Application of Derivatives
Application of Derivatives
UPMSP uses derivatives to describe change: rates, tangents, monotonicity, and extrema. For a differentiable function, f′(x)>0 on an interval means increasing, f′(x)<0 means decreasing, and f′(x)=0 at stationary points; endpoints of a closed interval must also be checked.
Core rules
- Tangent slope at x=a is f′(a); normal slope is -1/f′(a) when f′(a)≠0.
- First derivative test: sign change from + to − gives a local maximum, from − to + gives a local minimum.
- Second derivative test: f′(a)=0 and f″(a)>0 gives a local minimum; f″(a)<0 gives a local maximum.
- Concavity follows the sign of f″; an inflection point requires a change in concavity.
- For a word problem, write one variable in terms of another, state the domain, differentiate, and justify the extremum.
Worked example
For f(x)=x^3-3x, f′(x)=3x^2-3=3(x-1)(x+1). Critical points are x=-1 and x=1. Since f″(x)=6x, f″(-1)=-6<0 gives a local maximum f(-1)=2, and f″(1)=6>0 gives a local minimum f(1)=-2. The sign chart shows increasing on (-∞,-1) and (1,∞), decreasing on (-1,1).
Traps
Forgetting endpoint values on a closed interval; treating every critical point as an extremum; using the second derivative test when f″=0; losing the normal’s negative reciprocal; giving a maximum when the question asks for the point or the value.