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**“Teach A Level Maths” Vol. 2: A2 Core Modules**

Demo Disc “Teach A Level Maths” Vol. 2: A2 Core Modules © Christine Crisp

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The slides that follow are samples from the 55 presentations that make up the work for the A2 core modules C3 and C4. 2: Inverse Functions 22: Integrating the Simple Functions 3: Graphs of Inverse Functions 28: Integration giving Logs 4: The Function 31: Double Angle Formulae 13: Inverse Trig Ratios 15: More Transformations 43: Partial Fractions 16: The Modulus Function 47: Solving Differential Equations 18: Iteration Diagrams and Convergence 51: The Vector Equation of a Line

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**Explanation of Clip-art images**

An important result, example or summary that students might want to note. It would be a good idea for students to check they can use their calculators correctly to get the result shown. An exercise for students to do without help.

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**2: Inverse Functions Demo version note:**

The students have already met the formal definition of a function and the ideas of domain and range. In the following slides we prepare to introduce the condition for an inverse function.

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Inverse Functions One-to-one and many-to-one functions Consider the following graphs and Each value of x maps to only one value of y . . . Each value of x maps to only one value of y . . . and each y is mapped from only one x. BUT many other x values map to that y.

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Inverse Functions One-to-one and many-to-one functions Consider the following graphs and is an example of a many-to-one function is an example of a one-to-one function

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**3: Graphs of Inverse Functions**

Demo version note: By this time the students know how to find an inverse function. The graphical link between a function and its inverse has also been established and this example follows.

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**Graphs of Inverse Functions**

e.g. On the same axes, sketch the graph of and its inverse. N.B! Solution: x

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**Graphs of Inverse Functions**

e.g. On the same axes, sketch the graph of and its inverse. N.B! Solution: N.B. Using the translation of we can see the inverse function is

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**Graphs of Inverse Functions**

A bit more on domain and range The previous example used The domain of is . Since is found by swapping x and y, the values of the domain of give the values of the range of Domain Range

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**Graphs of Inverse Functions**

A bit more on domain and range The previous example used The domain of is . Since is found by swapping x and y, the values of the domain of give the values of the range of Similarly, the values of the range of give the values of the domain of

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**4: The Function Demo version note:**

The exponential function has been defined and here we build on earlier work to find the inverse and its graph.

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The Function More Indices and Logs We know that ( since an index is a log ) The function contains the index x, so x is a log. BUT the base of the log is e not 10, so Logs with a base e are called natural logs We write as ( n for natural ) so,

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The Function The Inverse of is a one-to-one function so it has an inverse function. We can sketch the inverse by reflecting in y = x. Finding the equation of the inverse function is easy! We’ve already done the 1st step of rearranging: So, Now swap letters: N.B. The domain is

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**13: Inverse Trig Ratios Demo version note:**

This presentation on inverse trig ratios revises the idea of a domain. It also reminds students that a one-to-one relationship is needed in order to find the inverse function.

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Inverse Trig Ratios What domain would you use? We need to make sure that we have all the y-values without any being repeated. The domain is defined as ( or, in degrees, as ).

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**15: More Transformations**

Demo version note: As new topics are introduced, the presentations revise earlier work. Here, the exponential function is used to show the result of combining transformations.

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**x has been replaced by 2x:**

More Transformations Combined Transformations e.g. 1 Describe the transformations of that give the function Hence sketch the function. Solution: x has been replaced by 2x: so we have a stretch of s.f. parallel to the x-axis 1 has then been added: so we have a translation of

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More Transformations We do the sketch in 2 stages: The point on the y-axis . . . doesn’t move with a stretch parallel to the x-axis

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More Transformations We do the sketch in 2 stages:

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**16: The Modulus Function Demo version note:**

Students are encouraged to sketch functions whenever possible rather than always using graphical calculators. A translation has been used to obtain the function shown and the inequality is found using the sketch.

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**e.g. 2 Solve the inequality .**

The Modulus Function e.g. 2 Solve the inequality Method 1: Sketch the graphs and x y is above 1 in two regions.

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**e.g. 2 Solve the inequality .**

The Modulus Function e.g. 2 Solve the inequality Method 1: Sketch the graphs and x y is above 1 in two regions. The sketch doesn’t show the x-coordinates of A and B, so we must find them. A B x x 1 3 The right hand branch of the modulus graph is the same as y = x – so B is given by The gradient of the right hand branch is +1 and the left hand is -1, so the graph is symmetrical. So,

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**18: Iteration Diagrams and Convergence**

Demo version note: By this stage the students understand how to rearrange equations in a variety of ways in order to find an iterative formula. They have met cobweb and staircase diagrams in simple cases of convergence and divergence.

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**Iteration Diagrams and Convergence**

In the next example we’ll look at an equation which has 2 roots and the iteration produces a surprising result. The equation is We’ll try the simplest iterative formula first : If you have Autograph or another graph plotter you may like to try to find both roots before you see my solution.

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**Iteration Diagrams and Convergence**

Let’s try with close to the positive root. Let

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**The sequence diverges rapidly.**

Iteration Diagrams and Convergence Let’s try with close to the positive root. The staircase moves away from the root. The sequence diverges rapidly. Using the iterative formula,

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**Iteration Diagrams and Convergence**

Suppose we try a value for on the left of the root.

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**Iteration Diagrams and Convergence**

Suppose we try a value for on the left of the root.

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**Iteration Diagrams and Convergence**

Suppose we try a value for on the left of the root.

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**Iteration Diagrams and Convergence**

Suppose we try a value for on the left of the root.

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**Iteration Diagrams and Convergence**

Suppose we try a value for on the left of the root. The sequence now converges but to the other root !

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**22: Integrating the Simple Functions**

Demo version note: The opportunity is taken here to remind the students about using integration to find areas, whilst applying the work to a trig integration.

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**Integrating the Simple Functions**

(b) Radians! The definite integral can give an area, so this result may seem surprising. However, the graph shows us why it is correct. This part gives a positive integral How would you find the area? Ans: Find the integral from 0 to and double it. This part gives a negative integral

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**28: Integration giving Logs**

Demo version note: Inspection is used to find integrals of the form In this example, the method is applied to a question where the integrand needs adjusting.

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**Integration giving Logs**

What do we want in the numerator? Ans: We now need to get rid of the minus and replace the 3. . . .

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**Integration giving Logs**

What do we want in the numerator? Ans: We now need to get rid of the minus and replace the 3. Always check by multiplying the numerator by the constant outside the integral:

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**Integration giving Logs**

What do we want in the numerator? Ans: We now need to get rid of the minus and replace the 3.

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**31: Double Angle Formulae**

Demo version note: Summaries are given at appropriate points in the presentations. The notebook icon suggests that students might want to copy the slide.

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Double Angle Formulae SUMMARY The double angle formulae are:

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Double Angle Formulae N.B. The formulae link any angle with double the angle. For example, they can be used for and and and and We use them to solve equations to prove other identities to integrate some functions

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**43: Partial Fractions Demo version note:**

Introductory exercises follow most sections of theory and examples. The next slide shows the first exercise on finding Partial Fractions. The full solution is given and the cover-up method used to check the result. Students would use their textbooks for further practice.

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Partial Fractions Exercises Express each of the following in partial fractions. 1. 2. 3. 4.

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Partial Fractions Solutions: 1. Multiply by : So, Check:

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Partial Fractions Solutions: 1. Multiply by : So, Check:

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Partial Fractions Solutions: 1. Multiply by : So, Check: ( You don’t need to write out the check in full )

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**47: Solving Differential Equations**

Demo version note: The slides here show the introduction to the method of separating the variables to solve some differential equations. Solutions are applied later to applications of growth and decay functions.

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**Solving Differential Equations**

e.g. (2) Before we see how to solve the equation, it’s useful to get some idea of the solution. The equation tells us that the graph of y has a gradient that always equals y. We can sketch the graph by drawing a gradient diagram. For example, at every point where y = 2, the gradient equals 2. We can draw a set of small lines showing this gradient. 2 1 We can cover the page with similar lines.

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**Solving Differential Equations**

We can now draw a curve through any point following the gradients.

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**Solving Differential Equations**

However, we haven’t got just one curve.

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**Solving Differential Equations**

The solution is a family of curves. Can you guess what sort of equation these curves represent ? ANS: They are exponential curves.

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**51: The Vector Equation of a Line**

Demo version note: By this time, the students have practised using vectors and are familiar with the notation. The equation of a straight line in vector form is developed.

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**The Vector Equation of a Line**

Finding the Equation of a Line In coordinate geometry, the equation of a line is e.g. The equation gives the value (coordinate) of y for any point which lies on the line. The vector equation of a line must give us the position vector of any point on the line. We start with fixing a line in space. We can do this by fixing 2 points, A and B. There is only one line passing through these points.

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**A and B are fixed points. a r1 The Vector Equation of a Line**

We consider several more points on the line. x a r1 We need an equation for r, the position vector of any point R on the line. x Starting with R1:

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**A and B are fixed points. a r2 The Vector Equation of a Line**

We consider several more points on the line. x a We need an equation for r, the position vector of any point R on the line. x x r2 Starting with R1:

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**A and B are fixed points. r3 a The Vector Equation of a Line**

We consider several more points on the line. r3 x a We need an equation for r, the position vector of any point R on the line. x x Starting with R1:

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**t is called a parameter and can have any real value.**

The Vector Equation of a Line x So for R1, R2 and R3 x x a x x For any position of R, we have t is called a parameter and can have any real value. It is a scalar not a vector.

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