Iteration for A Level Maths: staircase method & cobweb method

E.g. f(x) = x² - x - 4 When the above is rearranged in the form of x = g(x), it is said to be in iterative form. x² - x - 4 = 0 x = 1 + x/4 x n+1 = 1 + x n / 4 If x n is known, x n+1 can be calculated. The initial value to trigger off the iterative process is found by looking for change in sign of…

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Mechanics - A Level Applied Maths & Physics: really challenging questions

An object that falls off a tree travels 16/25 th of the height of the tree during the last second. Assuming that g = 10 m/s², find the total time taken for the fall and the height of the tree. A balloon is steadily rising at 5 m/s. After 4 seconds of its motion, a nail from the balloon falls off. Sk…

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Basic Integration: finding the area under a curve by a GeoGebra applet - fully interactive

With the following GeoGebra applet, you can find out the area under a curve by integration -  interactively.  All you need to do is  to move the two sliders to points of your choice and tick the checkbox for answer to appear. The function used in this case is a quadratic function. Enjoy, learn and pro…

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Linear and Geometric Sequences for GCSE, IGCSE & A Level

E.g.1 : Find the nth term of the sequence: 3, 7, 11, 15, ... The common difference is 4.  Let N = an + b, where N is the nth term and a an b are constants to be found. 3 = a(1) + b = a + b 7 =  a(2) + b = 2a + b Solving the two equations simultaneously, we get a = 4 and b = -1 So, the nth term, N = 4n - 1 …

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A Level Further Maths: Simple Harmonic Motion - SHM: Light Damping

The following example shows how to model light damping by using a second order differential equation. The second-order differential equation modelling light damping is given by: \[ \ddot{x} + 0.5 \dot{x} + 4 x = 0 \] The characteristic equation is: \[ m^2 …

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A Level Further Maths: Demystifying Polar Coordinates through Shapes

The above shape is known as cardioid in mathematics. It is produced by the polar equation, r = a(1 + cos(Θ)), where a can be an integer. You can practise a cardioid with the following simulation and see how r and Θ do the magic to produce the shape. Rose Petals You can practise the creation of shapes …

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The exponential graph

The exponential function is a special exponential function; the gradient of any point on the curve is the same as the value of the function at that point.

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Concave, convex functions and points of inflection

Concave functions If f''(x) ≤ 0 in a given interval of x, the function is said to be concave. Convex functions If f''(x) ≥ 0 in a given interval of x, the function is said to be convex. Point of Inflection The point at which a curve changes being concave to convex or vice versa is called…

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Differentiation: problem solving

Snell's Law When light refracts from one medium to another, the ratio of sine of the angles is equal to the ratio of refractive index of each corresponding medium. Snell's Law implies that light, when refracting from one medium to another, takes the shortest path. We can show by differentiatio…

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Straight Line Graph Generator for Maths Teachers: GCSE, IGCSE, GCE O Level, GCE A Level, A Level

Created with GeoGebra, this applet lets you generate straight line graphs at random, displaying the corresponding equation for each. In addition, you can find a parallel line - and its equation - for each line. Moreover, for each generated line, a random point is created. Then, the perpendicular line…

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