Roots in Intervals (Edexcel International AS Further Maths)

Roots in Intervals

What is a root of an equation?

• A root of the equation is a solution

  • If is a root of then

    • The Greek letter alpha is often used

• To find exactly, you need to solve the equation algebraically (analytically)

• Some equations cannot be solved algebraically

  • In which case you can approximate a root to a given accuracy

    • For example, to 3 decimal places

  • This is called solving equations numerically

What is an interval?

• If you don't know the root but know that it lies between and , where , then

  • is an interval containing the root

• Intervals can be written using bracket notation

  • is

  • is

How do I show that an interval contains a root?

• Use the sign-change and continuity test to show that the interval contains a root to the equation

• Show that and have different signs and check that is continuous across

  • Then the interval must contains a root

  • Continuous means no jumps or asymptotes in the interval

    • Check the equation does not divide by zero within the interval

    • (Jumps or asymptotes outside of the interval are fine)

  • You must write a conclusion to the test in words, for example:

    • " has a sign change in the interval and is continuous in the interval, so a root must lie in the interval "

How do I show that an equation has a root to a given accuracy?

• If asked to show that is the root of an equation to 2 decimal places

  • write an interval using the lower and upper bound of the root

    • 

  • use the sign-change and continuity test on this interval

• A suitable conclusion would be

  • has a sign change in the interval

  • and is continuous in the interval

  • so a root must lie in the interval

  • All values in the interval round to

  • so the root is to 2 decimal places

How do I know when the sign-change test fails?

• You need to know the three cases when the sign-change and continuity test fails to work properly:

  • If the curve only touches the -axis at the root, you will not see a sign change (even though there is a root)

  • If the curve has an asymptote in the interval, then you may see a sign change (but there is no root)

    • The asymptote means is not continuous

  • If the interval is too large, there may be more than one root in it

    • You may see a sign change (an odd number of roots)

    • Or you may not see a sign change (an even number of roots)