IB Maths AI SL 5 Calculus Questions

$\quad\left(g^{\prime}(x)=\right)-8 x^{-2}+x$
Note: Award A1 for -8 seen, A1 for $x^{-2}$ (or $\frac{1}{x^{2}}$ ) and A1 for second term being x. Award at most A1A1AO if additional terms are seen.

$x>2 \mathbf{OR}(2, \infty) \mathbf{OR} \quad 2<x<\infty$
Note: Award A1 for 2 seen and award A1 for correct inequality.

$\frac{\mathrm{d} V}{\mathrm{~d} x}=300 \sqrt{3}-\frac{27}{4} x^{2}$

Note: Award A1 for a correct term.

from the graph of V or $\frac{\mathrm{d} V}{\mathrm{~d} x}$ OR solving $\frac{\mathrm{d} V}{\mathrm{~d} x}=0$

from the graph of V OR substituting their value for x into V
$V_{\text {max }}=3040 \mathrm{~cm}^{3}(3039.34 \ldots)$

EITHER
wasted space / spheres do not pack densely (tesselate)

OR
the model uses exterior values / assumes infinite thinness of materials and hence the modelled volume is not the true volume

$\frac{1}{2} \times 1 \times((6+4.3)+2(1.7+1.3+3+4.7))$$=15.9(15.85)\left(\mathrm{cm}^{2}\right)$

$\quad \int_{0}^{5}\left(-\frac{x^{3}}{3}+3 x^{2}-7 x+6\right) \mathrm{d} x$

Note: Award A2 for an integral with correct integrand and correct limits.

$\quad 15.4\left(15.4166 \ldots, \frac{185}{12}\right) \quad\left(\mathrm{cm}^{2}\right)$

Note: If their limits are outside the domain, award at most A1AOFT.

$\frac{\mathrm{d} A}{\mathrm{~d} r}=4 \pi r-800 r^{-2}$
Note: Award A1 for $4 \pi r$ seen, and (M1) for expressing $\frac{800}{r}$ as $800 r^{-1}$ (can be implied through $\mp \frac{800}{r^{2}}$ seen), A1 for $-800 r^{-2}$. Award at most A1 (M1)A0 if any additional terms are seen.

$0=4 \pi r-800 r^{-2}$$r=3.99 \quad\left(3.99294 \ldots, \sqrt[3]{\frac{200}{\pi}}\right)(\mathrm{cm})$

Note: Award at most M1A0 if k is not found.