(5-9)  We are comparing these two cases, superposing waveforms y=cos(kx) over a range of k from 0 to 1 (on the left) and from 0 to 2 (on the right).

①  Just by looking at the pictures, first try this: 

If we ignore the small sidebands in the superposition function and just focus on the central peak, what is the approximate relationship between the limits of this peak and the wavelength, λ(kmax), of the y=cos(kmax) function?

②  Now recall the de Broglie equation,  λ = h/p.

Consider our superposition of all possible waveforms y = cos(kx), over a range of k from 0 to kmax. Which of these is associated with the biggest value of the momentum?

③  As we have lately been discussing, momentum is a vector quantity and so we can think of the momentum associated with any waveform as being, in effect ±p. 

That being so, what limits can we put on the range of the momentum that a particle might have, whose position probability density distribution is defined by this superposition?

④   So we have, from ①, an estimate of the uncertainty in the position of the particle (given by the range of x spanned by that central peak) and we have, from ③, an estimate of the uncertainty in the momentum of the particle (given by the range from -pmax to +pmax).

Now for the big moment: write down expressions for these two uncertainties, both in terms of λ, and deduce the relationship between them.