S6: Mathematics

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• UNIT1: COMPLEX NUMBERS

  Key unit competence

  Perform operations on complex numbers in different forms and use them to solve
  

  related problems in Physics, Engeneering, etc.

  Introductory activity

  Consider the extension of sets of numbers previously learnt from natural numbers
  to real numbers. It is actually very common for equations to be unsolved in one
  set of numbers but solved in another.
  Let us find the solution of the following quadratic equations in the set of real 
  

  numbers:

  

  1. 1 Algebraic form of Complex numbers and their geometric
  representation
  

  1.1.1 Definition of complex number

  
  

  
  
  
  

  Solution

  Each of these numbers can be put in the form a + ib where a and b are real numbers
  

  as detailed in the following table:

  
  

  Complex numbers are commonly used in electrical engineering, as well as in physics
  as it is developed in the last topic of this unit. To avoid the confusion between i
  representing the current and i for the imaginary unit, physicists prefer to use j to
  

  represent the imaginary unit.

  As an example , the Figure 1.1 below shows a simple current divider made up of a
  

  capacitor and a resistor. Using the formula, the current in the resistor is given by

  

  Figure 1. 1 A generator and the R-C current divider

  

  

  

  Figure 1. 2 Cycles of imaginary unit

  From the figure 1.2, the following relations may be used:

  

  

  1.1.2 Geometric representation of a complex number

  
  

  The complex plane consists of two number lines that intersect in a right angle at
  the point (0,0) . The horizontal number line (known as x − axis in Cartesian plane) is
  

  the real axis while the vertical number line (the y − axis in Cartesian plane) is the

  imaginary axis.
  Every complex number z = a + bi can be represented by a point Z (a,b) in the
  

  complex plane.

  The complex plane is also known as the Argand diagram. The new notation Z (a,b)
  of the complex number z = a + bi is the geometric form of z and the point Z (a,b)
  

  is called the affix of z = a + bi . In the Cartesian plane, (a,b)is the coordinate of the

  

  

  Figure 1. 3 The complex plane containing the complex number z = a + bi

  Complex impedances in series
  In electrical engineering, the treatment of resistors, capacitors, and inductors can
  be unified by introducing imaginary, frequency-dependent resistances for the latter
  two (capacitor and inductor) and combining all three in a single complex number
  called the impedance. If you work much with engineers, or if you plan to become
  one, you’ll get familiar with the RC (Resistor-Capacitor) plane, just as you will with
  the RL (Resistor-Inductor) plane.
  Each component (resistor, an inductor or a capacitor) has an impedance that can be
  represented as a vector in the RX plane. The vectors for resistors are constant
  

  regardless of the frequency.

  
  

  together when coils and capacitors are in series. Thus , X = X_L + X_C .
  In the RX plane, their vectors add, but because these vectors point in exactly
  opposite directions inductive reactance upwards and capacitive reactance
  downwards, the resultant sum vector will also inevitably point either straight up or
  

  down (Fig. 1.4).

  

  Figure 1. 4 Pure inductance and pure capacitance represented by reactance vectors that point straight
  up and down.
  

  Example 1.2

  

  Solution

  
  

  

  1.1.3 Operation on complex numbers

  1.1.3.1 Addition and subtraction in the set of complex numbers

  
  

  
  

  
  

  Adding impedance vectors
  

  If you plan to become an engineer, you will need to practice adding and subtracting
  

  complex numbers. But it is not difficult once you get used to it by doing a few sample
  

  problems. In an alternating current series circuit containing a coil and capacitor,
  

  there is resistance, as well as reactance.
  

  Whenever the resistance in a series circuit is significant, the impedance vectors
  

  no longer point straight up and straight down. Instead, they run off towards the
  

  “northeast” (for the inductive part of the circuit) and “southeast” (for the capacitive
  

  part). This is illustrated in Figure 1.5.

  
  

  Figure 1. 5 Resistance with reactance and impedance vectors pointing “northeast “or “southeast.”
  

  When vectors don’t lie along a single line, you need to use vector addition to be
  sure that you get the correct resultant. In Figure 1.6, the geometry of vector addition
  

  is shown by constructing a parallelogram, using the two vectors Z₁ = R₁ + jX₁ and

  Z₂=R₂+jX₂ as two of the sides. Then, the diagonal is the resultant

  
  

  Figure 1. 6 Vector addition of impedances  Z₁ = R₁ + jX₁ and Z₂ = R₂ + jX₂
  

  Formula for complex impedances in series RLC circuits
  

  
  

  
  

  Figure 1. 7 A series RLC circuit
  

  Example 1.4
  A resistor, coil, and capacitor are connected in series with  ,  R = 45Ω X_L= 22Ω and
  

   X_C = − 30Ω. What is the net impedance, Z ?

  Solution
  

  Consider the resistor to be part of the coil (inductor), obtaining two complex vectors,

  45 + 22 j and0 − 30 j. Adding these gives the resistance component of

  (45 + 0)Ω = 45Ω, and the reactive component of (22 j − 30 j)Ω = −8 jΩ. Therefore

  the net impedance is Z = (45 −8 j)Ω .

  Application activities 1.3

  Represent graphically the following complex numbers, and then deduce the

  numerical answers from the diagrams.

  

  1.1.3.2 Conjugate of a complex number

  Activity 1.5

  In the complex plane,
  1. Plot the affix of complex number z = 2 + 5i
  2. Find the image P'of the point P affix of z by the reflection across the real
       axis. What is the coordinate of P' ?
  3. Write the complex number z ' associated to P' and discuss the relationship
  between z and z ' of P' ?
  4. Write algebraically the complex number z ' associated to P' and discuss the
  

       relationship between z and z '

  Every complex number z a bi = + has a corresponding complex number z − called

  

  

  Figure 1. 8 Reflection of affix about the real axis

  

  

  1.1.3.3 Multiplication and powers of complex number

  
  

  
  

  
  

  

  
  

  
  

  1.1.4 Modulus of a complex number
  

  
  

  
  

  
  

  Figure 1. 9 Modulus of z = a + bi
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  Activity 1.11

  Given the quadratic equation z² − (1+ i) = 0 , you can write it as z² =1+ i . Calculate
  the square root of 1+ i to get the value of z and discuss how to solve equations of
  the form Az² +C = 0 where A and C are complex numbers and A is different
  

  from zero. Express in words the formula used.

  Solving simple quadratic equations in the set of complex numbers recalls the
  

  procedure of how to solve the quadratic equations in the set of real numbers

  

  

  

  

  

  

  

  From affix of a complex number z = a + bi , there is a connection between its
  modulus and angle between the corresponding vector and positive x − axis as
  illustrated in figure 1.10. This angle is called the argument of z and denoted by
  

  arg ( z) .

  

  

  

  

  

  

  

  

  

  

  

  

  

  1.3 Exponential form of complex numbers

  1.3.1 Definition of exponential form of a complex number

  
  

  
  

  In electrical engineering, the treatment of resistors, capacitors and inductors can be
  unified by introducing imaginary, frequency-dependent resistances for capacitor,
  inductors and combining all three (resistors, capacitors, and inductors) in a single
  complex number called the impedance. This approach is called phasor calculus. As
  we have seen, the imaginary unit is denoted by j to avoid confusion with i which
  is generally in use to denote electric current. Since the voltage in an AC circuit is
  

  oscillating, it can be represented as

  
  

  Example 1.15

  a) Express the complex numbers in exponential form

  
  

  Given that the same current must flow in each element, the resistor and capacitor
  are in series such that the common current can often be taken to have the reference
  

  phase.

  
  

  
  

  
  

  
  

  
  

  
  

  1.3.3 Application of complex numbers in Physics
  

  Activity 1.17

  Conduct a research from different books of the library or browse internet to
  discover the application of complex numbers in other subjects such as physics,
  

  applied mathematics, engineering, etc

  Complex numbers are applied in other subjects to express certain variables or
  facilitate the calculation in complicated expressions. They are mostly used in electrical
  engineering, electronics engineering, signal analysis, quantum mechanics, relativity,
  applied mathematics, fluid dynamics, electromagnetism, civil and mechanical
  engineering. Let look at an example from civil and mechanical engineering.
  An alternating current is a current created by rotating a coil of wire through a
  

  magnetic field.

  

  Figure1.11: generation of alternating current
  

  (Source: https://www.google.com/imgres?imgurl=https://image.pbs.org/poster_images)

  
  

  
  

  
  

• UNIT2: LOGARITHMIC AND EXPONENTIAL FUNCTIONS

  UNIT2: LOGARITHMIC AND
  

  EXPONENTIAL FUNCTIONS

  UNIT2: LOGARITHMIC AND EXPONENTIAL FUNCTIONS

  Key unit competence
  

  Extend the concepts of functions to investigate fully logarithmic and exponential
  

  functions and use them to model and solve problems about interest rates, population
  

  growth or decay, magnitude of earthquake, etc.

  Introductory activity
  The Accountant for a Health Center receives money from patients in an interesting
  way so that the money he/she earns each day doubles what he/she earned the
  previous day. If he/she had 200USD on the first day and by taking t as the number
  of days, discuss the money he/she can have at the t^th day through answering the
  following questions:
  a) Draw the table showing the money this Health Center Owner will have on
      each day starting from the first to the 10^th day.
  b) Plot these data in rectangular coordinates
  c) Based on the results in a), establish the formula for the Health Center
  Owner to find out the money he/she can earn on the nth day. Therefore, if t
     is the time in days, express the money F (t ) for the economist.
  d) Now the Health Center Owner wants to possess the money F under the
     same conditions, discuss how he/she can know the number of days
  

     necessary to get such money from the beginning of the business.

  From the discussion, the function F (t ) found in c) and the function Y (F ) found in
  d) are respectively exponential function and logarithmic functions that are needed
  to be developed to be used without problems. In this unit, we are going to study the
  behaviour and properties of such essential functions and their application in real life
  

  situation.

  2. 1 Logarithmic functions
  

  2.1.1 Domain of definition for logarithmic function

  
  

  

  If the base is 10, it is not necessary to write the base, and we say decimal logarithm
  or common logarithm or Brigg’s logarithm. So, the notation will become y = log x . If
  the base is e (where e =2.718281828…), we have Neperean logarithm or natural
  

  logarithm denoted by y = ln x instead of log_e y = x as we might expect.

  
  

  
  

  
  

  
  

  2.1.2 Limits and asymptotes of logarithmic functions
  

  Activity 2.2
  

  
  

  Consider the form of this graph then by using calculator, complete the table below to answer the
  

  questions that follow.

  
  

  
  

  
  

  
  

  
  

  2.1.3 Continuity and asymptote of logarithmic functions
  

  
  

  
  

  • The logarithmic function is increasing and takes its values (range) from
  

      negative infinity to positive infinity.

  Example 2.3

  Let us consider the logarithmic function y = log2 (x − 3)

  a) What is the equation of the asymptote line?

  b) Determine the domain and range

  c) Find the x − intercept.

  d) Determine other points through which the graph passes

  e) Sketch the graph

  Solution
  a) The basic graph of 2 y = log x has been translated 3 units to the right, so the line L ≡ x = 3
  is the vertical asymptote.
  b) The function  y = log₂ (x − 3) is defined for x − 3 > 0
  So, the domain is ]3,+∞[ .The range is 
  c) The intercept is (4, 0) since log₂  (x − 3) = 0⇔ x = 4
  d) Another point through which the graph passes can be found by allocating an arbitrary value
  to x in the domain then compute y.
  For example, when   x = 5, y = log₂ (5 − 3) = log₂ 2 =1 which gives the point (5,1) .
  Note that the graph does not intercept y-axis because the value 0 for x does not belong to
  

  the domain of the function.

  
  

  
  

  2.1.4. Differentiation of logarithmic functions
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  2. 3 Applications of logarithmic and exponential functions
  

  Logarithmic and exponential functions are very essential in pure sciences, social
  

  sciences and real life situations. They are used by bank officers to deal with interests
  

  on loans they provide to clients. Economists and demographists use such functions
  

  to estimate the number of population after a certain period and many researchers
  

  use them to model certain natural phenomena. We are going to develop some of
  

  these applications.

  
  

  
  

  
  

  
  

      When a person gets a loan (mortgage) from the bank, the mortgage amount M, the
      number of payments or the number t of years to cover the mortgage, the amount
      of the payment P, how often the payment is made or the number n of payments per
  

      year, and the interest rate r, it is proved that all the 5 components are related by

  

  Example 2.12
  A business woman wants to apply for a mortgage of 75,000 US dollars with an
  interest of 8% per month that runs for 20 years. How much interest will she pay over
  

  the 20 years?

  

  

  

  

  

  
  
  
  b) At the beginning, (t = 0), the number of fish is 500. After 4 months (t = 4),
  the number of fish is 4,1048.
  c) As the time x increases, the number of fish will be 10,000.
  d) The population is increasing most rapidly after 4 months. This is because
  

        the increment of fish after 1 month is greater.

  

  

  b) After 5 days, the calculator and the graph show that 54 students will be
        infected.
  c) According to this model, when the time increases without bound, the graph
       shows that all students can be infected. However, in real life, the infinite
       time is not possible. Therefore, all students cannot be infected.
  
  
  
  
  
  
  

  

  2.3.5 Earthquake problems

  Activity 2.15
  

  Do the research in the library or explore internet to find out how Charles Richter
  

  tried to compare the magnitude of two earthquakes by the use of logarithmic
  

  function.

  Seismographic readings are made at a distance of 100 kilometers from the epicenter
  of an earthquake. If there is no earthquake, the seismographic reading is x₀ = 0.001
  

  millimeter.

  

  

  

  

  

  Example 2.17
  A scientist determines that a sample of petrified wood has a carbon-14 decay rate of
  8.00 counts per minute per gram. What is the age of the piece of wood in years? The
  decay rate of carbon-14 in fresh wood today is 13.6 counts per minute per gram, and
  

  the half- life of carbon-14 is 5730 years.

  
  

  
  

  
  

  
  

  
  

  
  

  END UNIT ASSESSMENT
  

  
  

  
  

  
  

  
  

• Unit 3: INTEGRATION

  Unit 3: INTEGRATION
  

  Key unit competence
  

  Use integration as an inverse of differentiation and then apply definite integrals to
  

  find area of plane shapes.

  Introductory activity
  

  Two groups of students were asked to calculate the area of a quadrilateral field
  

  BCDA shown in the following figure:

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  3.4.3 Integration by parts
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  Many times, some functions can not be integrated directly. In that case we have
  to adopt other techniques in finding the integrals. The fundamental theorem in
  calculus tells us that computing definite integral of f(x) requires determining its
  antiderivtive, therefore the techniques used in determining indefinite integrals are

  also used in computing definite integrals.

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

  
  

• Unit 4. ORDNINARY DIFFERENTIAL EQUATIONS

  Unit 4. ORDNINARY DIFFERENTIAL EQUATIONS

  Key unit competence
  Use ordinary differential equations of first and second order to model and solve
  

  related problems in Physics, Economics, Chemistry, Biology, Demography, etc.

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  4.4.5 Differential equations in electricity (Series Circuits)
  

  Activity 4.8

  Let a series circuit contain only a resistor and an inductor as shown in Figure 4.3

  
  

  
  

  
  

  

  

  

  

  

  

  

  4.5. Introduction to second order linear homogeneous ordinary
  

  differential equations