Some important definitions:
|BINOMIAL EXPANSIONS |The binomial expansion formula is given in the formula book: |Example 3: [pic] |
| |[pic] |We can’t yet use the binomial expansion formula as it isn’t in the |
|Example 1: [pic] | |form [pic] |
|We can now find a binomial expansion for the bracket: |This expansion is only valid if [pic] |To get it in the correct form: |
| | |[pic] |
|[pic] | | |
| | |So, |
|=[pic] | |[pic] |
| | |=[pic] |
|= 4 – 8x + 12x2 – 16x3 + … | |= [pic] |
| | |This expansion is valid if [pic], i.e. if [pic] |
|The above expansion is valid if -1 < x < 1 | | |
| | | |
| |Example 2: [pic] | |
| |We can use the formula for the binomial expansion if we replace x by -4x: | |
| |[pic] | |
| |= [pic] | |
| |= [pic] | |
| |This expansion is valid if | |
| |-1 < 4x < 1, i.e -0.25 < x < 0.25 | |
|PARAMETRIC EQUATIONS | |CONVERTING PARAMETRIC EQUATIONS TO CARTESIAN FORM |
|Sometimes curves are defined in terms of a parameter (usually t or θ).|[pic] |We can eliminate a parameter from a parametric equation to give it in|
| | |Cartesian form. |
|Example: A curve is defined by the equations [pic]. Sketch the curve | | |
|for values of t from -3 to 2. | |Example: [pic] |
| | |We know that t = x + 1. |
|Step 1: Produce a table of values: | |Substituting this into the equation for y: |
| | |[pic] [pic] |
| | | |
| | |Example 2: [pic] |
| | |We use the result that [pic]. |
| | | |
|Step 2: Plot the values of y against x. | |But: [pic], [pic] |
| | |Therefore: [pic] |
| |[pic][pic] | |
-----------------------
|t |-3 |-2 |-1 |0 |1 |2 |
|x |-4 |-3 |-2 |-1 |0 |1 |
|y |6 |2 |0 |0 |2 |6 |
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