VCE Methods Integral Calculus Application Task 15

Question 1 [2019 Exam 2 Section B Q5]

Let \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = 1 - x^3\). The tangent to the graph of \(f\) at \(x=a\), where \(0 < a < 1\), intersects the graph of \(f\) again at \(P\) and intersects the horizontal axis at \(Q\). The shaded regions shown in the diagram below are bounded by the graph of \(f\), its tangent at \(x=a\) and the horizontal axis.

a. Find the equation of the tangent to the graph of \(f\) at \(x=a\), in terms of \(a\). 1 mark

b. Find the \(x\)-coordinate of \(Q\), in terms of \(a\). 1 mark

c. Find the \(x\)-coordinate of \(P\), in terms of \(a\). 2 marks

Let \(A\) be the function that determines the total area of the shaded regions.

d. Find the rule of \(A\), in terms of \(a\). 3 marks

e. Find the value of \(a\) for which \(A\) is a minimum. 2 marks

Consider the regions bounded by the graph of \(f^{-1}\), the tangent to the graph of \(f^{-1}\) at \(x=b\), where \(0 < b < 1\), and the vertical axis.

f. Find the value of \(b\) for which the total area of these regions is a minimum. 2 marks

g. Find the value of the acute angle between the tangent to the graph of \(f\) and the tangent to the graph of \(f^{-1}\) at \(x=1\). 1 mark