18M.1.AHL.TZ1.H_7

pestleMathematicsAAHLPaper 118M· sl-5-3-differentiating-polynomials-n-e-zsource ↗

Let $y = {\text{arccos}}\left( {\frac{x}{2}} \right)$

Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .

[2]

Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$ .

[7]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$y = {\text{arccos}}\left( {\frac{x}{2}} \right) \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{1}{{2\sqrt {1 - {{\left( {\frac{x}{2}} \right)}^2}} }}\left( { = - \frac{1}{{\sqrt {4 - {x^2}} }}} \right)$ M1A1

Note: M1 is for use of the chain rule.

[2 marks]

attempt at integration by parts M1

$u = {\text{arccos}}\left( {\frac{x}{2}} \right) \Rightarrow \frac{{{\text{d}}u}}{{{\text{d}}x}} = - \frac{1}{{\sqrt {4 - {x^2}} }}$

$\frac{{{\text{d}}v}}{{{\text{d}}x}} = 1 \Rightarrow v = x$ (A1)

$\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x} = \left[ {x\,\,{\text{arccos}}\left( {\frac{x}{2}} \right)} \right]_0^1 + \int_0^1 {\frac{1}{{\sqrt {4 - {x^2}} }}} dx$ A1

using integration by substitution or inspection (M1)

$\left[ {x\,\,{\text{arccos}}\left( {\frac{x}{2}} \right)} \right]_0^1 + \left[ { - {{\left( {4 - {x^2}} \right)}^{\frac{1}{2}}}} \right]_0^1$ A1

Note: Award A1 for ${ - {{\left( {4 - {x^2}} \right)}^{\frac{1}{2}}}}$ or equivalent.

Note: Condone lack of limits to this point.

attempt to substitute limits into their integral M1

$= \frac{\pi }{3} - \sqrt 3 + 2$ A1

[7 marks]

Examiners’ report

[N/A]