Mathematics · Ch 6 — Application of Derivatives

Summary

Rate of Change: The derivative $\frac{dy}{dx}$ measures the instantaneous rate of change of $y$ with respect to $x$ . For a quantity $s(t)$ , its velocity is $v(t) = s'(t)$ and acceleration is $a(t) = v'(t)$ .
Increasing/Decreasing Functions: A function $f$ is increasing on an interval if $f'(x) \ge 0$ (strictly increasing if $>0$ ), and decreasing if $f'(x) \le 0$ (strictly decreasing if $<0$ ), for all $x$ in that interval.
Tangents and Normals: The slope of the tangent to $y = f(x)$ at $(x_0, y_0)$ is $f'(x_0)$ . The normal is perpendicular, so its slope is $-\frac{1}{f'(x_0)}$ (provided $f'(x_0) \neq 0$ ).
Critical Points: Points where $f'(x) = 0$ or $f'(x)$ does not exist are candidates for local maxima/minima.
First Derivative Test: At a critical point $c$ , if $f'$ changes sign from positive to negative, $c$ is a local maximum; if negative to positive, a local minimum; if no sign change, neither.
Second Derivative Test: If $f'(c) = 0$ and $f''(c) > 0$ , then $c$ is a local minimum; if $f''(c) < 0$ , a local maximum; if $f''(c) = 0$ , the test fails.
Absolute Maxima/Minima: On a closed interval $[a, b]$ , evaluate $f$ at all critical points and at the endpoints $a$ and $b$ ; the largest value is the absolute maximum, the smallest the absolute minimum.
Approximations: For small $\Delta x$ , $\Delta y \approx dy = f'(x) \, dx$ , where $dx = \Delta x$ . This gives the linear approximation $f(x + \Delta x) \approx f(x) + f'(x) \Delta x$ . …