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SUBJECT:

MATHEMATICS ENGINEERING

SET 1

NAME OF LECTURER : DR.MARSYITA

GROUP :

DUE DATE : 24 NOVEMBER 2014

NAMA NO.

MATRIK

NO. KAD PENGENALAN

LIM JUN RONG 178463

KUAN POH YEE 176762 940202-08-5218

WONG SWEE YAO

LOW BEE SAN

LIM KOK SANG

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Question 1 (10 marks)

Find the following limits.

Solution

Since takes the form

at x =8, we apply L Hpitals rule to give

=2

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Using trigonometric identity,

Since

takes the form at x = , we apply L Hpitals rule to give

Question 2 (15 marks)

Sketch the graphs of the following functions IT tools for .

a)

||

b) ||c) ||

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Solution

a) || ,

b) ||

x f(x)

-2 4

-1 2

0 0

1 0

2 0

x f(x)

-2 -4

-1 -1

0 0

1 12 4

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c) ||

x f(x)-2 -1

-1 -1

0 undefined

1 1

2 1

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Question 3 (10 marks)

Two water waves,

and

, meet. Find the equation for the

resultant wave and compare its characteristics with those of the original waves. Discuss the

following special cases:

a)

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b) c)

Solution :

To find the equation for the resultant wave , sum the equation of first water wave () and secondwater wave ( .Equation for resultant wave

Let resultant wave be, +

+

+ By using the formula of trigonometric identities :

*[

Equation for resultant wave ,

Comparing the resultant wave with both the original waves.

RESULTANT WAVE ORIGINAL WAVE

Amplitude is higher than original wave. Amplitude is lower than resultant wave.

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Standing wave. Progressive wave.

a) When ( ) Since cos 0 = 1,

New original waves do not have any significant changes.

The new resultant wave

will now have a phase difference of

.

b) When ( ) ( ) Since , X = [ ] New original wave has no change but will become which

leads the old by . The new resultant wave will have a lower magnitude.

c) When

( )

( ) Since ,

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New original wave has no change but will become which

lags the old by . The new resultant wave

will become 0.

Question 4 (5 marks)

A projectile is launch from level ground at with initial velocity at an angle to thehorizontal. Subsequently, its position is described by ,

Find the equation of trajectory of the projectile. If , find where the projectile hits theground on its returns.

Solution :

y

0 x

= =

Since velocity, v =

Horizontal distance: x =) tx =

Vertical distance: Sy= t + 2

u

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Since a = -g, so y= ()t - 2Trajectory equation:

x = t =

y = () t - 2

= ( ) - 2

= y = - (1 + )

time of flight, t =

since , t

=

Position of x = )

= ( )( )

x=

Trigonometric identities:

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Question 5 (10 marks)

Express the following in partial fraction:

a)

b)

Solution :

a)

[

]

When

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By substituting and into

,

*Answer

Solution :

b )

+ When = 2 ,

+

+

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When = 3 ,

+ +

By substituting 22 and 16into ,

*Answer

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