on the chart below represent monthly online sales in Australia.

The variable \( y \) represents sales in millions of dollars.

The variable \( t \) represents the month when the sales were made, where \( t = 1 \) corresponds to January 2021, \( t = 2 \) corresponds to February 2021 and so on.

a. A cubic polynomial \( p : (0, 12] \rightarrow \mathbb{R}, \, p(t) = at^3 + bt^2 + ct + d \) can be used to model monthly online sales in 2021.

The graph of \( y = p(t) \) is shown as a dashed curve on the set of axes above.

It has a local minimum at (2, 2500) and a local maximum at (11, 4400).

i. Find, correct to two decimal places, the values of \( a \), \( b \), \( c \) and \( d \). 3 marks

ii. Let \( q : (12, 24] \rightarrow \mathbb{R}, \, q(t) = p(t - h) + k \) be a cubic function obtained by translating \( p \), which can be used to model monthly online sales in 2022.

Find the values of \( h \) and \( k \) such that the graph of \( y = q(t) \) has a local maximum at (23, 4750). 2 marks

b. Another function \( f \) can be used to model monthly online sales, where

\( f : (0, 36] \rightarrow \mathbb{R}, \quad f(t) = 3000 + 30t + 700\cos\left(\frac{\pi t}{6}\right) + 400\cos\left(\frac{\pi t}{3}\right) \)

Part of the graph of \( f \) is shown on the axes below.

i. Complete the graph of \( f \) on the set of axes above until December 2023, that is, for \( t \in (24, 36] \).
Label the endpoint at \( t = 36 \) with its coordinates. 2 marks

ii. The function \( f \) predicts that every 12 months, monthly online sales increase by \( n \) million dollars.
Find the value of \( n \). 1 mark

iii. Find the derivative \( f'(t) \). 1 mark

iv. Hence, find the maximum instantaneous rate of change for the function \( f \), correct to the nearest million dollars per month, and the values of \( t \) in the interval \( (0, 36] \) when this maximum rate occurs, correct to one decimal place. 2 marks