Application of Derivatives - BSEB Inter Study Guide
Application of Derivatives
BSEB tests rates of change, monotonicity, tangents and normals, and local maxima and minima. The work is mostly sign analysis of f′(x), so a clear number line prevents errors.
Core rules
- Rate of change: if s(t) is position, velocity is ds/dt and acceleration is d²s/dt².
- f is increasing where f′(x)>0 and decreasing where f′(x)<0 on an interval.
- Tangent at (x1,y1): y−y1=f′(x1)(x−x1). Normal slope = −1/f′(x1).
- First derivative test: f′ changing + to − at c gives local maximum; − to + gives local minimum.
- Second derivative test: f′(c)=0 and f″(c)<0 gives local maximum; f″(c)>0 gives local minimum.
Worked example
Find and classify the stationary points of f(x)=x³−3x²+4. f′(x)=3x²−6x=3x(x−2), so x=0,2. f″(x)=6x−6. At x=0, f″(0)=−6<0, so local maximum f(0)=4. At x=2, f″(2)=6>0, so local minimum f(2)=8−12+4=0.
Traps
- Treating f′(c)=0 as automatically a maximum or minimum.
- Testing the sign of f, not f′, when deciding monotonicity.
- Missing endpoints when a question asks for absolute maximum on a closed interval.
- Using the normal slope as f′(x) instead of −1/f′(x).
- Ignoring the domain restriction on the variable.