MRS EZRINDA MOHD ZAIHIDEE

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CHAPTER 2 ENGINEERING FUNCTIONS
MRS EZRINDA MOHD ZAIHIDEE FAKULTI KEJ. ELEKTRIK
ELEKTRONIK
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LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• sketch a graph for several functions.
• solve all the questions.

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APPLICATIONS

• ? the intensity of sound
• ? the intensity of earthquakes
• ? power gains (electrical transmission
  lines)
• ? in studying population growth (biology)
• ? to calculate compound interest (business)
• ? used extensively in electronics,
  mechanical systems, thermodynamics, nuclear
  physics etc.

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EXPONENTIAL FUNCTIONS
One-to-one function, so exists.
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EXPONENTIAL FUNCTIONS
The value of can be found to any level of
precision decision from the series expansion
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EXPONENTIAL FUNCTIONS
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EXPONENTIAL FUNCTIONS
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EXPONENTIAL FUNCTIONS

• EXAMPLE 2
• The decay of voltage, volts, across a
  capacitor at time
• seconds is given by
  Draw a graph showing
• the natural decay curve over the first
  seconds. From the
• graph, find
• the voltage after
• the time when the voltage is

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EXPONENTIAL FUNCTIONS
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EXPONENTIAL FUNCTIONS
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LOGARITHMIC FUNCTIONS
2 types of logarithm (a) (b)
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LOGARITHMIC FUNCTIONS
y
1
x
0
1
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LOGARITHMIC FUNCTIONS
where are constants
change to logarithmic form (natural logarithm)
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LOGARITHMIC FUNCTIONS
EXAMPLE 2
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LOGARITHMIC FUNCTIONS
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LOGARITHMIC FUNCTIONS
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LOGARITHMIC FUNCTIONS
Laws of logarithms
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LOGARITHMIC FUNCTIONS
EXAMPLE 3 Simplify each expression (a) (b) (c) (d
) (e)
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LOGARITHMIC FUNCTIONS
EXAMPLE 4 Solve the following equations (a) (b) (
c) (d) (e)
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CHAPTER 2 ENGINEERING FUNCTIONS
MRS EZRINDA MOHD ZAIHIDEE FAKULTI KEJ. ELEKTRIK
ELEKTRONIK
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LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• sketch a graph of hyperbolic functions.
• solve all the questions.

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HYPERBOLIC FUNCTIONS
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HYPERBOLIC FUNCTIONS
Graphs of hyperbolic functions
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HYPERBOLIC FUNCTIONS
EXAMPLE 1 Find the value of (a) (b)
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HYPERBOLIC FUNCTIONS
Hyperbolic identities
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HYPERBOLIC FUNCTIONS
EXAMPLE 2 Express (a) in terms
of and (b)
in terms of and
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INVERSE HYPERBOLIC FUNCTIONS
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INVERSE HYPERBOLIC FUNCTIONS
EXAMPLE 3 Evaluate (a) (b) (c)
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CHAPTER 2 ENGINEERING FUNCTIONS
MRS EZRINDA MOHD ZAIHIDEE FAKULTI KEJ. ELEKTRIK
ELEKTRONIK
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LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• find trigonometrical ratios of some special
  angles without using calculator.
• sketch graph of sin x, cos x, and tan x.
• solve questions using trigonometric identities.
• (4) differentiate all 7 terminologies for
  modelling waves using sin t and cos t.
• (5) solve trigonometric equations.

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TRIGONOMETRIC FUNCTIONS
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TRIGONOMETRIC FUNCTIONS
B
90-?
y
r
?
O
A
x
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TRIGONOMETRIC FUNCTIONS
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TRIGONOMETRIC FUNCTIONS
For any angle ?
A
S
?
?
C
T
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TRIGONOMETRIC FUNCTIONS
Trigonometrical ratios of some special angles
B
B
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60
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A
O
A
O
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TRIGONOMETRIC FUNCTIONS
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TRIGONOMETRIC FUNCTIONS
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TRIGONOMETRIC FUNCTIONS
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TRIGONOMETRIC IDENTITIES
Basic identities
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TRIGONOMETRIC IDENTITIES
The compound angle formula
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TRIGONOMETRIC IDENTITIES
The double angle formula
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TRIGONOMETRIC IDENTITIES
The half-angle formula
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TRIGONOMETRIC IDENTITIES
The factor formula
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TRIGONOMETRIC IDENTITIES
Other formula
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TRIGONOMETRIC IDENTITIES
EXAMPLE 1 Show that
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TRIGONOMETRIC IDENTITIES
EXAMPLE 2 Simplify each of the following. (a)
(d) (b) (e) (c)
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MODELLING WAVES USING sin t AND cos t
f(t)
A
t
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MODELLING WAVES USING sin t AND cos t

• maximum displacement of the wave from its mean
  position.
• measured in radians per second (rad s-1)

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MODELLING WAVES USING sin t AND cos t

• time taken to complete one full cycle.
• number of cycles completed in 1 second.
• measured in Heartz (Hz).

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MODELLING WAVES USING sin t AND cos t

• allows wave to be shifted along the time axis
  (x-axis).
• the actual movement of the wave along
  the time axis.

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MODELLING WAVES USING sin t AND cos t

• sine and cosine functions repeat themselves at
  regular intervals.

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MODELLING WAVES USING sin t AND cos t
EXAMPLE 3 State the amplitude, angular frequency
and period of each of the following waves. (a) (b)
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COMBINING WAVES
There are many situations in which engineers
need to combine two or more waves together to
form a single wave.
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COMBINING WAVES

• EXAMPLE 4
• Two voltage signals, and , have
  the following mathematical expressions
• State the amplitude and angular frequency
  of the two signals.
• Obtain an expression for the signal,
  , given by
• Reduce the expression obtained in part (b)
  to a single sinusoid and hence state its
  amplitude and phase.

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COMBINING WAVES

• EXAMPLE 5
• Two current signals, and , have
  the following mathematical expressions
• State the amplitude and angular frequency
  of the two signals.
• Obtain an expression for the signal,
  , given by
• Reduce the expression obtained in part (b)
  to a single sinusoid and hence state its
  amplitude and phase.

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COMBINING WAVES
EXAMPLE 6 Express as
a single cosine wave.
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TRIGONOMETRIC EQUATIONS
EXAMPLE 7 Solve for (a) sin t 0.6105 (b)
cos t - 0.3685 (c) tan t 1.3100
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CHAPTER 2 ENGINEERING FUNCTIONS
MRS EZRINDA MOHD ZAIHIDEE FAKULTI KEJ. ELEKTRIK
ELEKTRONIK
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LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• sketch graph for the given functions.
• differentiate continuous function and piecewise
  continuous function by refer to the graph.

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CONTINUOUS AND PIECEWISE CONTINUOUS FUNCTIONS

• a function is said to be continuous if its graph
  can be drawn over each interval of its domain
  with a continuous motion of the pen without
  lifting the pen.
• a piecewise continuous function has a finite
  number of discontinuities in any given interval.

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CONTINUOUS AND PIECEWISE CONTINUOUS FUNCTIONS
EXAMPLE Determine whether each of the given
function is continuous or not. (a) (d) (b) (
e) (c)
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CHAPTER 2 ENGINEERING FUNCTIONS
MRS EZRINDA MOHD ZAIHIDEE FAKULTI KEJ. ELEKTRIK
ELEKTRONIK
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LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• sketch graph and determine the discontinuity for
  the given functions.

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UNIT STEP FUNCTIONS, u(t)
Discontinuity at t 0
u(t)
1
t
0
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UNIT STEP FUNCTIONS, u(t)
The position of the discontinuity may be shifted
u(t-d)
1
t
0
d
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UNIT STEP FUNCTIONS, u(t)
EXAMPLE Sketch the following functions. (a) (b
) (c) (d) (e)
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CHAPTER 2 ENGINEERING FUNCTIONS
MRS EZRINDA MOHD ZAIHIDEE FAKULTI KEJ. ELEKTRIK
ELEKTRONIK
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LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• sketch graph of the given functions.

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DELTA FUNCTIONS OR UNIT IMPULSE FUNCTIONS
Rectangle function as h approaches 0
R(t)
t
0
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DELTA FUNCTIONS OR UNIT IMPULSE FUNCTIONS
? an impulse of strength k at the origin
? an impulse of strength k at t d
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DELTA FUNCTIONS OR UNIT IMPULSE FUNCTIONS
EXAMPLE Sketch the impulse train given by
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