VCE Methods Differential Calculus Application Task 5

Question 1 [2023 Exam 2 Section B Q3]

Consider the function \( g : \mathbb{R} \rightarrow \mathbb{R} \), \( g(x) = 2^x + 5 \).

a. State the value of \( \lim_{x \to -\infty} g(x) \). 1 mark

b. The derivative, \( g'(x) \), can be expressed in the form \( g'(x) = k \times 2^x \). Find the real number \( k \). 1 mark

c. i. Let \( a \) be a real number. Find, in terms of \( a \), the equation of the tangent to \( g \) at the point \( (a, g(a)) \). 1 mark

ii. Hence, or otherwise, find the equation of the tangent to \( g \) that passes through the origin, correct to three decimal places. 2 marks

Let \( h : \mathbb{R} \rightarrow \mathbb{R} \), \( h(x) = 2^x - x^2 \).

d. Find the coordinates of the point of inflection for \( h \), correct to two decimal places. 1 mark

e. Find the largest interval of \( x \) values for which \( h \) is strictly decreasing. Give your answer correct to two decimal places. 1 mark

f. Apply Newton’s method, with an initial estimate of \( x_0 = 0 \), to find an approximate x-intercept of \( h \). Write the estimates \( x_1 \), \( x_2 \) and \( x_3 \) in the table below, correct to three decimal places. 2 marks

\( x_0 \)	0
\( x_1 \)
\( x_2 \)
\( x_3 \)

g. For the function \( h \), explain why a solution to the equation \( \log_e(2) \times 2^x - 2x = 0 \) should not be used as an initial estimate \( x_0 \) in Newton’s method. 1 mark

h. There is a positive real number \( n \) for which the function \( f(x) = n^x - x^n \) has a local minimum on the x-axis. Find this value of \( n \). 2 marks