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).

There are resources on the Arithmetic of Fractions on the Summertime Maths webpage.

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).

Week 1, Lectures 3-4 (Chapter 1 : indices, surds and algebraic fractions).

There are resources on Indices, Surds, and Algebraic Fractions on the Summertime Maths webpage.

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/a^b.
   2. In questions such as simplify 8^1/3 and 84^1/2 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. 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).
4. True or False? x = qy^2/(q-y) = y^2/(1-y).

You are now able to do the practice questions at the end of chapter 1 of your notes. The topics covered are: indices, surds and algebraic fractions. You should be doing these questions as part of your weekly workload for this subject!

Week 2, Lectures 1 & 2 (Chapter 2 - logarithms).

There are resources for Logarithms on the Summertime Maths webpage.

1. x = log_b N means that b^x = 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). Rule (iv) (the change of base rule) will only be infrequently used.

Week 2, Lecture 3 (Chapter 2).

We worked our way through the exercises on logarithms in the book. You need to know how to use the log rules!

Week 2, Lecture 4 (Practical Class).

We worked out way through practice questions on indices and logarithms. You need to know how to use the rules for indices and logs! In your own time, you need to finish all the questions in sections 1.5 and 1.6.

Week 3, Lecture 1 & 2 (Chapter 3 - Function Notation).

There are resources for Functions on the Summertime Maths webpage. (Some of the material are this site is more advanced than what is taught in MATH151).

1. Functions as rules, notation, evaluation. Celsius/Fahrenheit example.
2. Composing functions. Application to converting scales.
3. Solving equations, roots.
4. Roots of linear functions, roots of quadratic functions - the quadratic formula.
5. "Bad" values for functions - reciprocals and square roots.
6. Vertical line test for the graph of a function.

Week 3, Lecture 3 (Chapter 3 - Function Notation).

Science Question. I started 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.

There are resources for Functions on the Summertime Maths webpage. (Some of the material are this site is more advanced than what is taught in MATH151).

Functions continued...

1. What is the vertical line test?
2. How is the vertical line test used to decide if the graph y = f(x) represents a function?

Practice makes perfect. I also gave you some practice questions. In each case, rearrange to make x the subject.

1. ln(x) = 7.
2. ln(x²) = y.
3. ln(xt) = y .
4. exp(x) = 3.
5. exp(-6x) = h.
6. exp(x²) = y.

Week 3, Lecture 4 (Practical Class).

We worked our way through practice questions on logarithms. In your own time you need to finish all the questions in chapter 2.

Week 4, Lecture 1 (Practice Test).

You did the practice version of the first class test.

Week 4, Lecture 2.

You went through the solutions to the practice version of the first class test (section A: 1-8).

Week 4, Lecture 3 (Practice Test).

Week 4, Lecture 4 (Class Test 1).

You did the first class test.

Week 5 Lecture 1 - (Chapter 4 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 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?

Week 5 Lecture 2 (Chapter 4 - Mostly About Straight Lines).

1. 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.
   4. Find the equation of the straight line that passes through a designated point and is perpendicular to a specified line.
2. 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!).

Week 5 Lecture 3 (Chapter 4 - Mostly About Straight Lines).

1. We discussed the solutions of the system of linear equations
   A₁x +B₁ +C₁ = 0,
   A₂x +B₂ +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.
2. 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.
3. The distance between two points P₁(x₁,y₁) and P₂(x₂,y₂) is given by
   D_P1P2 = ( (x₂-x₁)² +(y₂-y₁)²)^1/2

Week 5 Lecture 4 (Practical Class).

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 questions.

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 $\beta$-Carotene (the controlled variable, x)?

Week 12, Lecture 1 (differentiation continued).

1. If the average rate of change is negative, then the quantity is decreasing as x increases. If the average rate of change is positive, then the quantity is increasing as x increases.
2. Don't forget that in scientific calculations the average rate of change will have units.
3. To find the instantaneous rate of change we take the limit of the average rate of chance in the limit lim κ→0. (This is the key idea of the chapter).
4. A point where the instantaneous derivative is equal to zero is known as a critical point. Finding the location of critical points is crucially in correctly sketching graphs.

Week 12, Lecture 2 (differentiation continued),

1. Critical points can be of three types:
   1. the slope of the function changes sign from positive to negative (a relative maximum)
   2. the slope of the function changes sign from negative to positive (a relative maximum)
   3. the slope does not change sign
2. The degree of a polynomial is the highest power of x that appears in the polynomial.
3. Differentiation Rule 1: Power: If y = cx^p, where c and p are real numbers then dy/dx = pcx^p-1.
4. Differentiation Rule 2:. If u = f(x) and v = g(x) are two differentiable functions and c is a number, then u+v and cu are differentiable functions. Further,
   d(u+v)/dx = du/dx + dv/dx and d(cu)/dx = cdu/dx.
5. Differentiation Rule 3: The Product Rule. If u = f(x) and v = g(x) are two differentiable functions, then uv is a differentiable function and
   d(uv)/dx = vdu/dx +udv/dx.
6. Differentiation Rule 4: The Quotient Rule. If u = f(x) and v = g(x) are two differentiable functions, then u/v is a differentiable function and
   d(u/v)/dx = (vdu/dx -udv/dx)/v².

Week 12, Lecture 3 (differentiation continued).

1. Differentiation Rule 5: The Chain Rule. If y = f(x) and u = g(x) are two differentiable functions, then y = f(u) is a differentiable function and
   d(y)/dx = dy/du×du/dx.
2. Differentiation Rule 6: Trig, Exponential and Log Functions .
   1. If y = sin(x) then dy/dx = cos(x).
   2. If y = cos(x) then dy/dx = -sin(x).
   3. If y = e^x then dy/dx = e^x.
   4. If y = ln(x) (x>0) then dy/dx = 1/x.
3. Note, by using the chain function you are expected to be able to differentiate functions such as y = sin(x²).

Week 13, lecture 1 (differentiation continued, antidifferentiation)

Differentiation

1. Suppose that you are asked to sketch the graph y = f(x). Finding dy/dx will give you some good `pointers' for the sketch, e.g. identifying regions where the function is increasing (dy/dx>0) and decreasing (dy/dx<0).

Antidifferentiation (or integration)

1. Antidifferentiation is the opposite to integration. The more you understand differentiation, the easier you will find antidifferentiation. Given the function dy/dx = f(x) we are essentially asking "What function y = g(x) did we differentiate to obtain dy/dx = f(x)
2. Suppose that when we differentiate y = g(x) we obtain dy/dx = f(x). Does that mean that the answer to the question `integrate f(x)' is g(x)? No! The answer is g(x) +c, where c is a constant. To see, differentiate g(x) +c: we obtain f(x) because when we differentiate a constant we get zero.
3. The symbol ∫ is the `antiderivative symbol'.
4. Integration Rule 1: Powers. If u = f(x) = x^p, where p is a real number, then

   ∫ xpdx	=	1/(p+1)xp+1 +c,	p ≠ -1
   	=	ln(x) + x,	p = -1

5. Integration Rule 2: If u = f(x) and v = g(x) are two functions such that ∫ f(x)dx and ∫ g(x)dx both exist, and c is a number, then
   ∫ (f(x) +g(x))dx = ∫ f(x)dx + ∫ g(x)dx, and
   ∫ cf(x)dx = c ∫ f(x)dx.
6. The Definite Integral. If u = g(x) is any antiderivative of u = f(x), then
   ∫_{ab f(x)dx = [g(x)]a^b = g(b) -g(a)}.
7. The definite integral given above finds the area bounded by the graph of the function and the x-axis.

Week 13, lecture 2 (antidifferentiation continued)

1. More difficult types of antiderivatives can be solved using the technique of substitution. For the purposes of this course, if you see an integral involving the product or quotient of functions then you should immediately think "substitution".
   1. Make a substitution (u=f(x)).
   2. Differentiate the substitution.
   3. Replace dx by du. (You might need to use the `trick' of multiplying by 1).
   4. Integrate.
   5. Your final answer should only involve x: back-substitute to remove u.

Week 13, Lecture 3 (antidifferentiation continued)

1. Integration Rule 3: Trig, Exponentials and Log Functions.

   For u = sin(x),	∫ sin(x)dx	= -cos(x) +c,
   For u = cos(x),	∫ cos(x)dx	= sin(x) +c,
   For u = ex,	∫ exdx	= ex +c
   For u = ln(x),	∫ ln(x)dx	= xln(x) -x +c.

2. That's all folks!