Differentiation and Integration
Differentiation and Integration covers limits, derivatives, tangents, monotonicity, extrema, indefinite and definite integrals, and areas. EJU Course 2 problems often connect a derivative sign table to a graph and then to an area.
Core rules
- Derivative: for f(x)=x^n, f'(x)=nx^{n-1}; sums and constant multiples differentiate termwise.
- Tangent at x=a: y-f(a)=f'(a)(x-a).
- f'>0 on an interval means increasing; f'<0 means decreasing. Critical points occur where f'=0 or f' is undefined.
- Indefinite integral: ∫x^n dx=x^{n+1}/(n+1)+C for n≠-1. Definite integral gives signed area; geometric area uses absolute value on subintervals.
- The second derivative can confirm concavity, but sign of f' is usually enough for monotonicity.
Worked example
For f(x)=x^3-3x, f'(x)=3x^2-3=3(x-1)(x+1). The sign chart is positive on (-∞,-1), negative on (-1,1), positive on (1,∞). Thus f has a local maximum at x=-1, f(-1)=2, and a local minimum at x=1, f(1)=-2.
Traps
Do not confuse f'=0 with an extremum; check the sign change. Do not treat a definite integral as always positive area-it is signed. When finding total area between curves, split at intersection points. Include endpoints when a problem asks for absolute extrema on a closed interval.