IB Questionbank

Loading [MathJax]/jax/element/mml/optable/Latin1Supplement.js

Updates to Questionbank

User interface language: English | Español

Date	November 2019	Marks available	2	Reference code	19N.1.AHL.TZ0.H_10
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Show and Sketch	Question number	H_10	Adapted from	N/A

Question

Consider $f\left( x \right) = \frac{{2x - 4}}{{{x^2} - 1}}{\text{, }} - 1 < x < 1$ .

For the graph of $y = f\left( x \right)$ ,

Find $f'\left( x \right)$ .

[2]

a.i.

Show that, if $f'\left( x \right) = 0$ , then $x = 2 - \sqrt 3$ .

[3]

a.ii.

find the coordinates of the $y$ -intercept.

[1]

b.i.

show that there are no $x$ -intercepts.

[2]

b.ii.

sketch the graph, showing clearly any asymptotic behaviour.

[2]

b.iii.

Show that $\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}$ .

[2]

c.

The area enclosed by the graph of $y = f\left( x \right)$ and the line $y = 4$ can be expressed as ${\text{ln}}\,v$ . Find the value of $v$ .

[7]

d.

Markscheme

attempt to use quotient rule (or equivalent) (M1)

$f'\left( x \right) = \frac{{\left( {{x^2} - 1} \right)\left( 2 \right) - \left( {2x - 4} \right)\left( {2x} \right)}}{{{{\left( {{x^2} - 1} \right)}^2}}}$ A1

$= \frac{{ - 2{x^2} + 8x - 2}}{{{{\left( {{x^2} - 1} \right)}^2}}}$

[2 marks]

a.i.

$f'\left( x \right) = 0$

simplifying numerator (may be seen in part (i)) (M1)

$\Rightarrow {x^2} - 4x + 1 = 0$ or equivalent quadratic equation A1

EITHER

use of quadratic formula

$\Rightarrow x = \frac{{4 \pm \sqrt {12} }}{2}$ A1

OR

use of completing the square

${\left( {x - 2} \right)^2} = 3$ A1

THEN

$x = 2 - \sqrt 3$ (since $2 + \sqrt 3$ is outside the domain) AG

Note: Do not condone verification that $x = 2 - \sqrt 3 \Rightarrow f'\left( x \right) = 0$ .

Do not award the final A1 as follow through from part (i).

[3 marks]

a.ii.

(0, 4) A1

[1 mark]

b.i.

$2x - 4 = 0 \Rightarrow x = 2$ A1

outside the domain R1

[2 marks]

b.ii.

A1A1

award A1 for concave up curve over correct domain with one minimum point in the first quadrant
award A1 for approaching $x = \pm 1$ asymptotically

[2 marks]

b.iii.

valid attempt to combine fractions (using common denominator) M1

$\frac{{3\left( {x - 1} \right) - \left( {x + 1} \right)}}{{\left( {x + 1} \right)\left( {x - 1} \right)}}$ A1

$= \frac{{3x - 3 - x - 1}}{{{x^2} - 1}}$

$= \frac{{2x - 4}}{{{x^2} - 1}}$ AG

[2 marks]

c.

$f\left( x \right) = 4 \Rightarrow 2x - 4 = 4{x^2} - 4$ M1

( $x = 0$ or) $x = \frac{1}{2}$ A1

area under the curve is $\int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x}$ M1

$= \int_0^{\frac{1}{2}} {\frac{3}{{x + 1}} - \frac{1}{{x - 1}}{\text{d}}x}$

Note: Ignore absence of, or incorrect limits up to this point.

$= \left[ {3\,{\text{ln}}\,\left| {x + 1} \right| - {\text{ln}}\,\left| {x - 1} \right|} \right]_0^{\frac{1}{2}}$ A1

$= 3\,{\text{ln}}\frac{3}{2} - {\text{ln}}\frac{1}{2}\left( { - 0} \right)$

$= {\text{ln}}\frac{{27}}{4}$ A1

area is $2 - \int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x}$ or $\int_0^{\frac{1}{2}} {4\,{\text{d}}x} - \int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x}$ M1

$= 2 - {\text{ln}}\frac{{27}}{4}$

$= {\text{ln}}\frac{{4\,{{\text{e}}^2}}}{{27}}$ A1

$\left( { \Rightarrow v = \frac{{4\,{{\text{e}}^2}}}{{27}}} \right)$

[7 marks]

d.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

[N/A]

c.

[N/A]

d.

Syllabus sections

Topic 2—Functions » AHL 2.13—Rational functions

Show 40 related questions

21N.2.AHL.TZ0.10d:
Sketch the graph of $f$ for $- 30 \leq x \leq 30$ , clearly indicating the points of intersection with each axis and any asymptotes.
EXM.1.AHL.TZ0.5b:
Write down the equation of the vertical asymptote.
EXM.1.AHL.TZ0.6c:
With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.
EXM.1.AHL.TZ0.4b:
Find the equation of the horizontal asymptote.
EXM.1.AHL.TZ0.4c:
Find the exact value of $\int\limits_0^1 {f\left( x \right)} \,dx$ , giving the answer in the form ${\text{ln}}\,q{\text{,}}\,\,q \in \mathbb{Q}$ .
EXM.1.AHL.TZ0.6a:
Find the co-ordinates of all stationary points.
18N.2.AHL.TZ0.H_11a:
Find an expression for ${\theta _p}$ in terms of $p$ .
18N.2.AHL.TZ0.H_11b:
Show that $p = \frac{{{q^2} + q - 1}}{{2q + 1}}$ .
19N.1.AHL.TZ0.H_10b.i:
find the coordinates of the $y$ -intercept.
19N.1.AHL.TZ0.H_10b.ii:
show that there are no $x$ -intercepts.
EXM.1.AHL.TZ0.6b:
Write down the equation of the vertical asymptote.
18N.2.AHL.TZ0.H_11c:
By sketching the graph of $p$ as a function of $q$ , determine the range of values of $p$ for which there are possible values of $q$ .
17N.2.AHL.TZ0.H_6b:
Find the expected number of weeks in the year in which Lucca eats no bananas.
22M.2.AHL.TZ1.6b:
Solve the inequality $f (x) \geq g (x)$ .
20N.1.AHL.TZ0.H_10a.i:
Write down an expression for $f' (x)$ .
20N.1.AHL.TZ0.H_10b.i:
Show that $g^{- 1}$ exists.
SPM.2.AHL.TZ0.9:
Consider the graphs of $y = \frac{{{x^2}}}{{x - 3}}$ and $y = m\left( {x + 3} \right)$ , $m \in \mathbb{R}$ .

Find the set of values for $m$ such that the two graphs have no intersection points.
21M.2.AHL.TZ1.11d:
Sketch the graph of $y = f (x)$ for $- 3 \leq x \leq 3$ , showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.
21M.2.AHL.TZ1.11c:
Find the $x$ -coordinate of the point of inflexion.
21M.2.AHL.TZ1.11a:
Find the value of $p$ and the value of $q$ .
20N.1.AHL.TZ0.H_10b.iii:
Hence find $g^{- 1} (x)$ .
20N.1.AHL.TZ0.H_10a.ii:
Hence, given that $f^{- 1}$ does not exist, show that $b^{2} - 3 a c > 0$ .
17N.1.AHL.TZ0.H_6a:
Sketch the graph of $y = \frac{{1 - 3x}}{{x - 2}}$ , showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.
EXM.1.AHL.TZ0.5c:
As $x \to \pm \infty$ the graph of $f\left( x \right)$ approaches an oblique straight line asymptote.

Divide $2{x^2} - 5x - 12$ by $x + 2$ to find the equation of this asymptote.
17N.2.AHL.TZ0.H_6a:
Find the probability that Lucca eats at least one banana in a particular day.
21N.2.AHL.TZ0.10a.ii:
$y$ -axis.
22M.1.AHL.TZ2.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
21N.2.AHL.TZ0.10c:
The oblique asymptote of the graph of $f$ can be written as $y = a x + b$ where $a, b \in ℚ$ .

Find the value of $a$ and the value of $b$ .
EXM.1.AHL.TZ0.4a:
Show that $f\left( x \right)$ has no vertical asymptotes.
20N.1.AHL.TZ0.H_10b.ii:
$g (x)$ can be written in the form $p (x - 2)^{3} + q$ , where $p, q \in ℝ$ .

Find the value of $p$ and the value of $q$ .
20N.1.AHL.TZ0.H_10d:
Sketch the graphs of $y = g (x)$ and $y = g^{- 1} (x)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.
21M.2.AHL.TZ1.11b:
Find an expression for $f' (x)$ .
EXM.1.AHL.TZ0.5a:
Find all the intercepts of the graph of $f\left( x \right)$ with both the $x$ and $y$ axes.
21M.2.AHL.TZ1.11e:
Find the equations of all the asymptotes on the graph of $y = g (x)$ .
20N.1.AHL.TZ0.H_10c:
State each of the transformations in the order in which they are applied.
21M.2.AHL.TZ1.11f:
By considering the graph of $y = g (x) - f (x)$ , or otherwise, solve $f (x) < g (x)$ for $x \in ℝ$ .
21N.2.AHL.TZ0.10e.i:
Express $\frac{1}{f (x)}$ in partial fractions.
21N.2.AHL.TZ0.10e.ii:
Hence find the exact value of $\int_{0}^{3} \frac{1}{f (x)} d x$ , expressing your answer as a single logarithm.
21N.2.AHL.TZ0.10a.i:
$x$ -axis.
21N.2.AHL.TZ0.10b:
Write down the equation of the vertical asymptote of the graph of $f$ .

Hide related questions

Topic 2—Functions