Alan

if you have n values: x1, x2, ... xn, the geometric mean is the n'th root of the product of all the values.  That is:

geometric mean = ⁿ√(x1*x2*....*xn)

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If mass density is ρ then you need to find ∫ρ.dV  where dV is infinitesimal volume and the integral goes from zero to infinity.

dV = 4pi.s².ds, so

 $$mass=\int_0^{\infty}\frac{4\pi s^2 }{s(1+s)^3}ds$$ 

or

$$mass = 2\pi$$

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Do you mean quadrature?

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Look at the bottom two pictures. Imagine the bolt starts to drop just as the floor of the elevator is at the same level as the observer's feet  By the time the bolt hits the floor of the elevator, the floor will have moved upwards.  

Imagine the observer has a large vertical ruler 3m long.  He positions the top of the ruler at the height of the top of the elevator, when the elevator floor is at his feet.  The bottom of the ruler, level with the floor of the elevator at the instant the bolt starts to drop, would be 3m below the top.  Because the elevator is rising, by the time the bolt hits the elevator floor, the floor, and hence the bolt, will be at a level above the bottom of the ruler (which isn't in the elevator and so doesn't move).  This means the ruler will measure a distance less than 3m for the drop.

Yes, the linear approximation they are talking about isn't the bisection method I've illustrated above (Newton's method is the same).  

 Their linear method uses a tangent line, as does Newton's method, but it's just used to find an approximation to the value of the function at some "ugly" number (where "ugly" means it is not easy to calculate the function at that point) by using a nearby "non-ugly" number!  I suspect this is a very old-fashioned method, where calculations were done either in your head, or with a slide rule, or with log tables.  In these days of calculators and computers, there aren't really any such "ugly" numbers! 

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If, by linear, you mean linear bisection, then the following pictures might help:

In the linear bisection method, two x values are chosen, such that the signs of the corresponding y-values are different, and a straight line is drawn between them.  The next value of x is chosen to be where this line hits the y-axis.  It replaces one or other of the previous x values (depending on the sign of the new y-value), and the process repeated to convergence.  Just one step is illustrated in the first graph above - the open circle indicates the next x-value.

In Newton's method, just one point is chosen and the gradient of the curve at that point is calculated.  The gradient is then extended until it crosses the y-axis.  The x-value at that point is used as the next x-value and the process repeated etc.  Again just one step is illustrated in the second graph above.

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1.  On the cliff.

Resolving forces parallel to the surface we have m.g.sinθ - μ.m.g.cosθ down the slope where θ is 30.9° and μ is 0.241;  m is mass and g ia gravitational acceleration.

From Newton's 2nd law of motion this must equal m.a where a is (constant) acceleration.  

This means a = g.(sinθ - μ.cosθ)

Hence velocity, v, at the edge of the cliff is v = √(2.a.s)  where s is 19 metres.

2.  Off the cliff.

The horizontal velocity, vh, is constant (ignoring air resistance etc.) at  vh = v.cosθ.

The initial vertical velocity, vv, is vv = v.sinθ.

For the vertical motion we have vvf²= vv² + 2.g.sv  where vvf is final vertical velocity and sv is 3.76m.

3. Landing

The magnitude of the resultant velocity, vr, when she lands is given by:  vr = √(vvf² + vh²) 

You can insert the numbers.

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Covariance between two variables X and Y is defined as E(X*Y) - E(X)*E(Y)

A standard correlation coefficient is defined as:

Does that help?

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Oops!  Just noticed that I assumed zero on the right-hand side for c), but it should have been 3.  Also, no inequality was asked for!

I see Chris has got it right though.

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The versine is defined as geno has written it.