Understanding Engineering Mathematics

3.The following summation is used in the multiplication ofmatrices(see Chapter 13):

∑n

k= 1

aikbkj

whereaik,bkj, are elements of a matrix. Write out the summations in full in the

following cases.

(i) n= 2 (ii) n=3, i=1, j= 2

(iii) n=4, i = 3, j=3(iv)n=4, i=2, j= 4

Repeat the exercise for

∑n

l= 1

ailblj

withkreplaced byl.

4.A spring vibrates 50 mm in the first oscillation and subsequently 85% of its previous
value in each succeeding oscillation. How many complete oscillations will occur before
the vibration is less than 10 mm and how far will the spring have travelled in this time?
Assuming that the spring can perform an infinite sequence of oscillations, how far does
it move before finally stopping? Repeat these calculations with 85% replaced by a
general factor ofk% and various initial and final conditions.

5.The emitter efficiency,γ, in an n-p diode is given by

γ=

In

In+Ip

whereIpis the hole current andInis the electron current crossing the emitter-base

junction. In practice,Ipis made much smaller thanIn. Show that in this case

γ 1 −

Ip

In

6.In the next chapter we will meet theexponential function. The definition of this
important function is perhaps a little more sophisticated than you are used to, either by
alimitor by aseries. The connection between these definitions relies on a use of a
binomial expansion of a particular kind. Show that ifnis an integer (➤126):

(

1 +

x

n

)n

= 1 +x+

1

(

1 −

1

n

)

2!

x^2 +

1

(

1 −

1

n

)(

1 −

2

n

)

3!

x^3 +···

By lettingn‘go to infinity’ – i.e. lettingnbecome as large as we like – obtain a series

expression for the limit

lim

n→∞

(

1 +

x

n

)n

This is in fact the definition of the exponential functionex.