# Derivative applications

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Solve real world problems (and some pretty elaborate mathematical problems) using the power of differential calculus.

The method of linear approximation (also called local linearization) allows us to approximate a function at hairy x-values using the line tangent to the function's graph at strategic points.

Solve problems about motion along a line using the power of differential calculus. For example, given the position of a particle as a function of time s(t), find the particle's maximum velocity.

Solve problems about motion on a 2-dimensional plane using the power of differential calculus. For example, given the (x,y) position of a particle as a function of time (x(t),y(t)), find the particle's position when its acceleration is 0.

Solve geometrical and real-world problems that involve finding the maximum (or minimum) possible value of a quantity. For example, find the maximum area of a rectangle whose perimeter is given.

Solve various rate of change problems where a real-world situation is modeled by an algebraic function.

The mean value theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].

L'Hôpital's rule provides us with an easy, almost magical way of finding indeterminate limits of quotients of functions using the functions' derivatives. In short, the rule says that if the limits of functions f and g at x=a are 0 (or ထ) and the limit of f'(x)/g'(x) at x=a is equal to L, then the limit of f(x)/g(x) at x=a is also equal to L.

Review your understanding of the various applications of differential calculus with some challenge problems.