Applications of Derivatives
Applications of Derivatives examines how f'(x) and f''(x) describe a graph and solve optimisation problems. Questions cover tangents and normals, monotonicity, maxima and minima, small approximations, and the hypotheses of Rolle's theorem and Lagrange's mean value theorem.
Core rules
- Tangent at x=a: slope f'(a), equation y - f(a) = f'(a)(x-a).
- Normal slope is -1/f'(a) when f'(a) ≠ 0; a vertical tangent gives a horizontal normal.
- f is increasing where f'(x) > 0 and decreasing where f'(x) < 0, on the stated interval.
- Critical points occur where f'(x)=0 or f'(x) is undefined, and also at endpoints when a closed interval is given.
- First derivative test: sign change + to - gives a local maximum; - to + gives a local minimum.
- Second derivative test: f'(c)=0 and f''(c)<0 gives a local maximum; f''(c)>0 gives a local minimum.
Worked example
For f(x)=x^3-3x^2+4, f'(x)=3x^2-6x=3x(x-2), so critical points are x=0,2. f''(x)=6x-6. At x=0, f''=-6<0, so f(0)=4 is a local maximum. At x=2, f''=6>0, so f(2)=0 is a local minimum.
Mark-losing traps
- Reporting a stationary point as an extremum without checking the sign change or f''.
- Using the tangent slope for the normal instead of the negative reciprocal.
- Ignoring the domain or endpoints; an absolute extremum may occur at an endpoint.
- Treating f'(x)=0 as automatically a maximum.
- Dropping units or failing to state the maximum/minimum value, not just the x-value.