Overview of MATH151 lectures

On this page I try to summarise the main ideas of each lecture.

  1. Week 1.
  2. Week 2.
  3. Week 3.
  4. Week 4.
  5. Week 5.
  6. Week 6.
  7. Week 7.
  8. Week 8.
  9. Week 9.
  10. Week 10.
  11. Week 11.
  12. Week 12.
  13. Week 13.

Week 1, Lecture 1 (Subject outline).
You must read through the subject outline and the generic subject outline. The university (and I) will assume that you are familiar with the content of these documents.
Week 1, Lecture 2 (Chapter 0 - arithmetic of fractions).
Additional resources for Arithmetic of fractions.
You should understand how to add and subtract fractions.
2/4 + 3/5
2/7 - 5/14
Understanding the ideas of this process are important because (latter) we will apply them to simplify expressions such as
1/(x-3) + x/(x-2)
x/(x^2+5) -1/(x-3).
I mentioned in the lecture ``irrational numbers''. Here is a youtube video about the square root of 3, which is an irrational number. (Video link supplied by Lachlan Mitchell).
Week 1, Lectures 3-4 (Chapter 1 : indices and surds).
Additional resources on Indices, Surds, and Algebraic Fractions.
  1. You need to know the rules for manipulation of indices and how to use them.
    1. Remember that in a-b the minus sign means `1 divided by' so that this expression is the same as 1/ab. (Useful Idea)
    2. In questions such as simplify 81/3 and 841/2 the key is to first factorise the number. In particular, if you are trying to simplify the square root of a number you want to factorise using numbers such as 4, 9, 16...
    3. Is is often useful to write √x as x0.5, i.e., to use fractional indices. (Useful Idea).
    4. Evaluate (2m)0 +5.
    5. True or false? (Justify your answer). (a2 +b2)1/2 = a +b.
    6. Sometimes there is more than one correct way to write an answer. Don't assume that your answer is wrong if it is different from the answer given in a textbook.
  2. (a-b)(a+b) = a2-b2. (This is called the difference of two squares).
  3. Mathematicians don't like to see expressions which have square root signs on the denominator of a fraction! To eliminate the square root sign we have to multiply both sides of the fraction by the conjugate. (This reason why this is works is because we are using the difference of two squares).
    1. Remove the surd from the denominator of the expression 2/(√5 +√3).
You are now able to do the practice questions at the end of chapter 1 for the topics indices and surds. You should be doing these questions as part of your weekly workload for this subject!

Week 2, Lectures 1-2 (Chapter 1 : algebraic fractions, expansions, binomial expansion and factorisation).
Additional resources on Algebraic Fractions.
  1. True or False? (Justify your answer). x = qy^2/(q-y) = y^2/(1-y).
Additional resources on Expansion of algebraic factors.
  1. Expand (-x-1)(x-3).
Additional resources on Pascal's triangle.
  1. Write down the first four lines of Pascal's triangle and explain how to use it.
  2. Expand (a+b)3.
  3. Expand (a-b)4.
  4. True or false? (Justify your answer). (a+b)c = ac +bc.
Additional resources on Factorisation.
  1. Factorise x2 +2x -15.
  2. Factorise (c+1) -a(c+1).
  3. Factorise:
    1. a2 -b2.
    2. a2 +b2.
    3. a3 -b3.
    4. a3 +b3.

Week 2, Lecture 3 (Chapter 2 - Logarithms).
Additional resources for Logarithms

Science Question. I started the lecture with a science question.
The concentration of H+ ions in solution is related to the pH of the solution by the formula
pH = -log10 H+.
If the pH of an aqueous solution is 3.12 what is the corresponding concentration of H+ ions?

  1. x = logb N means that bx = N.
  2. We cannot find logarithms of negative numbers or zero!
  3. To solve questions such as log3(1/9) they key is to factorise the number. In this case (1/9) = 3-2 so that log3(1/9) = -2.
  4. You need to know the four rules for manipulation of logarithms and how to use them. You will frequently use rules (i)-(iii).
    1. logb xy = logbby.
    2. logb x/y = logbby.
    3. logb xp = p*logbx.
    Rule (iv) (the change of base rule) will only be infrequently used.
  5. The following special cases are useful to know:
    1. logbb = 1.
    2. logb1 = 0.
    3. logbbp = p.
    4. logb(1/x) = -logbx.
    5. logbn√x = (1/n)*logbx.
Week 2, Lecture 4 (Practical Class).
You worked through practice questions on Indices, Surds and Algebraic Fractions. In your own time, you need to finish the following exercises from chapter one of the MATH151 book.
  1. Problem Set A on page 17.
  2. Problem Set B on page 21.
  3. Problem Set C on page 23.
  4. Multiple-Choice Questions in section 1.10.1 on page 26.
  5. Standard questions in section 1.10.2 on page 27.
  6. Practice questions in section 1.10.3 on page 29.

You should also

  1. Complete the week one "assignment". (Worked solutions are available on the e-learning site).
  2. Complete the week two tutorial sheet. (Worked solutions will be available on the e-learning site).

Anyone who finishes all these questions will have a very strong foundation in the basic mathematics that will be used extensively in the rest of the course.

Anyone who does none of these questions will be heading for a fail on the course.

Week 3, Lecture 1 (Chapter 2 - logarithms).
Additional resources for Logarithms
Science Question. I started the lecture with a science question.
The concentration of H+ ions in solution is related to the pH of the solution by the formula
pH = -log10 H+.
If the pH of an aqueous solution is 3.12 what is the corresponding concentration of H+ ions?

  1. We reviewed the key properties about logarithms that are used in solving equations.
    1. If logby = x then y = bx.
    2. If x = y then logbx = logb y.
    3. If logbx = logb y then x = y.
  2. I reminded you that on the e-learning site you can generate a multiple-choice quiz on logarithms. (You can do this for many of the topics taught in this subject).
  3. We finished the lecture by using properties of logarithms to solve the science problem that was stated at the start of the lecture.
    [H+] = 10-3.12 mol/l = 0.00076 mol/l.
Week 3, Lecture 2 (Chapter 3 - Function Notation).
Additional resources on Functions.
  1. Think of a function as a rule for converting an input into an output. Celsius/Fahrenheit example.
  2. Composing functions. What does the notation f(g(x)) mean?
Week 3, Lecture 3 (Chapter 3 - Function Notation).
Additional resources on Functions.
Science Question. I started the lecture with the following science question.

The concentration of H+ ions in solution is related to the pH of the solution by the formula
pH = -log10 H+.
A solution of trifluoroethanoic acid has a concentration of 10-4 mol dm-3. What is the pH of the acid?

  1. Application of function composition to converting scales.
  2. Finding zeros or roots of equations
  3. Definition of linear and quadratic functions.
  4. Roots of linear and quadratic functions - the quadratic formula.
  5. Functions are not necessarily defined for all inputs. "Bad" values for functions - reciprocals and square roots.
  6. Vertical line test for the graph of a function.
Week 3, Lecture 4 (Practical Class).
Additional resources on Logarithms and Functions.
Science Question. I started the practical class with a science question.

The Ideal Gas Law can be written
PV = nRT.
In this formula P is the pressure of the gas, R is the ideal gas constant, T is the temperature of the gas (in Kelvin), V is the volume of the gas, n is the number of moles of gas,

In a particular experiment the values for T and n are kept constant and the volume is doubled. How does the pressure change?

You worked through practice questions on logarithms. If you finish these then you worked through practice questions on functions. In your own time, you need to finish the following exercises from chapter one of the MATH151 book.
  1. Multiple Choice Question - Logarithms - in section 2.8.1 on page 41.
  2. Practice Questions - Logarithms - in section 2.8.2 on page 41.
  3. Standard Questions - Logarithms - in section 2.8.3 on page 43.
  4. Practice Questions - Functions - in section 3.10.1 on page 56.
You should also
  1. Complete the week three tutorial sheet. (Worked solutions will be available on the e-learning site.

Week 4, Lecture 1 (Chapter 3 - Function Notation).
Additional resources on Functions.
I started the lecture by going over some basic ideas from this chapter: We then finished going through:
  1. The vertical line test - when is a graph a function or not a function?
  2. Two examples of graphs that are not functions are those give by:
    1. x=3.
    2. x2 +y2 = 1.
Science Question. We finished the lecture with a science question on the dissociation of an acid. The mathematical questions boil down to the following.
  1. Rearrange the equation
    K = x2/(1-x)
    to obtain a quadratic equation for x.
  2. What is the value of x if K=0.6? Hint. We are only interested in the positive value.
Week 4, Lecture 2.
We start this lecture by going through some practice questions that are not in the notes
  1. If 2x = 2.5 and 2y=3, then 2x+y is equal to
    1. 6.5.
    2. 7.5
      3. 5.5
    3. 1/5
    4. 25.5
  2. Simplify 2/(1-x) -1/(1+x).
    1. (1+3x)/(1-x2).
    2. (1+x)/(1-x2).
    3. (1+2x)/(1-x2).
    4. (3+3x)/(1-x2).
    5. 1/(1-x).
We went through the solutions to some of the questions on the practice version of the first class test. The practice test paper and written solutions are available on the e-learning page. Look in the folder
Home Page > Lecture Material > Assessment Tests in week 4 > In Class Tests
Week 4, Lecture 3 (Chapter 4 Mostly About Straight Lines).
Additional resources on Mostly About Straight Lines.
I started the lecture by going through some of the aims of the chapter. These include
  1. The equation of a straight line is
    y = mx +b
    1. What does the symbol m mean?
    2. Sketch an example of a straight line corresponding to:
      1. m<0
      2. m>0
      3. m=0
    3. What does the symbol b mean?
    4. Sketch an example of a straight line corresponding to:
      1. b<0
      2. b>0
      3. b=0
  2. Let y=m1x + b1 and y=m2x + b2 be the equations of two straight lines.
    1. What do you know if the two lines are parallel?
    2. What do you know if the two lines are perpendicular?
  3. Let y=m1x + b1 and y=m2x + b2 be the equations of two straight lines.
    1. What does it mean to ``solve'' these two equations?
    2. What does it mean geometrically to ``solve'' these two equations?
    3. Under what conditions will there be:
      1. A single solution to the pair of equations?
      2. An infinite number of solutions to the pair of equations?
      3. No solutions to their pair of equations?
  4. Consider an equation of the form Ax +By +C.
    1. Show that this equation can be transformed into the equation of a straight line (y = mx + b).
    2. Sketch the equation when A=B=0. Is the resulting graph a function?
    3. Sketch the equation when A=0. Is the resulting graph a function?
  5. Given the points P1 = (x1,y1) and P2 = (x2,y2) what is the gradient of the straight line connecting them?
  6. We learnt how to find the equation of a line given two pieces of information. Typical questions are:
    1. Find the equation of a straight line with a specified gradient going through a designated point.
    2. Find the equation of the straight line connecting two points.
    3. Find the equation of the straight line that passes through a designated point and is parallel to a specified line.
  7. A useful idea is to try to sketch the straight line from the specified information before you find its equation. This is sometimes a useful way to discover if your equation is incorrect (provided that your sketch was correct!).
Some practice questions. In each case, rearrange to make x the subject.
  1. ln(x) = 7.
  2. ln(x)2 = y.
  3. ln(xt) = y.
  4. ex = 4.
  5. e-6x = h.
  6. exp(x2) = y.
Week 4, Lecture 4 (Class Test 1).
You did the first class test.
In your own time, you should do the following.
  1. Mark sure that you have completed the Tutorial Sheet for week 4. (solutions will go onto the e-learning page).
  2. In 2009 approximately 48% of students who scored 10.0 or lower on the first test failed the course. (In 2010 the corresponding figure was about 55%). If you scored 10.0 or lower on the first test you are therefore at a high risk of failing this course unless you start doing some serious work.

Week 5
Week 5 Lecture 1 (Chapter 4 - Mostly About Straight Lines).
Additional resources on Mostly About Straight Lines.
    1. Find the equation of the straight line that passes through a designated point and is perpendicular to a specified line.
  2. We discussed the solutions of the system of linear equations
    A1x +B1y +C1 = 0,
    A2x +B2y +C2 = 0.
    The important points are that this system has:
    1. no solution, when the two lines are parallel but distinct
    2. one solution, when the two lines are not parallel
    3. an infinite number of solutions, when the two lines are identical.
  3. There are two methods to solve a system of linear equations: elimination and substitution. Which method is best? It depends upon the particular problem that you are looking at.
  4. The distance between two points P1(x1,y1) and P2(x2,y2) is given by
    DP1P2 = ( (x2-x1)2 +(y2-y1)2)1/2
Science Question. We finished the lecture with a science question on the dissociation of an acid. The mathematical questions boil down to the following.
  1. Rearrange the equation
    K = x2/(1-x)
    to obtain a quadratic equation for x.
  2. What is the value of x if K=0.6? Hint. We are only interested in the positive value.
We went through this question in week four and a similar question was on the first assignment... but most of you can not do it!
Week 5 Lecture 2 (Chapter 5 - Trigonometry).
I started the lecture by giving an overview of some of the aims of this chapter. One of the more interesting aims is being able to describe periodic experimental data using the formula
y = a*cos[w*(t-t0)]
where a is the amplitude of the oscillations, w is the frequency of the oscillations and t0 is the time to the first maximum.
  1. Pythagoras' Theorem.
  2. Definition of the trigonometric ratios: sin, cos and tan.
  3. Right-angled triangles with 30, 45, 60 degree angles.
  4. Table of `standard' values of the basic trigonometric ratios.
Week 5 Lecture 3.
  1. Radian measure of angle, converting degrees to radians and radians to degrees.
    2π radians = 360o
  2. Using the circle of radius one: a counterclockwise direction is a positive angle whereas a clockwise direction is a negative direction.
  3. In the Cartesian Plane where are: Quadrant I, Quadrant II, Quadrant III, Quadrant IV, the x-axis and the y-axis?
Week 5 Lecture 4.
You worked your way through the questions at the end of chapter 5. In addition to doing these questions I gave you the following science question.
An analytical chemist wants to prepare a calibration graph, relating the amount of the natural pigment β-Carotene with its optical absorbance when in solution. The analyst dissolves 0.01 g of β-Carotene and obtains an optical absorbance of 0.8, then weighs a mass of 0.03 g and obtains a higher absorbance of 2.0.

What is the relationship between the absorbance (the observed variable, y) and the mass of β-Carotene (the controlled variable, x)?

This week your first assignment was returned to you.
  1. Use the written solutions to mark the questions on the first assignment that were not marked by your tutor.
  2. Look at the written solution for any questions on the first assignment that you got wrong, both those marked by your tutor and those marked by you.

Week 6 Lecture 1 - (Chapter 5 Trigonometry).
Let t be an angle measured in radiants. To find the trigonometric ratio of sin(t), cos(t) or tan(t):
  1. Locate the quadrant that the angle t is in. Predict the algebraic sign of the answer using the `jingle': All Stations To Central.
  2. Find the related angle θ.
    Quadrant Formula
    I &theta = t
    II &theta = π t
    III &theta = t-π
    IV &theta = 2π -t
    Note. It is always true that: 0<= θ <= Π/2.
  3. Determine answer using steps (1) and (2). (You will almost certainly need to use the table of exact values).
Week 6 Lecture 2 (Chapter 5 - Trigonometry).
  1. Basic Facts about Trigonmetric Functions
    1. sin (-t) = - sin (t), cos (-t) = cos (t), tan (-t) = -tan (t).
    2. The periods of cos(x) and sin(x) are both 2π.
    3. The period of tan(x) is π.
  2. Solving Trigonometric Equations.
    1. How many solutions does the equation sin(t) = 1/2 have?
    2. Which quadrant should I expect a solution in?
    3. Solve cos(x) = 1/2 where 0<= x<= 2π.
    4. Solve cos(x) = -1/2 where 0<= x<= 2π.
Winner of the Dalek video competition (Michelle Diab): http://www.youtube.com/watch?v=w-d6vWMseVo.
Week 6 Lecture 3 (Chapter 5 - Trigonometry)
Graphs of Trig (sin and cos) Functions
  1. The functions sin(x) and cos(x) have period (P) P=2π.
  2. The functions sin(x) and cos(x) have amplitude 1.
  3. The amplitude (a) of a periodic function is defined by
    a = (1/2)*(ymax-ymin)
    where ymax and ymin are the maximum and minimim of the function.
  4. The periodicity (P) of the functions cos(ωx) and sin(ωx) are given by
    P = 2π/ω.
Revision Questions on functions.
For what values of x are the following functions not defined?
  1. ln(x).
  2. (x-2)0.5.
  3. 1/(3x+2).
  4. ln (x-5).
Revision Questions on Straight Lines.
Deduce the equations of the straight lines connecting the following pairs of points.
  1. (1,2) and (2,4).
  2. (0,-2) and (3,11).
  3. (9,12) and (28,50).
  4. (7,9) and (6,10).
  5. (1,2) and (-3,-4).
  6. (-3,-3) and (-2,6).
Week 6 Lecture 4 (Practical Class)
You worked through
  1. 5.8 Practical Class - Evaluating Trigonometric Functions.

Week 7 Lecture 1 (Chapter 5 - Trigonometry)
Lecture material.
  1. Graphs of Trig (tan) Functions
    1. The function tan(x) has period (P) P=π. Note the period!
    2. The function tan(x) does not have amplitude.
    3. The periodicity (P) of the function tan(ωx) is given by
      P = π/ω.
  2. The amplitude of a trig function a*cos(b*x) or a*sin(b*x) is a.
  3. The period of a trig function a*cos(b*x) or a*sin(b*x) is 2*π/b.
  4. We learnt how to plot the function
    y = a*cos(b*x).
    1. Determine the amplitude of the function.
    2. Determine the period (P) of the function.
    3. Determine the critical values of the function. These are the values for y when:
      1. x=0 (the start of the period).
      2. x=P/4 (one-quarter of the period).
      3. x=P/2 (one-half of the period).
      4. x=3P/4 (three-quarters of the period).
      5. x=P (the period).
      Note. If the value for y is not the same when x=0 as it is when x=P then you have made a mistake!
    4. Plot the critical values.
Week 7 Lecture 2 (Chapter 5 - Trigonometry).
We practised sketching graphs of trigonometric functions!
Week 7 Lecture 3 (Chapter 5 - Trigonometry)
I started the lecture with some revision questions.
  1. Write down the exact values of:
    1. cos(11π/6),
    2. tan(2π/3),
    3. sin(5&pi/4).
  2. Write down all the solutions to:
    1. sin(t) = 1/2,
    2. cos(t) = -(2)0.5/2,
    3. tan(t) = -(3)0.5.
We finished chapter 5.
  1. 5.10 Application of Trigonometric Graphs.
    The key idea is to learn how to fit periodic experimental data to a function of the form
    y(t) = a +bcos(ω(t-t0)).
    In this equation:
    • a is the average value of the function over one period. This is given by
      a = (ymax+ymin)/2.
    • b is the amplitude of the function. This is given by
      b = (ymax-ymin)/2.
    • ω is the frequency of the oscillations. This is given by
      ω = 2π/P
      where P is the period of the oscillations.
    • t0 is the time at which the first maximum is observed.
    It's highly probably that there will be an exam question on this topic.
Week 7 Lecture 4 (Practical Class)
I had some extra examples for you to work through
Example One.
What is the period and amplitude of the function y= f(x) = 5cos(x/8)?
Example Two
A certain biological variable y is observed to vary approximately in a sinusoidal manner, oscillating between the values y=1 and y=2 on a 24-hr cycle. It reaches its maximum at 3 o'clock P.M. and its minimum at 3 o'clock A.M. every day. Find a formula for y as a function of t.
Example Three
A variable y varies sinusoidally between the values -1 and +3 with a period of 2 sec. Taking t=0 to coincide with an instant at which y=3, obtain a formula for y as a function of t.
Example Four
The respiratory cycle of a resting human being has a duration of approximately 5 sec. Assuming that the rate of flow y of air into the lungs varies sinusoidally as a function of time with a maximum value of 0.5 liters/sec, express y as a function of t. Take t=0 first to correspond to the instant when the lungs are empty at the end of expiration, and then repeat the question with t=0 corresponding to the instant when the lungs are full at the end of inspiration.

You finished the hour with a practical class on solving trigonometric equations. (5.12 Practice Questions - Trigonometry).

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Page Created: 13th May 2008.
Last Updated: 21st April 2011.