ACS123 Functions Dr Viktor Fedun Automatic Control and Systems Engineering, C09 Based on lectures by Dr Anthony Rossiter.

ACS123Functions

Dr Viktor Fedun

Automatic Control and Systems Engineering, C09

Based on lectures by Dr Anthony Rossiter

Why is mathematics important?

Why do engineers need to be good at mathematics?

Is it sufficient to memorise key results?

Just because a learning technique worked at school, does that make it the best method now?

What mathematics do I need to be good at?

Mathematics is a tool-kit

A good engineer:

1. knows which is the best tool to use?

2. Is proficient in using the tool?

3. Can adapt the tool to a new use.

It is not good enough to memorise key results as the most important skill is abstraction. You must put your effort into understanding.

Module assessment

• 3 in class tests in weeks 4, 7, 11, 13

These will be similar to exam questions.• An exam in May or June

If you want feedback on an answer you have done, ask in a tutorial.

20 credit module, similar pattern in semester 2.

Module organisation

I will teach the first semester

Lectures and tutorials

Of the 5 timetabled hours, 2-3 will be used for lectures (these times may vary each week).

MOLE

Please use the discussions board to ask questions. Then everyone can see the question and answer.

I will not respond to email queries unless of a personal or private nature.

Resources

Learning is only effective where students engage in self-discovery.

1. What you hear, you will usually forget.2. You only really understand something when

you use it.3. We will provide ample materials, but YOU will

only learn if you use these properly. [5-6 hours per week]

Lecturers are here to guide – NOT TO TEACH! We will answer queries and be as helpful as possible, but only you can do the work.

Be a function

Stand up. 1. Use your arms to illustrate y=x.2. What about y=-x?3. Can you do y=x2, or even x3.4. What about sine(x) – you may need a

partner. Now do cosine(x).5. Can you y=mod(x)? Or even y=sqrt(x2)?6. Can you think of any more?

Common functions

1. sine, cosine, tangent (and their inverses)

2. logarithm, exponential

3. sinh and cosh

4. straightline, quadratic, general polynomial

5. combinations of above as products, composites and fractions.

You should be familiar with shapes of common functions and be able to sketch quickly.

Example 1 (Page 136, Kuldeep and Singh, Example 3)

[Mechanics]

The displacement, φ(t), of a particle at time t is given by:

φ(t)= 2t3 + t2 - 10t + 10

– Evaluate φ(2), φ(3), φ(5).– Find simplified expressions for:

(i) φ(t2)

(ii) φ(t + 1)

Example 1 Solution

Solution:

(a) We have

(a)φ(2) = (2 x 23) + 22 – (10 x 2) + 10 = 10

(a)φ(3) = (2 x 33) + 32 – (10 x 3) + 10 = 43

(b)φ(5) = (2 x 53) + 52 – (10 x 5) + 10 = 235

Example 1 Solution

(b) (i) For φ(t2) we replace the t with t2 in φ(t)= 2t3 +

t2 - 10t + 10:

φ(t2) = 2(t2)3 + (t2)2 – 10(t2) + 10

= 2t6 + t4 – 10t2 + 10

(ii) For φ(t + 1) we replace t with t+1 in φ(t)= 2t3 + t2 - 10t + 10:

φ(t+1)= 2(t+1)3 + (t+1)2 – 10(t+1) + 10:

What is a function?

1. A rule which translates an input, usually to a single output.

2. What are the functions for:i. Double the input

ii. Shift the input by 3

iii. Cube the input and subtract 1.

3. Write down in words the functions for

2)1(;4;25 xyxyxy

What variables can a function have?

What is the difference between the functions f(x), g(w), h(y) and k(x)

A function describes a relationship, the variable names are unimportant.

Engineers typically use variable names that relate to the topic: W for weight, h for height, L for length, etc.

22 )2()()2()(

)sin()()sin()(

xxkwyyhx

wwgzxxfy

What is a function argument?

The part that appears in the brackets;

• For y=f(x), x is the argument.

• For z=g(w), w is the argument.

Thus argument is another word for the input to the function.

Independent and dependent variables: what do you think these are? Use common sense.

Composition of functions

What do the following statements mean?

;3)()));(((

)sin()(;)());((

)sin()(;)());((2

2

xxhzfghw

xxgxxfxfgy

xxgxxfxgfy

Evaluate the following

Find y when x=pi/2.

Find w when z=1.

;3)()));(((

)sin()(;)());(( 2

xxhzfghw

xxgxxfxgfy

Function products

Evaluate A given that:A = y2h with x=2 and z=3

Write down a detailed function expression to express A.

;3)()));(((

)sin()(;)());(( 2

xxhzfghw

xxgxxfxgfy

Example 2 (Page 152 Kuldeep Singh, Example 16)

[Reliability Engineering]The failure density function, f(t), for a component is given by:

f(t) = 1/8 where 0 < t < 8 years.

Find F(t), R(t) and h(t) where these are defined as:F(t) = tf(t) (Failure Distribution function)R(t) = 1-F(t) (Reliability function)h(t) = f(t) / R(t) (hazard Rate function)

and 0 < t < 8 years.

Example 2 (Page 152 Example 16)

SolutionWe have:

F(t) = tf(t) = t(1/8) = t/8.

R(t) = 1-F(t) = 1- t/8

h(t) = f(t) / R(t) = (1/8)/(1-t/8)

= 1/(8-t)

Graphs and sketching

By first producing a suitable table, sketch the graphs of the following functions in the domain -3 to 3.

)tan(

)2)(1(

)2

sin(

2)sin(

xy

xxy

xy

xy

Domain is the

values allowed to the argument or independent variable.

Range is the values the output (dependent variable) can take. What is the range of these?

Example 3 (Page 110 Example 7)

[Fluid Mechanics]

The streamlines of fluid flow are given by:

y = x2 + c

where c is constant.

Sketch the streamlines for c = 0, -1 ,1, -2, 2, -3 and 3.

Example 3 (Page 110 Example 7)

Solution

The graphs of y = x2 + c for c = 0, -1 ,1, -2, 2, -3 and 3 are:

(c=0) y = x2 (c=-1) y = x2 - 1 (c=1) y = x2 + 1(c=-2) y = x2 -2(c=2) y = x2 +2(c=-3) y = x2 – 3(c=3) y = x2 + 3

y

x

-3-2-10

123

c = -3

c = -2c = -1c = 0c = 1c = 2c = 3

-1-2 1 2

Notice how the graph of y=x2 + c varies as c changes. The c is where the curve cuts the y axis.

Inverse functionmean that all we’ve done is made a switch in

emphasis

Inverse functionmean that all we’ve done is made a switch in

emphasis

7 – 4 = 3 3 + 4 = 7

Inverse functionmean that all we’ve done is made a switch in

emphasis

7 – 4 = 3 3 + 4 = 7

Both of this statements say the same thing, but with a change in emphasis

Inverse functionmean that all we’ve done is made a switch in

emphasis

7 – 4 = 3 3 + 4 = 7

Both of this statements say the same thing, but with a change in emphasis

Inverse functionmean that all we’ve done is made a switch in

emphasis

7 – 4 = 3 3 + 4 = 7

Both of this statements say the same thing, but with a change in emphasis

-1

Inverse functionexample

y=2x-7; y=f(x)=2x-7

Inverse functionexample

y=2x-7; y=f(x)=2x-7

Identity function

Inverse functionexample

y=2x-7; y=f(x)=2x-7

Identity function

If and inverse function

Inverse functionexample

y=2x-7; y=f(x)=2x-7

Identity function

If and inverse function

Composition of functions

Inverse function

A function f and its inverse f . Because f maps 1 to 4,

the inverse f  maps 4 back to 1.

-1

-1

-1 f f

One to one function

For every value of x, there is a distinct value of y and for every value of y there is a distinct value of x.

Which of the following is one to one?

)sin(

13

242

xy

eyxxy

xy

x

Draw the graph and it should be obvious.

Inverse function

What about?

3

223

)(sin)sin( 1

yxxy

yxxy

)(cos)sin( 12 xxxy

Proof

Inverse functionexample

Inverse functionexample

Sometimes the inverse of a function cannot be expressed by a formula

with a finite number of terms. For example, if f is the function

then f is one-to-one, and therefore possesses an inverse function f . The formula for this inverse has an infinite number of terms:

-1

Inverse functionexamples

link

Many-to-one and one-to-many

Give some examples of many-to-one and one-to-many functions.

The logic goes from independent variable to dependent variable.

Notation

Get into groups and decide three example functions with the following properties [3 for each item].

1. Continuous

2. Discontinuous

3. Periodic (Why are these important?)

4. Odd

5. Even

Odd Even

Summary

Independent variable (domain)Dependent variable (range)FunctionMany-to-one (one-to-one,…)Odd, even, periodicInverse functionContinuous/discontinuousComposite functionStraight lines

Exponential functions

On some rough paper, do a sketch of the following functions.

In what sense are the functions equivalent?

23

2

1

5

3

2

x

x

x

y

y

y

With a suitable rescaling of x, they are all the same shape.

Functions of this form are called exponentials.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

9

y1

y2y3

Exponential properties

If you double the value of the independent variable, you square the value of the dependent variable.

There is a constant ratio which depends solely on the difference of the argument:

3

2

)]([)3(

)]([)2(

xfxf

xfxf

)]([)(

)(xbaf

bxf

axf

For all x!

Exponential properties

Exponential properties

Exponentiation is not commutative

4 + 5 = 5 + 4 4 * 5 = 5 * 4 but 4 = 55 4

256 = 625

Exponential properties

Exponentiation is not commutative

4 + 5 = 5 + 4 4 * 5 = 5 * 4

but

4 = 55 4

256 = 625Exponentiation is not associative

(2 + 3) + 4 = 2 + (3 + 4) (2 * 3) * 4 = 2 * (3 * 4)

but

2 = 40963( )

42 = 2.417.851.639.229.258.349.412.353

3( )4

Exponential convention

1. When dealing with exponential functions it is usual to assume the same base – ALWAYS!

2. The assumed base is `e’.

3. It will become clearer later why `e’ is chosen because this makes a lot of common algebra much simpler.

4. `e’ is irrational, but has a value near 2.7

Exponential convention

1. When dealing with exponential functions it is usual to assume the same base – ALWAYS!

2. The assumed base is `e’.

3. It will become clearer later why `e’ is chosen because this makes a lot of common algebra much simpler.

4. `e’ is irrational, but has a value near 2.7and more precisely

Common exponential

The most common functions you will deal with are:

A positive exponent gives an increasing function with increasing argument.

A negative exponent gives a decreasing argument with exponent.

bt

at

x

x

ety

ety

xey

xey

)(

)(

)exp(

)exp(

-1 -0.5 0 0.5 1 1.5 2 2.5 30.5

1

1.5

2

exp(0.2t)exp(-0.2t)

Exponentials and systems engineering

The behaviour of real systems is often described as an exponential decay.

1. Radioactivity follows a curve of the form.

2. An explosion might be an increasing function.

3. Many systems have dynamics with 2 exponentials.

ctehth )0()( btat BeAetz )(

atertr )0()(

If behaviour has a positive exponent – BEWARE!

Note

Logarithms and exponentials are inverse functions of one another.

1log

)(log3

log)(log)()exp()(

log

log

3

1

e

AeorAe

zwez

xxfxxf

yxey

e

AAe

ew

e

ex

e

Even/odd and hyperbolic Functions

Even functions

An even function is one whereby the vertical axis is equivalent to a mirror.

In mathematical terms, this means that

f(-x)=f(x)

Examples of even functions

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x2-2cos(x)

sin(x)2

Notice symmetry about x=0

Odd functions

An odd function is one whereby the vertical axis reverses the value of the function.

In mathematical terms, this means that

f(-x)=-f(x)

Examples of odd functionsNotice asymmetry about x=0

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-6

-4

-2

0

2

4

6

x3-2xsin(x)tan(x)

Constructing even and odd functions

Every function can be made up of even and odd functions. This can make some engineering problems easier to handle.

EVEN FUNCTIONS

f(x)=f(-x)

ODD FUNCTIONS

f(x)=-f(-x)

)]()([2

1)]()([

2

1)(

)]()([2

1)]()([

2

1)(

xfxfxfxfxf

xfxfxfxfxf

Constructing even and odd functions

EVEN ODD

EVEN FUNCTIONS

f(x)=f(-x)

ODD FUNCTIONS

f(x)=-f(-x)

)()]()([2

1)(

)()]()([2

1)(

)]()([2

1)]()([

2

1)(

xhxfxfxh

xgxfxfxg

xfxfxfxfxf

Construct even and odd functions to make up the following

wewwwg

zzh

xxxf

2tan)(

)3

sin()(

)cos(3)(

2

Construct even and odd functions to make up the following

22tan

22tan

sin3

coscos3

sin)3

sin(

3)cos()cos(3

22wwww

w eeww

eeeww

zzz

xxxx

)]()([2

1

)]()([2

1

xfxfODD

xfxfEVEN

USE

Simple rules

• EVEN*EVEN = EVEN

• ODD*ODD = EVEN

• EVEN*ODD=ODD

• EVEN+EVEN=EVEN

• ODD+ODD=ODD

• ODD+EVEN=NEITHER ODD NOR EVEN

Can you prove these?

Common even/odd functions

EVEN

cos

x2n

cosh

ODD

sin

x2n+1

tan

sinh

Today we focus on cosh and sinh

2sinh;

2cosh

xxxx eex

eex

Plots of cosh and sinh

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-4

-3

-2

-1

0

1

2

3

4

cosh(x)sinh(x)

2sinh;

2cosh

xxxx eex

eex

Engineering examples of cosh and sinh

• Some examples taken from the following book:– “Engineering Mathematics through

Applications”• Kuldeep Singh

– Published by: Palgrave MacMillan– ISBN 0-333-92224-7

Example 1 (Page 251 Example 21)

[Electrical Principles]

A transmission line of length L has voltage V. At a distance x from the sending end, the voltage is given by:

VL = ½(V + IZ0)e-φx + ½(V - IZ0)eφx (*)

Where I is the current, Z0 is the characteristic impedance and φ is the propagation coefficient.

Show that at x = L:

VL = Vcosh(φL) - IZ0sinh(φL)

VL = ½(V + IZ0)e-φx + ½(V - IZ0)eφx (*)

Solution:

Putting x = L into (*) gives:

VL = ½(V + IZ0)e-φL + ½(V - IZ0)eφL

= V(e-φL + eφL)/2 + IZ0(e-φL - eφL)/2

= Vcosh(φL) + IZ0(e-φL - eφL)/2

so VL = Vcosh(φL) + IZ0sinh(φL)

Example 2 (Page 251 Example 22)

[Electronics]

In a semiconductor, a force, F, exerted on an electron is given by:

F = Qcke-kx/(1+e-kx)2 (*)

Where c and k are constants, x is the distance from the pn junction and Q is the charge.

Show that

F = Qck/2[1+cosh(kx)]

Solution:

F = Qcke-kx/(1+e-kx)2 (*)

so F = Qcke-kx/(1 + 2e-kx + e-2kx) = Qcke-kx / e-kx (ekx + 2 + e-kx) = Qck /(ekx + 2 + e-kx) = Qck /(2 + ekx + e-kx) = Qck /(2 + 2(ekx + e-kx)/2) = Qck /(2 + 2cosh(kx))

hence F = Qck/2[1+cosh(kx)]

Show that F = Qck/2[1+cosh(kx)]

Solution to ODES

Where an ODE takes the form

The solution can be represented in two similar ways.

022

2

xadt

xd

atDatCxORBeAex atat sinhcosh;

Identities

You should be familiar with common identities using cosh and sinh.

Prove the following:

xxxxxxx

xyyxyxyxyxyx

xx

coshsinh22sinh1cosh2sinhcosh2cosh

coshsinhcoshsinh)sinh(sinhsinhcoshcosh)cosh(

1sinhcosh

222

22

Use of hyperbolic equations with parametric descriptions

Some simple curves lend themselves to parametric descriptions. Consider:

1. Circle

2. Ellipse

3. Hyperbola 222

22222

222

ryx

rybxa

ryx

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

222 1 yx

22

2

2

2

16.02

yx

222 1 yx

Parametric descriptions

sin,cos1222 yxyx

sin6.0,cos216.02

22

2

2

2

yxyx

tytxyx sinh,cosh1222

ENGINEERING APPLICATION: Space orbits can be either elliptical or hyperbolic (often called a sling shot).

the sine and cosine functions give a parametric equation for the ellipse

the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola

Questions

1. Simplify the following expressions.

2. Find parametric expressions for x,y satisfying the following hyperbola.

)sinh()cosh(

)2sinh2(cosh2cosh3

2cosh4sinh5.0 2

yxyxh

fxxgx

xxf

2162 22 yx

Link between cosine, sine, cosh and sinh

You may find the following useful.

ixee

xi

ixee

x

xixe

ixix

ixix

ix

sinh2

)sin(

cosh2

)cos(

sincos