## Question 7:

### Part (a):
$\frac{x^6}{8} \geq 0$ or $x^6 \geq 0$ | **B1** | Do not accept "$\frac{x^6}{8}$ is always positive (ie $> 0$)"

### Part (b):
The expansion with two terms is a better approximation than the one with three terms but it should be the other way round. | **E1** | O.E. See exemplars

### Part (c):
$1 + \frac{1}{2}(-x^3) + \frac{1}{2}\left(-\frac{1}{2}\right)\frac{(-x^3)^2}{2!} + \frac{1}{2}\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)\frac{(-x^3)^3}{3!} + \ldots$ | **M1** | Working for 3rd or 4th term correct; must see the negative $x^3$

$1 - \frac{x^3}{2} - \frac{x^6}{8} - \frac{x^9}{16}\ldots$ | **A2** | A2 all terms correct or A1 for three terms correct

### Part (d):
$|x| < 1$ oe | **B1** |

### Part (e):
Goes through $2.5$ on $y$-axis and $(1,0)$ | **B1** |
Right shape for values of $x$ from $-1$ to $1$ | **B1** | Curve reaches between $(-1, 3)$ to $(-1, 4.5)$

### Part (f):
$2.5\int_{-0.75}^{0.75}\left(1 - \frac{x^3}{2} - \frac{x^6}{8} - \frac{x^9}{16}\right)dx$ | **M1** | For their expression from 7(c) in an integration; condone $dx$ missing; 2.5 and limits needed but may be seen later

$[2.5]\left[x - \frac{x^4}{8} - \frac{x^7}{56} - \frac{x^{10}}{160}\right]_{-0.75}^{0.75}$ | **M1** | For integrating their expression (3 terms or more); allow one error; limits could be wrong or missing here

$[2.5]\left[\left(0.75 - \frac{0.75^4}{8} - \frac{0.75^7}{56} - \frac{0.75^{10}}{160}\right) - \left(-0.75 - \frac{(-0.75)^4}{8} - \frac{(-0.75)^7}{56} - \frac{(-0.75)^{10}}{160}\right)\right]$ | **M1** | For attempt at their limits substituted into their integrand; substitution must be seen

$3.74\ \text{m}^2$ | **A1** | AWRT 3.74 from correct working; must include units; 1.495 will probably get **M2** or **M3**

**Special Case:** If trapezium rule used on original function allow **M1 A1** maximum | | Likely answers: 2 strips: $3.71\text{m}^2$; 3 strips: $3.72\text{m}^2$; 6 strips: $3.73\text{m}^2$

---