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 concern multiple quantities that change at different, but related, rates. For example, given the rate of change of a circle's radius, find the rate of change of the circle's area.
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.