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AS and A Level: Core & Pure Mathematics

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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.

Quadratics and circles

  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.

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  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)

    • Word count: 683

  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.

    • Word count: 3016

  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.

    • Word count: 872
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