Numerical Solutions of Equations

locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change

understand the idea of, and use the notation for, a sequence of approximations that converges to a root of an equation

understand how a given simple iterative formula of the form x_n+1 = F(x_n) relates to the equation being solved

use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy.

Substitute numbers for letters in complicated formulae and algebraic expressions

Evaluate the following expression when x = 1.5

 $\frac{2}{3}\left(x+\frac{1}{x^2}\right)\$  

 $\sqrt{\ln x+3}$  

Rearrange complicated formulae and equations.

Rearrange each formula so that x

x is the subject.

 $2=\frac{y-3x+7}{w}$              ---->  $x=\frac{2w-7-y}{3}$  

 $y=2x^2+13,\ x>0$     ---->  $x=\sqrt{\frac{y-13}{2}}$  

Understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations.

Know that the values of x that make both sides of an equation equal are called the roots of the equation.

Find the roots of the equation   $x^2-7x+10=0$  

You are used to solving equations using direct, algebraic methods such as factorising or using the quadratic formula.

ou might be surprised to learn that not all equations can be solved using a direct, algebraic process.

Numerical methods are ways of calculating approximate solutions to equations. They are extremely powerful problem-solving tools and many are available.

They are widely used in engineering, computing, finance and many other application

understand when the results are likely to be reasonable

understand how to use available software correctly

select an appropriate method when choices are available

write our own programs when we need to do so, if we have the programming skills!

cos x = 2x−1 so cos x − 2x + 1 = 0

Let f(x)=cosx−2x+1 and so f(x)=0 then

f(0.8) = cos 0.8−2(0.8)+1=0.0967…

f(0.9) = cos0.9−2(0.9)+1=−0.1783…

Change of sign indicates the presence of a root.