Section A: Calculus

1.  [20 marks]

(a) LEEDS1005 Optimise f (x, y, z) = xy + yz + zx subject to x + y + z = A for the following values of A: A = 1 and A = 1.01.

(b) LEEDS1005 Let f*   : R - R be defined as f* (A) is the optimal value of f as found in part a for a given value of A. Using your answers from part a, compute

f* (1.01) O f* (1)

(c) LEEDS1005 Investigate at least two further cases of values of A and A + 0.01, computing the ratio

f* (A + 0.01) O f* (A)

You need not show full working for this investigation, but summarise your results in a table structured as follows.  Record all your answers to at least two decimal places.

A     λ     f* (A)      

1    . . .      . . .                . . .

Look carefully over your workings and your results: do you spot any pattern?

(d) LEEDS1005 Solve the general case of the optimisation problem in part b, where the constraint is x + y + z = A, and hence explain why the pattern holds.

(e) LEEDS1005 Generalise your result from the above investigation (parts b to d), applicable to all Lagrange multiplier problems with an objective function f : Rg  -

R and a single constraint function, g : Rg  - R, with g(x) = A.

(f) LEEDS1005 Prove your generalisation from part e above.

(g) LEEDS1005 Your firm runs an industrial process that requires three inputs, labelled x, y and z . The income your firm derives from this process, in millions of pounds, is given by f (x, y, z) = 20xy z  , and the amount it spends on the process, also in millions of pounds, is given by the function g(x, y, z) = 3x2 + 4y2 + 2z2 . Currently, your firm allocates a budget of g(x, y, z) = 10 to running the process, and derives an income from the process of around 22.10.

At the current budget level, is it profitable to increase the budget allocated to this process?  For each extra pound invested, how much profit (or loss) should your firm expect to generate?

2.  [10 marks]

(a) LEEDS1005 Let f : Rg  - R, and a e Rg .  Prove that the following two state- ments are equivalent:

. Du [f](a) = 0 for all u e Rg ; and

. Vf (a) = 0.

(b) LEEDS1005 Give an example of a function f and a point a such that Vf (a) = 0 and det Hf (a) = 0 for each of the following two cases:

i. where f (a) is either a minimum or a maximum; and

ii. where f (a) is a saddle point.

Explain why your examples meet the criteria specified.

(c) LEEDS1005 What fact from this semester’s calculus studies has your pair of ex- amples demonstrated? Explain your answer.

Section B: Differential equations and Mechanics

1.  [50 marks]

(a) LEEDS1005 Find the solution of the following initial value problems.

.

 +  O 2y = 6et ,    y = 0 and  = 0 at x = 0.

ii.

d2 y         dy                                                    dy

dx2           dx                                              dx

(b) LEEDS1005 For the following homogeneous differential equation, given that y1 (x) = et  is a solution, find the other independent solution y2 . Then, check explicitly that y1  and y2  are independent.

d2y                    dy

dx2                            dx

(c) LEEDS1005 Consider the following differential equation x2   + bx  + y = 0.

where b is a real parameter.   Recall the solution procedure for a homogeneous Cauchy differential equation:  the general solution is a linear combination of two functions of the form xλ  unless the auxiliary equation has a repeated root. Find the allowed range of values for b such that the general solution is a linear combination two functions of the form xλ  with real exponent λ .

(d) LEEDS1005 Find the functions a(x) and b(x) such that y1 (x) = sin ^x is a solu- tion of the following linear second order differential equation

d2y              dy

dx2                   dx

Then, find the other independent solution y2 . (Hint: you may find the antiderivative

^t sin(dt)2 ^t  = O2 cot ^x useful.)

2.  [20 marks]

(a) LEEDS1005 Recall exercise T3  in  Example sheet  15.   A  particle of mass  m  is projected vertically upwards with velocity v0 from the surface of the Earth at z = 0. Gravity acts downwards in the negative z direction, with expression

GmM 

F1  = O             k

where R is the radius of the Earth M its mass, and G is the gravitational constant.

Now suppose that, in addition, there is a constant friction force F2  = OmF0 k,

v2  = v0(2) +  O  O 2F0 z .

(b) LEEDS1005 Is there an escape velocity in this case? Justify. If yes, find its expres- sion.  If no, find the maximum height reached by the particle.  Answers should be given in terms of R, M , v0 , G and F0 . Check the dimensions of your answer.

3.  [30 marks]

(a) LEEDS1005 Consider the simple pendulum as presented in Section 7.4 of the notes. We assume exactly the same setting for the mass m attached to a rod of length a except that the other end of the rod is no longer attached to the fixed origin O but to a moving point P with position vector OP = x(t)j.  We recall that, in the setup of Section 7.4, the basis vector j gives the horizontal direction while the basis vector i direct the vertical direction and is pointing downwards.  These are just convenient choices.  Thus, we now assume that the pendulum is attached to a moving point that is allowed to move horizontally only.

Show that the equations of motion (7.14a) for the particle are modified to the following equations

,m(O)lθ¨(m)mg(O)os(O)sin θ ,                                (1)

Note that of course, if the point P is fixed, so that  = 0, we recover exactly (7.14a).

(b) LEEDS1005 We now assume that the point P is forced to move periodically in such a way that x(t) = A cos pt where A > 0 is the amplitude of the motion and p its angular frequency. We suppose that p  ω with ω =尸 . Consider the second equation written as

θ¨ + ω2 sin θ = O  cos θ .

Assume θ remains small so that we may use sin θ ~ θ and cos θ ~ 1.  Find the general solution θ(t) under these assumptions.

(c) LEEDS1005 Suppose now p = ω and all other assumptions remain the same. Find the new form of the general solution.  As time evolves, does the small amplitude assumption remain valid in this case? Which phenomenon discussed in the lectures notes is this signaling?

(d) LEEDS1005 This is an open type question. Find an example of motion of the point P (not necessarily horizontal) which results in the second equation of motion being

modified to

aθ¨+ (g O a) sin θ = 0 ,

for some constant number a.

Before you submit your solutions remember to attach a completed Academic Integrity form. We recommend www′Ylovdpc』′aom for signing and merging pdf documents.