Fp1 Revision Notes
Further Pure 1 Revision Notes • Proof by induction To prove by induction that a statement (C) involving an integer n is true for all [pic], prove that, i) (C) is true for [pic]; ii) if (C) is true for [pic]then (C) is true for [pic]. And then bring these parts together to complete the proof.
• Summing series using formulae To sum series, use a combination of the following formulae. Second two are given in formula book. [pic] [pic] [pic] e.g. Q: Find [pic] By substituting in formulae find, [pic] • Summing series using method of differences e.g. Q: Find [pic] A: First split into partial fractions [pic] Write out the first three terms and last two or three terms. Last term will be the one where you substitute in r=n. [pic] See what cancels [pic]
• Properties of the roots of polynomial equations If [pic] and [pic] are the roots of quadratic equation [pic], then [pic] [pic] If [pic], [pic] and [pic] are the roots of cubic equation [pic], then [pic] [pic] [pic] If [pic], [pic], [pic] and [pic] are the roots of quartic equation [pic], then [pic] [pic] [pic] [pic] • Finding equations with related roots Q: The roots of the cubic equation [pic] are [pic], [pic] and [pic] where [pic]. Find the cubic equation with roots, [pic]. A: Method 1 (Let ‘ denote new coefficient) [pic] [pic] [pic] Let [pic] [pic] and [pic] So [pic] Method 2 (By substitution) Let [pic], then [pic] are roots of [pic] iff [pic] are the roots of [pic] So [pic] • Sketching graphs of rational functions To sketch the graph of [pic]:- 1. Find the intercepts i.e. substitute in [pic] (to find y intercepts) and make [pic] (to find x intercepts). 2. Find the asymptotes (a) Vertical, when [pic] (b) Horizontal, To find horizontal asymptotes complete a long division to
• Summing series using formulae To sum series, use a combination of the following formulae. Second two are given in formula book. [pic] [pic] [pic] e.g. Q: Find [pic] By substituting in formulae find, [pic] • Summing series using method of differences e.g. Q: Find [pic] A: First split into partial fractions [pic] Write out the first three terms and last two or three terms. Last term will be the one where you substitute in r=n. [pic] See what cancels [pic]
• Properties of the roots of polynomial equations If [pic] and [pic] are the roots of quadratic equation [pic], then [pic] [pic] If [pic], [pic] and [pic] are the roots of cubic equation [pic], then [pic] [pic] [pic] If [pic], [pic], [pic] and [pic] are the roots of quartic equation [pic], then [pic] [pic] [pic] [pic] • Finding equations with related roots Q: The roots of the cubic equation [pic] are [pic], [pic] and [pic] where [pic]. Find the cubic equation with roots, [pic]. A: Method 1 (Let ‘ denote new coefficient) [pic] [pic] [pic] Let [pic] [pic] and [pic] So [pic] Method 2 (By substitution) Let [pic], then [pic] are roots of [pic] iff [pic] are the roots of [pic] So [pic] • Sketching graphs of rational functions To sketch the graph of [pic]:- 1. Find the intercepts i.e. substitute in [pic] (to find y intercepts) and make [pic] (to find x intercepts). 2. Find the asymptotes (a) Vertical, when [pic] (b) Horizontal, To find horizontal asymptotes complete a long division to