C2 - Introduction

C2 is the second module, of the Alevel Maths which is core. It is in my opinion the hardest module of AS, for Edexcel, it is one of the longer modules of the whole course.The topics are :

Edexcel

1. Algebra and Functions - Algebraic Division, Factor Theorem, Remainder Theorem.
2. Coordinate Geometry - Equation of Circle, Use of certain circle properties
3. Sequences and Series - Geometric Series {Sum of a g. series, Sum to infinity, Proof of sum formula}, Binomial Expansion for (1+x)^n and (a+b) ^n.
4. Trigonometry - Sine and Cosine Rule, Area of Triangle, Radians, Arc length, area of a sector and segment, Sin,Cos and Tan Graphs, Use of Trig. Identities, Solving simple equations.
5. Exponentials and Logarithms - y=a* and the graph, Laws of Logarithms and Change of base formula.
6. Differentiation - Maxima and Minima problems, Stationery points, Increasing and Decreasing Functions.
7. Integration - Definite Integrals, Area under a curve is a definite integral, Trapezium Rule, Area bounded between a curve and line.

AQA

1. Algebra and Functions - Laws of Indices, Simple transformations on graphs
2. Sequences and Series - Arithmetic series {nth term, Sum of a. series}, Geometric series {Sum of finite g. series, sum to infinity}, Binomial Expansion (1+x)^n.
3. Trigonometry - Sine and Cosine Rule, Area of Triangle, Radians, Arc length, area of a sector and segment, Sin,Cos and Tan Graphs, Use of Trig. Identities, Solving simple equations.
4.  Exponentials and Logarithms - y=a* and the graph, Laws of Logarithms and Change of base formula.
5. Differentiation - Differentiate functions of form x^n. where n is a rational number.
6.  Integration - Integration of functions of form x^n, Trapezium Rule

OCR

1. Algebra - Factor theorem, Remainder theorem, algebraic division, a* graph, laws of logarithms, change of base of logs, and solving log equations.
2. Sequences and Series - Sigma Notation, Arithmetic & Geometric Series, Sum of arithmetic and geometric series, sum to infinity, Binomial Expansion.
3. Trigonometry - Sine and Cosine Rule, Area of Triangle, Radians, Arc length, area of a sector and segment, Sin,Cos and Tan Graphs, Use of Trig. Identities, Solving simple equations.
4.  Integration - Indefinite Integration, Evaluation of definite integrals, Finding area of a region bounded by a curve and line,. Trapezium Rule.

There is alot of new content, and alot which is expanded from the topics learnt in C1. To Achieve a top grade, you should ensure you can apply everything used in C1, these modules are synoptic, and they can ask you a c1 question in c2, or ask a question which relates to c1 content.

Indefinite Integration

•  Edexcel - C1, Integration
• AQA - C1, Integration
• OCR - C2, Integration

We already discussed one of the branches of calculus. The other is Integration, it's general purpose is to find the area between some given intervals. It can be used to find the area bounded by a curve, you will learn this in C2. Right now, we need to know what Indefinite Integration is. It is simply the reverse of Differentiation, if dy/dx is the derivative of some function, we can use integration to obtain the function y. It's also called Anti differentiation. When will integrate some function, we will add a constant c. (The constant of Integration).

That image is the general notation used for Integration. When we integrate, we first draw the S kind of line, then write the function, and then dx. (for now anyway). This shows we are integrating. We need to know how to integrate functions in the form : x^n.

As I said Integration is the reverse of Differentiation. So when we differentiate we multiply the power by the coefficient of the function, and subtract the power by 1. When integrating, we firstly raise the power by 1, and divide by this new power.


a)

      *We usually write c (instead of constant) [  + c ]


I will also post some further examples in my image notes. In my image notes, I will also show you how to derive a the equation of the curve, when given dy/dx and a point on the curve, these question are usually 5/6 easy marks.

With this we end C1 here. I shall start C2 next week... I have done all model solutions for the Edexcel C1 Papers (From Jan 2005 - June 2011), these are available on request. I will start working on the model solutions for Solomon Papers aswell.

Differentiating Functions with Examples and Applications of Differentiation

I will use two examples from the specification itself, which you should be able to differentiate :

1) y= (2x+5)(x-1)
First we expand the brackets, and simplify :

2x^2 - 2x + 5x -5
y = 2x^2 + 3x -5, now we can differentiate so :
dy/dx = 4x + 3

I will show the 2nd example, in my set of image notes.

Application of Differentiation to Equations of Tangents

If we have a curve, and a tangent to some point on the curve. We can find the equation of the tangent. How ?

If we know the coordinates of that point. Firstly we differentiate the curve, and substitute the x-coordinate to get the gradient of the curve at the point. Say, if the tangent is perpendicular to the curve. We know that the gradient of this tangent, will be -1 divided by the gradient of the curve, we just got. Using the gradient, and the coordinates, we can use the y -y1 = m(x-x1) formula to find out the equation.

Example

This question is taken from the Edexcel C1 Jan 2006 Paper. (Question 9)

Sorry for the bad quality

Model Solution 

I have ignored part a).. the rest of the question is more relevant to the post.

Notes on Differentiation with Four Examples

Application of Differentiation for Tangents,Gradients and Normals

Differentiation

• Edexcel- C1 Differentiation
• AQA- C1 Differentiation
• OCR- C1 Differentiation

One of the new topics, we learn in A-Level Maths is Differentiation. It is one of the branches of Calculus, which is a major field in Mathematics, and almost is a useful application in loads of other fields.. Engineering, Economics, Physics and Chemistry. It is concerned with how one thing changes, as a result of another quantity changes. E.g. How displacement changes, as time changes (dd/ dt) would be the velocity.... We shall look for now at how y changes with respect to x (we call this dy/dx) = d (delta which means change)

We will look at curves, where the gradient is changing at each point on the curve. So dy/dx on each point is different, and not constant. First, i'll introduce you to a tangent. A tangent, is a straight line which touches a point on the curve, it only touches that point though.

The green line is the tangent to the curve( in black), this tangent only touches the yellow point on the curve. Note the gradient at the yellow point, is different to the rest of the points on the curve, because the gradient changes as x changes.

The derivative of a curve is the same as the dy/dx of a function e.g.

The derivative of x^3 = 3x^2
dy/dx (x^3) = 3x^2
This is the first derivative, if differentiate again, we would get the second derivative, again.. the third.. and so on..

dy / dx means differentiating y with respect to x. (what is happening to y, as x changes)

How to Differentiate

To differentiate a function, you reduce the power by 1, and multiply by the new power :

Function          Derivative
axⁿ                 anx<sup>n-1

e.g. x^2 
dy/dx = 2x


x^3 
dy/dx = 3x^2


* If we had to differentiate anything to the first power e.g. x , 3x, 5x... it would be 1,3 and 5 respectively. Why ? 
Because it is to the power 1, reducing the power to 0.. anything to the power of 0 equals 1 .. so we just multiply the coefficient of x by 1.. which is the same as taking away the x.

*Differentiating a number .. gives 0. Think about it, if draw a graph of say y = 5, the gradient is 0.

Other Notation 

A function can be written as y =... or f(x) = ..., if we have a function defined as f(x) =...., then the derivative of that is f'(x)=...

f(x) = 5x ... f '(x) = 5 (this is called f prime)

*For the first derivative we use one dash.. second derivative two..etc 
We can only differentiate functions in the form of</sup> axⁿ    , so if it looks any different, we have to rearrange to get in that form, using rules of indices.

C1 Introduction

C1. The first module of AS, of any Alevel Maths Course. Suppose to be the easiest, it generally forms on from the A-A* topics of gcse. However, you don't have a calculator... so brush up on general arithmetic skills, and fractions. Topics :

1)  Algebra and Functions
*Laws of Indices, Surds, Rationalising the Denominator, Quadratic Equations (Discriminant, Completing the Square, Quadratic Formula,Factorisation), Sketching Graphs and Transformations of Graphs.

2) Coordinate Geometry
*Equation of a Straight Line, Gradients, Perpendicular Lines

3) Sequences and Series
*Terms, nth Term, Arithmetic Series, Sum of Terms and Stigma Notations.

4)Differentiation
*Differentiating basic functions, and relation to Gradients.

5)Integration
*Indefinite Integration