VCE Methods Integral Calculus Application Task 10

Question 1 [2020 Exam 2 Section B Q1]

Let \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = a(x+2)^2(x-2)^2\), where \(a \in \mathbb{R}\). Part of the graph of \(f\) is shown below.

a. Show that \(a = \frac{1}{4}\). 1 mark

b. Express \(f(x) = \frac{1}{4}(x+2)^2(x-2)^2\) in the form \(f(x) = \frac{1}{4}x^4 + bx^2 + c\), where \(b\) and \(c\) are integers. 1 mark

Part of the graph of the derivative function \(f'\) is shown below.

c.

i. Write the rule for \(f'\) in terms of \(x\). 1 mark

ii. Find the minimum value of the graph of \(f'\) on the interval \(x \in (0, 2)\). 2 marks

Let \(h: \mathbb{R} \to \mathbb{R}\), \(h(x) = -\frac{1}{4}(x+2)^2(x-2)^2 + 2\). Parts of the graphs of \(f\) and \(h\) are shown below.

d. Write a sequence of two transformations that map the graph of \(f\) onto the graph of \(h\). 1 mark

e.

i. State the values of \(x\) for which the graphs of \(f\) and \(h\) intersect. 1 mark

ii. Write down a definite integral that will give the total area of the shaded regions in the graph above. 1 mark

iii. Find the total area of the shaded regions in the graph above. Give your answer correct to two decimal places. 1 mark

f. Let \(D\) be the vertical distance between the graphs of \(f\) and \(h\).
Find all values of \(x\) for which \(D\) is at most 2 units. Give your answers correct to two decimal places. 2 marks