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AS and A Level: Core & Pure Mathematics

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Get help from 80+ teachers and hundreds of thousands of student written documents Differentiation and intergration

1. 1 It is easy to get differentiation and integration the wrong way round. Remember that the power gets smaller when differentiating.
2. 2 Differentiation allows you to find the gradient of a tangent at any point on a curve. The first derivative describes the rate of change.
3. 3 If a function is increasing then the first derivative is positive, if a function is decreasing, then the first derivative is negative.
4. 4 When asked to find the area under a curve, it is asking you to integrate that curve between two points. Even if you don’t know the points, pick two numbers. You’ll get marks for methods.
5. 5 When referring to a min/max/stationary point, the gradient equals 0. Differentiate the curve and set this to equal 0. The second derivative tells you whether it is a maximum or minimum. If the second derivative is positive, the point is a minimum, if the second derivative is negative, then the point is a maximum.

1. 1 When solving a quadratic inequality, always draw a picture. The inequality is less than 0, where the curve is below the x-axis and bigger than 0 when the curve is above the x-axis.
2. 2 Sometimes in part (a) of a question you are asked to find something, for example a radius. In part (b) you might be then asked to use the radius that you found. If you couldn’t do part (a), don’t give up, choose a random radius.

Straight lines

1. 1 To find the distance of a straight line, draw the straight line with the co-ordinates. Then make a right angle triangle, find the lengths of the horizontal and vertical lines, then use Pythagoras.
2. 2 When a question asks you for a straight line. The first thing to do is to write down the equation of a straight line. Then find out what information you know, and what information you need. Even if you don’t understand the whole question, it is important to start.

1. Fixed point iteration: Rearrangement method explained

g(x) = 3? (9x+11-2x2). The steps I followed to derive this were: f(x) = x3 +2x2 ? 9x ? 11 0 = x3 +2x2 ? 9x ? 11 9x + 11 ? 2x2 = x3 So: 3? (9x+11-2x2) Now I will need to create a table to see if my ?x? values and g(x) values converge, where they do is the value of a root. Hence the ?x? values should be the same as the g(x) values or extremely close. In this case I will quote my answer to 5 s.f. I need to repeat my iterations until x and g(x)

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2. Fractals. In order to create a fractal, you will need to be acquainted with complex numbers. Complex numbers on a graph are characterized by the coordinates of (x,y)

An example of a complex number can be: (5 + 4i). Complex numbers look ambitious, but they are not. In order to add complex numbers, you just need to add like terms:. In order to multiply complex numbers, you need to use the distributive law: . If you are faced with i2, please note that it is equal to -1! We shall start with Mandelbrot?s set. The Mandelbrot set was popularized due to its aesthetic appeal to many, and the simple rules that were applied in order to generate a complex structure.

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3. Solving Equations Using Numerical Methods

This allowed me to work out that the three roots were between the integers of -4 and 1. These three roots are: -4 < ? > -3 -1 < ? > 0 0 < ? > 1 For one root, I worked out where the change of sign took place to 5 sig.fig. The first step to complete this process was to start with entering -4 to -3 (in 0.1?s) into Excel and the formula that I was using into the column next to it to show me where the change of sign was ? between -3.5 and -3.4.

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