IB Mathematics Analysis and Approaches Topic AHL…

(c) The function $g$ is given by $g(x) = -x^{3}$ , for $x \in \mathbb{R}$ . Find , find the coordinates of all points where the graphs of and meet, and which of these points do not lie on the line .

This question investigates iterated functions and their long-term behaviour.

Given a function $f$ , the iteration sequence starting at $x_0$ is defined by

x_{n+1} = f(x_n), \quad n = 0, 1, 2, \ldots

(a) Starting with $x_0 = 1$ , find the values of $x_1$ , $x_2$ and , giving your answers correct to three decimal places.

(b) Suppose the iteration sequence converges to a limit $\alpha$ . why $\alpha$ must satisfy $\cos\alpha = \alpha$ .

(d) The behaviour of such iteration sequences can be visualised using a cobweb diagram. On the same axes, one draws $y = f(x)$ and $y = x$ . Starting from $(x_0, 0)$ , vertical and horizontal lines are drawn alternately between the two graphs, tracing the path of the iteration.

Let $f(x) = x^3 - 6x^2 + 9x + 1$ , for .

(a) Find $f'(x)$ and hence find the coordinates of the stationary points of $f$ .

(b) Use $f''(x)$ to classify the stationary points.

(c) the graph of $y = f(x)$ , labelling the stationary points, the $y$ -intercept and the $x$ -intercept (giving values to three significant figures where necessary).

(d) Find the $x$ -coordinate of the point of inflexion of the graph of $y = f(x)$ .

Let $f(x) = \dfrac{2x + 3}{3x - 4}$ , for $x \neq \dfrac{4}{3}$ .

(a) For the graph of $y = f(x)$ , find the coordinates of:

the $x$ -intercept;

the $y$ -intercept.

(b) For the graph of $y = f(x)$ , find the equation of:

(i) the horizontal asymptote;

(ii) the vertical asymptote.

(d) The graphs of $f(x)$ and $f^{-1}(x)$ have two points of intersection.

Find the coordinates of the two points of intersection. Give your answers in exact form.

The function $f$ is given by $f(x) = 0.5x^{3} + 0.4$ , for $x \in \mathbb{R}$ .

(a) Show that $f$ is strictly increasing, and hence that any point where the graphs of $f$ and $f^{-1}$ meet must lie on the line $y = x$ .

(b) Find the coordinates of all points where the graphs of $f$ and $f^{-1}$ meet.

(f) To understand why some fixed points attract nearby iterates, we study the behaviour of the error $e_n = x_n - L$ , where $L$ is a fixed point.

For $x_n$ close to $L$ , we may use the approximation $f(x) \approx f(L) + f'(L)(x - L)$ .

Using this approximation, show that $e_{n+1} \approx f'(L) \cdot e_n$ .

(h) Show that $g'(x^*) = 2 - k$ , where $x^* = \dfrac{k-1}{k}$ is the non-zero fixed point of $g$ .

(j) Verify that your result from part 7 is consistent with the convergence observed in part 3 for $f(x) = \cos x$ .

(k) For $k = 3.2$ , the non-zero fixed point is no longer attracting. Starting with $x_0 = 0.2$ , use your GDC to iterate the function $g(x) = 3.2,x(1 - x)$ and the long-term behaviour of the sequence. the values the sequence approaches, correct to three decimal places.

(g) Find the value of $g^{-1}(2.5)$ , giving your answer to three significant figures.

(e) Hence find the equation of the perpendicular bisector of the segment joining the two points of intersection.

(e) For $k = 2.5$ and $x_0 = 0.1$ , $x_1$ , $x_2$ and $x_3$ , and on the behaviour of the sequence in relation to the non-zero fixed point.

(g) Hence why the fixed point $L$ attracts nearby iterates when $|f'(L)| < 1$ , and repels them when $|f'(L)| > 1$ .

(i) Hence the range of values of $k$ for which the non-zero fixed point $x^*$ is an attracting fixed point in the interval $(0, 1)$ .

(e) The function $g$ is defined by $g(x) = f(x)$ , for $1 \leq x \leq 3$ .

why $g^{-1}$ exists.

(f) the domain and range of $g^{-1}$ .

IB Mathematics Analysis and Approaches AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank

IB Mathematics Analysis and Approaches AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank

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IB Mathematics Analysis and Approaches AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank

IB Mathematics Analysis and Approaches AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 2.14—odd and Even Functions, Self-inverse, Inverse and Domain Restriction Questionbank

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