Modelling involving Numerical Methods (Edexcel A Level Maths: Pure): Flashcards

A method that finds an approximate solution to an equation, usually by improving an estimate step by step until it is accurate enough.

Numerical methods are used where an analytical method, solving by algebra to reach an exact answer, is difficult or impossible. Real models such as for a planet's orbit often cannot be rearranged to make the unknown the subject at all.

Rearrange it so that one side is zero, then treat the other side as a function whose root you are looking for:

So the value you want is the root of , with .

Every numerical method here finds a root, so a model has to be written in that form before any of them can start.

The context puts limits on which values are possible, and those limits rule parts of the domain in or out.

The depth of a stream cannot rise above ground level, and a population cannot fall below zero. If a model predicts a value outside such a limit, that solution is not a real one, whatever the algebra says.

Questions set in context expect this reasoning alongside the calculation, not instead of it.

False.

It gives an approximation, which you can make as accurate as the question requires by taking more steps.

That is why these questions specify a degree of accuracy, such as correct to 3 decimal places, and often go on to ask you to verify that the approximation really is accurate to it.

Because the model goes on behaving in the same way while the real situation changes.

Some cars become more valuable as they age, once they become classics, and a model whose main term only ever decreases cannot describe that.

It also fails on its own terms: tends to zero as grows, so the term eventually dominates and the predicted value becomes negative, which no car's value can be.

The distance travelled, in metres.

Speed multiplied by time gives distance, and each strip of the area is a speed multiplied by a time interval. The units of the area are the units of the two axes multiplied together.

So check the axes before saying what an area means: the area under a graph represents a compound measure fixed by the units on the axes.

Look at how the speeds change. The increases are , so they are getting larger.

That makes the speed-time graph convex, so between any two readings the straight top of a trapezium lies above the curve. Every strip adds slightly too much, and the estimate is an overestimate.

A concave graph, whose increases were getting smaller, would give an underestimate.

Because the two methods do different jobs.

Newton-Raphson produces an approximation but says nothing about how accurate it is. Change of sign confirms that a root really does lie inside the interval your rounded answer stands for.

Context questions often chain methods like this, one to find a value and another to verify it, so read the whole question before starting on any part of it.