VCE Methods Integral Calculus Application Task 18

Question 1 [2018 Exam 2 Section B Q5]

Consider functions of the form

\( f : \mathbb{R} \rightarrow \mathbb{R}, f(x) = \frac{81x^2(a-x)}{4a^4} \)

and

\( h : \mathbb{R} \rightarrow \mathbb{R}, h(x) = \frac{9x}{2a^2} \)

where \(a\) is a positive real number.

a. Find the coordinates of the local maximum of \(f\) in terms of \(a\). 2 marks

b. Find the x-values of all of the points of intersection between the graphs of \(f\) and \(h\), in terms of \(a\) where appropriate. 1 mark

c. Determine the total area of the regions bounded by the graphs of \(y = f(x)\) and \(y = h(x)\). 2 marks

Consider the function \( g : \left[0, \frac{2a}{3}\right] \rightarrow \mathbb{R}, g(x) = \frac{81x^2(a-x)}{4a^4} \), where \(a\) is a positive real number.

d. Evaluate \(\frac{2a}{3} \times g\left(\frac{2a}{3}\right)\). 1 mark

e. Find the area bounded by the graph of \(g^{-1}\), the x-axis and the line \(x = g\left(\frac{2a}{3}\right)\). 2 marks

f. Find the value of \(a\) for which the graphs of \(g\) and \(g^{-1}\) have the same endpoints. 1 mark

g. Find the area enclosed by the graphs of \(g\) and \(g^{-1}\) when they have the same endpoints. 1 mark