AQA A Level Maths: Pure复习笔记4.2.4 Approximating Values

Approximating Values

 

What is the general binomial expansion?

  • The general binomial expansion, as given in the formula booklet, is

 

4.2.4-Approximating-values-Notes-Diagram-1

  • If n ∈ ℕ then the expansion is finite (see Binomial Expansion)
  • Otherwise the expansion is infinitely long
    • It is only valid for |x| < 1 (-1 < x < 1)
    • Only the first few terms of an expansion are usually needed

     

How do I use a binomial expansion to approximate a value?

4.2.4-Approximating-values-Notes-Diagram-2

  •  Ignoring higher powers of x leads to an approximation
  • The more terms the closer the approximation is to the true value
  • For most purposes, squared or cubed terms are accurate enough

4.2.4-Approximating-values-Notes-Diagram-3

 

STEP 1  Compare the value you are approximating to (a + bx)n
STEP 2  Solve the appropriate equation to find the value of x
STEP 3  Substitute this value of x into the expansion to find the approximation

Exam Tip

  • You can get a good idea if your approximation is correct by working out the “real” answer using your calculator.
  • Sometimes it helps to factorise out a number before approximating

Worked Example

4.2.4-Approximating-values-Example-Diagram-1 4.2.4-Approximating-values-Example-Diagram-2 4.2.4-Approximating-values-Example-Diagram-3

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