Applications of derivatives
A derivative is a rate at one instant. What can that one number tell you about the shape of a graph, the motion of an object, or the best possible choice?
Where you are 11 inside
What this area is about
The sign of f' says where f rises or falls, the sign of f'' says which way the graph bends, and the tangent line stands in for the curve near a point. So the derivative estimates, finds peaks and valleys, and describes motion.
How these fit together
Lines from the derivative. Tangent and normal lines turn a slope into a line. Linear approximation and differentials use that line to estimate values and errors, and Newton’s method repeats it to close in on a root.
Shape. Concavity and inflection points read the sign of f''. Maxima, minima and critical points finds where peaks and valleys can sit and tests each one. Curve sketching puts it all into one picture.
Theorems. The mean value theorem says that if f is continuous on [a, b] and differentiable on (a, b), then \displaystyle f'(c) = \frac{f(b) - f(a)}{b - a} for some c in (a, b). L’Hôpital’s rule settles limits of the forms \displaystyle \frac{0}{0} and \displaystyle \frac{\infty}{\infty} through derivatives.
Applied problems. Optimization and related rates build a function or an equation from a situation and read the answer from derivatives; position, velocity and acceleration reads motion along a line from the signs of s' and s''.
A path through it
Tangent lines, linear approximation, concavity, then extrema. Newton’s method needs linear approximation; optimization, motion, the mean value theorem and curve sketching need extrema. Related rates can follow implicit differentiation. L’Hôpital’s rule is proved from linearization and Rolle’s theorem, so it goes last.
Next come the fundamental theorem of calculus, Euler’s method and Taylor polynomials.