Algebra and Elementary Calculus Foundations
Algebra and Elementary Calculus Foundations
This domain supplies the algebraic and limiting skills used across the paper: quadratic equations, progressions, binomial expansions, and basic limits and derivatives. A quadratic ax^2+bx+c=0 has discriminant D=b^2−4ac; D>0 gives two distinct real roots, D=0 equal real roots, and D<0 no real roots. The roots satisfy sum=−b/a and product=c/a.
Core rules
- Quadratic formula: x=[−b±√(b^2−4ac)]/(2a).
- AP: nth term a+(n−1)d; sum n/2[2a+(n−1)d].
- GP: nth term ar^{n−1}; sum a(1−r^n)/(1−r) for r≠1; infinite sum a/(1−r) for |r|<1.
- Binomial theorem: (a+b)^n=ΣC(n,r)a^{n−r}b^r; general term C(n,r)a^{n−r}b^r.
- Limit of an indeterminate form often follows factoring, rationalising, or standard limits such as lim_{x→0}(sin x)/x=1.
- Basic derivative: d/dx(x^n)=nx^{n−1}; use it after simplifying.
Worked example
Evaluate lim_{x→1}(x^2−1)/(x−1). Direct substitution gives 0/0, so factor: x^2−1=(x−1)(x+1). For x≠1, the expression equals x+1. Therefore the limit is 1+1=2. Cancelling is valid only after noting x≠1; the original function is undefined at x=1 but the limit still exists.
Traps
Cancelling factors without stating the excluded value; using the discriminant incorrectly for repeated roots; applying the infinite GP sum when |r|≥1; losing the binomial coefficient; substituting into a 0/0 limit before simplifying; dropping the domain restriction in a rational expression.