17M.1.AHL.TZ2.H_9

pestleMathematicsAAHLPaper 117M· sl-5-8-testing-for-max-and-min-optimisation-points-of-inflexionsource ↗

Consider the function $f$ defined by $f(x) = {x^2} - {a^2},{\text{ }}x \in \mathbb{R}$ where $a$ is a positive constant.

The function $g$ is defined by $g(x) = x\sqrt {f(x)}$ for $\left| x \right| > a$ .

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = f(x)$ ;

[2]

a.i.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = \frac{1}{{f(x)}}$ ;

[4]

a.ii.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = \left| {\frac{1}{{f(x)}}} \right|$ .

[2]

a.iii.

Find $\int {f(x)\cos x{\text{d}}x}$ .

[5]

By finding $g'(x)$ explain why $g$ is an increasing function.

[4]

Markscheme / solution

M17/5/MATHL/HP1/ENG/TZ2/09.a.i/M

A1 for correct shape

A1 for correct $x$ and $y$ intercepts and minimum point

[2 marks]

a.i.

M17/5/MATHL/HP1/ENG/TZ2/09.a.ii/M

A1 for correct shape

A1 for correct vertical asymptotes

A1 for correct implied horizontal asymptote

A1 for correct maximum point

[??? marks]

a.ii.

M17/5/MATHL/HP1/ENG/TZ2/09.a.iii/M

A1 for reflecting negative branch from (ii) in the $x$ -axis

A1 for correctly labelled minimum point

[2 marks]

a.iii.

EITHER

attempt at integration by parts (M1)

$\int {({x^2} - {a^2})\cos x{\text{d}}x = ({x^2} - {a^2})\sin x - \int {2x\sin x{\text{d}}x} }$ A1A1

$= ({x^2} - {a^2})\sin x - 2\left[ { - x\cos x + \int {\cos x{\text{d}}x} } \right]$ A1

$= ({x^2} - {a^2})\sin x + 2x\cos - 2\sin x + c$ A1

$\int {({x^2} - {a^2})\cos x{\text{d}}x = \int {{x^2}\cos x{\text{d}}x - \int {{a^2}\cos x{\text{d}}x} } }$

attempt at integration by parts (M1)

$\int {{x^2}\cos x{\text{d}}x = {x^2}\sin x - \int {2x\sin x{\text{d}}x} }$ A1A1

$= {x^2}\sin x - 2\left[ { - x\cos x + \int {\cos x{\text{d}}x} } \right]$ A1

$= {x^2}\sin x + 2x\cos x - 2\sin x$

$- \int {{a^2}\cos x{\text{d}}x = - {a^2}\sin x}$

$\int {({x^2} - {a^2})\cos x{\text{d}}x = ({x^2} - {a^2})\sin x + 2x\cos x - 2\sin x + c}$ A1

[5 marks]

$g(x) = x{({x^2} - {a^2})^{\frac{1}{2}}}$

$g'(x) = {({x^2} - {a^2})^{\frac{1}{2}}} + \frac{1}{2}x{({x^2} - {a^2})^{ - \frac{1}{2}}}(2x)$ M1A1A1

Note: Method mark is for differentiating the product. Award A1 for each correct term.

$g'(x) = {({x^2} - {a^2})^{\frac{1}{2}}} + {x^2}{({x^2} - {a^2})^{ - \frac{1}{2}}}$

both parts of the expression are positive hence $g'(x)$ is positive R1

and therefore $g$ is an increasing function (for $\left| x \right| > a$ ) AG

[4 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

a.iii.

[N/A]