Assessment task

Please ensure that all work is clearly communicated and any answers clearly labelled in the simulation file(s).

1. Consider the problem of evaluating the value of the definite integral

* x*^{2}

e sin1/2 dx .

x

i) Using the substitution y ex and 5000 independent

1 ex

realisations of a U[0,1] variable, estimate the value of the integral.

ii) Find an alternative suitable substitution and obtain a second estimate of the value of the integral using the same 5000 realisations of a U[0,1] variable.

Clearly state your change of variable and derive the resulting equivalent definite integral.

iii) Which estimate – from part i) or from part ii) – would you expect to converge more quickly to the true numerical value of the integral? Justify your answer.

2. Apply the Monte Carlo method using 2000 independent realisations of a U[0,1] variable to estimate the value of the definite integral

3.96 ( x 2)2

2π

1 e2 dx .

0.04

Clearly state any changes of variable required and derive the resulting equivalent definite integral.

3. A mass hangs on a spring. Let x ( t ) be the displacement

(in cm) at time t (in seconds) of the mass from its equilibrium position.

Assuming no energy is lost (to friction, to air resistance or to other reasons), the velocity of the mass for

t 0 is given by x ( t ) Aω cos( ωt ) Bω sin( ωt ) where

ω is the angular frequency of the system.

Assume that ω 1 rad/sec and the system released at time zero from x(0) 8 with zero velocity so that the equation simplifies to x ( t ) 8 sin( t ) .

i) Using 5000 independent realisations of a U[0,1] variable, estimate the total distance that the mass moves in the first five minutes.

A more realistic model includes damping (i.e. the loss of energy from the system to friction and air resistance). The velocity of the mass for

t 0 is now described by x (t ) e Dt

Aω cos(ωt ) Bω sin(ωt ) where

D 0 is the damping coefficient governing the rate of energy loss in the system. Incorporating the same assumptions and initial conditions simplifies the velocity equation to x (t ) 8e Dt sin(t ) .

ii) Using the same 5000 realisations of a U[0,1] variable in each case, plot a graph of the damping coefficient (on the horizontal axis) against the total distance travelled in the first five minutes (on the vertical axis)

Include datapoints for at least ten values of D.

iii) Estimate the largest value of D for which the mass moves a total distance of at least 1m.

Hint: Remember that if the mass travels upwards for 8cm and then downwards for 8cm, the change in its displacement over that time period is zero, although the distance it has travelled is 16cm.

4. (Extension task, worth 2 of potential 10 marks.)

Consider the problem of evaluating the value of the definite integral

0 10

dx .

1 x

i) Using the substitution y 1 and 5000 independent

1 x2

realisations of a U[0,1] variable, estimate the value of the integral.

Hint: Think carefully when calculating x ( y ) for the required substitution.

ii) Find an alternative suitable substitution and obtain a second estimate of the value of the integral using the same 5000 realisations of a U[0,1] variable.

Clearly state your change of variable and derive the resulting equivalent definite integral.

Note: z denotes the floor function which, for any z , takes the

value of the largest integer z . For example, if z3.5 , then

10 10 2 .

1 ( 3.5) 4.5

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