VCE Methods Integral Calculus Application Task 2

Question 1 [2023 Exam 2 Section B Q5]

Let \( f : \mathbb{R} \rightarrow \mathbb{R} \), \( f(x) = e^x + e^{-x} \) and \( g : \mathbb{R} \rightarrow \mathbb{R} \), \( g(x) = \frac{1}{2} f(2-x) \).

a. Complete a possible sequence of transformations to map \( f \) to \( g \). 2 marks

• Dilation of factor \( \frac{1}{2} \) from the x-axis.

Two functions \( g_1 \) and \( g_2 \) are created, both with the same rule as \( g \) but with distinct domains, such that \( g_1 \) is strictly increasing and \( g_2 \) is strictly decreasing.

b. Give the domain and range for the inverse of \( g_1 \). 2 marks

Shown below is the graph of \( g \), the inverses of \( g_1 \) and \( g_2 \), and the line \( y = x \).

The intersection points between the graphs of \( y = x \), \( y = g(x) \) and the inverses of \( g_1 \) and \( g_2 \), are labelled P and Q.

c. i. Find the coordinates of \( P \) and \( Q \), correct to two decimal places. 1 mark

ii. Find the area of the region bound by the graphs of \( g \), the inverse of \( g_1 \) and the inverse of \( g_2 \). Give your answer correct to two decimal places. 2 marks

Let \( h : \mathbb{R} \rightarrow \mathbb{R} \), \( h(x) = \frac{1}{k} f(k-x) \), where \( k \in (0, \infty) \).

d. The turning point of \( h \) always lies on the graph of the function \( y = 2x^n \), where \( n \) is an integer. Find the value of \( n \). 1 mark

Let \( h_1 : [k, \infty) \rightarrow \mathbb{R} \), \( h_1(x) = h(x) \). The rule for the inverse of \( h_1 \) is \( y = \log_e\left( \frac{k}{2}x + \frac{1}{2} \sqrt{k^2 x^2 - 4} \right) + k \).

e. What is the smallest value of \( k \) such that \( h \) will intersect with the inverse of \( h_1 \)? Give your answer correct to two decimal places. 1 mark

It is possible for the graphs of \( h \) and the inverse of \( h_1 \) to intersect twice. This occurs when \( k = 5 \).

f. Find the area of the region bound by the graphs of \( h \) and the inverse of \( h_1 \), when \( k = 5 \). Give your answer correct to two decimal places. 2 marks