Differential Calculus: Applications I
Differential Calculus: Applications I tests the derivative as a rate of change, tangent and normal lines, monotonicity, and local extrema. The exam expects a sign chart and a clear statement of intervals.
Core rules
The slope of the tangent at x0 is f'(x0); the normal slope is −1/f'(x0) when f'(x0)≠0, and the normal is vertical when f'(x0)=0. A differentiable function is increasing on an interval where f'≥0 and decreasing where f'≤0. Critical points occur where f'=0 or f' does not exist. A sign change of f' from + to − gives a local maximum; from − to + gives a local minimum. The second-derivative test gives a minimum when f''>0 and a maximum when f''<0 at a stationary point. Related-rate problems differentiate a relation with respect to time, using the chain rule.
Worked example
For f(x)=x³−3x²+4, f'(x)=3x²−6x=3x(x−2). The critical points are x=0 and x=2. The sign chart gives f'>0 on (−∞,0), f'<0 on (0,2), and f'>0 on (2,∞). Thus f is increasing on (−∞,0) and (2,∞), decreasing on (0,2); f(0)=4 is a local maximum and f(2)=0 is a local minimum.
Traps
f'(x)=0 does not guarantee an extremum: x³ has f'(0)=0 but no extremum. Do not include endpoints when asked for local extrema. When f'(x0)=0, the normal is vertical, not horizontal. In related rates, differentiate before substituting numerical values that change with time.