# Derivative applications

Contents

Why know how to differentiate function if you don't put it to good use? Learn about the various ways in which we can use differential calculus to study functions and solve real-world problems.

Critical points are points where a function may obtain their minimum or maximum value. They play a critical role (pun intended) in analyzing the increasing and decreasing intervals of functions, and in finding their minimum and maximum points.

Analyze functions to find the intervals in which they are increasing or decreasing.

Analyze functions to find their relative extrema (i.e. their relative minimum and maximum points).

Analyze functions to find their absolute extrema (i.e. their absolute minimum and maximum points).

Review your understanding of increasing and decreasing intervals, and extremum points, with some challenge problems.

Concavity describes the shape of a graph as it increases or decreases: a graph that's concave up is shaped like a cup, U, and a graph that's concave down is shaped like a cap, ∩. Learn more about concavity and how it relates to a function's second derivative.

Points of inflection are points where the function changes concavity. Learn how to find and analyze them.

Combine the tools provided by differential calculus with other algebraic tools in order to obtain detailed sketches of various functions.

Review your understanding of concavity and points of inflections with some challenge problems.

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.