The diagram shows the graph of $y=f(x)$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2003 - Paper 1

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

To sketch this graph, note that the original function $f(x)$ has peaks and troughs. The reciprocal will approach infinity near zeros of $f(x)$ and will be close to zero where $f(x)$ has larger values.

(ii) $y=f(x)+f'(x)$

This sketch involves adding the value of the derivative to the original function. The graph will show points of inflection and steepness changes relative to $f(x)$, highlighting steep areas.

(iii) $y=(f(x))^2$

Since squaring eliminates negative values, the sketch will reflect $f(x)$ upwards, and all previously negative regions will be pushed to the x-axis.

(iv) $y=e^{f(x)}$

As an exponential function, this sketch will produce a graph that grows significantly as $f(x)$ increases, leading to rapid increases above the x-axis.

Find the eccentricity, foci and the equations of the directrices of the ellipse

For the ellipse defined by [ \frac{x^2}{9} + \frac{y^2}{4} = 1 ]:

1. Eccentricity:
   [ e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} ]
2. Foci: The foci are located at [ (\pm ae, 0) = (\pm 3 \cdot \frac{\sqrt{5}}{3}, 0) = (\pm \sqrt{5}, 0) ]
3. Equations of the directrices: [ x = \pm \frac{a}{e} = \pm \frac{3}{\frac{\sqrt{5}}{3}} = \pm \frac{9}{\sqrt{5}} ]

Show that the area of the annulus at height $y$ is given by $4\pi \sqrt{1-y}$:

To find the area of the annulus at height $y$, determine the outer and inner radii based on the given bounds:

1. Arch the outer function $y = 3-\frac{y}{2}$ (for the area at $y$) and the inner function $y=0$: The outer radius will be: [ R_{outer} = 3-\sqrt{1-y} ] The inner radius is simply [ R_{inner} = 0 ]
2. Find the area: [ A(y) = \pi R_{outer}^2 - \pi R_{inner}^2 = \pi (R_{outer}^2) = \pi ((3-\sqrt{1-y})^2) = 4\pi \sqrt{1-y}. ]

Find the volume of the solid:

Using the washer method, the volume can be expressed as:

1. [ V = \int (R(y)^2 - r(y)^2) dy ]
2. Substitute the respective functions for bounds and then integrate to gain the volume.