MATH141: Mathematics 1C Part 1

Overview of Lectures on fundamentals

In this section I will try to summarise the main ideas of each lecture.

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. (Basic skills test)

Week 1, Lecture 3 (Indices and Surds)

There are worksheets on indices (solutions) and surds (solutions).

There are resources for Indices and Surds on the summertime Summertime Maths webpage.

Exam questions on indices and surds.

1. You need to know the rules for manipulation of indices and how to use them.
2. In questions such as simplify sqrt(20)+sqrt(45) the key is to first factorise the numbers. 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. (a-b)(a+b) = a²-b². (This is called the difference of two squares).
4. The conjugate of an expression such as a +sqrt(b) is a -sqrt(b).
5. 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 the numerator and the denominator of the fraction by the conjugate. (The reason why this works is because we are using the difference of two squares).

Week 1, Lecture 4 (Logarithms, Factorisation).

There are worksheets on logarithms (solutions) and factorisation (solutions).

Exam questions on logarithms.

There are resources for logarithms and factorisation on the Summertime Maths webpage.

1. You need to know the rules for manipulation of logarithms and how to use them.
2. Common logarithms use base 10. The common logarithm of a number N is usually written as log₁₀ N or, simply, log N. Natural logarithms use base e. The natural logarithm of a number N is usually written as log_e N or, most frequently, ln N.
3. Look out for the difference of two squares. The following can be easily simplified using the difference of two squares.
   (a) a⁴ -b⁴.
   (b) (2x+3y)² -(x-4y)².

Week 2, Lecture 1 (Algebraic Fractions).

There is a worksheet on algebraic fractions (solutions).

There are resources for Algebraic Fractions on the Summertime Maths webpage.

Exam questions on algebraic fractions.

You need to be able to add/subtract/multiply and divide algebraic fractions.

Week 2, Lecture 1 (continued) (Function notation, Zeroes of Functions and Quadratic Equations).

There are worksheets on functions (solutions) and quadratic equations (solutions).

There are resources for functions on the Summertime Maths webpage. (This page also contains some material relating to functions that will be covered latter in the course).

Exam questions on function notation, zeroes of functions , and quadratic functions.

1. What is a function?
2. What does it mean for x to be a zero of the function f(x)?
3. Give two ways to find the zeroes (or roots) of a quadratic equation?
4. What condition must hold for the zeroes of a quadratic equation to be real?
5. Some equations are quadratic equations in disguise.
   4x⁴ +7x² -15 = 0
   4^x -9(2^x) +8 = 0
6. How do you sketch the function y = ax² +bx +c?

Week 3, Lecture 1 (continued) (Geometry, graphs of functions, trig ratios, radian measure, angles of any magnitude, graphing trig functions).

There are a large number of facts in this lecture. (It's not one of my favourites). I had to move at a quick pace. Now, you need to learn them.

There are worksheets on geometry (solutions) and trigonometry (solutions).

There are resources for elementary trigonometry (all work that can be done in a right-angle triangle, including exact values for 30, 45 and 60 degrees), intermediate trigonometry (extending trigonometry to angles of any size and introducing radian measures. Also, using the inverse trig functions on the calculator.) and advanced trigonometry (trig functions and trig identities) on the Summertime Maths webpage. (At the moment I don't think there are any resources on the advanced trigonometry page).

Exam questions on geometry.

1. In the equation y = mx + c what is the geometric interpretation of the constants m and c?
2. Plot the graph y = |x-2|.
3. In the equation (x-x₁)² + (y-y₁)² = r² what is the geometric interpretation of the constants x₁, y₁ and r?
4. Define the following functions: cosec(w), sec(w), cot(w).
5. What are the two most important things to show when plotting a trig function?
6. How do you convert from an angle measured in degrees to an angle measured in radians?
7. What are the equivalent angles in the second, third and fourth quadrants for an angle w in the first quadrant?
8. In which quadrant is which trig function positive?

Week 3 Lecture 2 (Notation, Pascals Triangle and the Binomial Theorem, Long Division of Polynomials)

Exam questions on factorial notation and polynomials.

1. Pascal's triangle to expand expressions of the form (a+b)ⁿ.
2. Long division of polynomials in analogy with long division of numbers.
3. In the question (4x³ +13x +33)÷(2x+3) don't forget to write out the missing powers of x, i.e. write the question as (4x³ +0x² +13x +33)÷ (2x+3)

Week 4 Lecture 1 (Basic Skills test)

Week 4 Lecture 2 (Remainder Theorem, Factor Theorem)

Exam questions on this material are probably classified under polynomials.

1. Determine the remainder when the polynomial x³ +5x² -2x -24 is divided by the polynomial x-1. (Hint. Remainder Theorem).
2. Suppose that x-a is a factor of the polynomial P(x). What is the remainder when P(x) is divided by x-a?
3. How do we set about trying to factorise the polynomial x³ +5x² -2x -24?

Week 5 Lecture 1 (Basic trigonometric identities)

Exam questions on trigonometry.

1. There are a large number of trigonometry identities in this lecture. Don't try to memorise them - put them on your formula sheet!
2. There isn't enough time in the lecture to go through more than one or two examples for each class of identities. It's your responsibility to work through additional examples to master the use of these identities. (Look at exercise 1.13.7)
3. When asked to prove a formula remember the following handy hints.
   1. Don't Panic!
   2. Do what's obvious - such as expanding brackets.
   3. Do what's obvious - such as cross-multiplying to eliminate fractions.
   4. Do what's obvious - such as replacing double-angle expressions by single-angle expressions using the double-angle formulae.

      Replace sin(2x) by 2sin(x)cos(x) and cos(2x) by either 2cos²(x -1 or 1 -2sin²(x).

4. I won't set a question in the mid-session test or the final exam that uses the half-angle tangent formula. However, your tutor may put a such a question on an assignment sheet.