Calculus, the study of limits, derivatives, integrals and other functions, is fundamental to mathematical economics. It is so fundamental that almost all other mathematical economic methods rely on it. The ability to take a derivative of an economic function is essential for researchers in mathematical economics as this allows you to find the solutions to optimization problems. By taking the derivative of a function and setting it equal to zero, you can find the points at which the function obtains its maximum or minimum value. Another highly useful, basic application of calculus in mathematical economics is differential equations, which lets a researcher mathematically represent the relationships of changes in economic functions to each other.
Linear programming is the strategy of formulating an optimization problem in mathematical terms. This method of mathematical economics is oriented around linear inequalities as opposed to equations. These linear inequalities can represent linear constraints in economic systems or situations. Through linear programming, an economist can graphically and mathematically represent the possible solutions to a constrained system; these possible solutions are called the feasible region of the system of inequalities. To arrive at a solution quickly, economists use the simplex method, which is an algorithm for finding the optimal points in the feasible region. This allows economists to find the optimal solutions to realistic situations where resources are limited.
Nonlinear programming is a method similar to linear programming in idea. However, economists only use this method in situations where it is not possible to represent constraints of a problem in a linear manner. For example, think about an economic situation where the cost of a resource increases in a quadratic manner instead of linearly with respect to production. In this situation, limits are written as higher-dimension inequalities and solved through nonlinear programming methods, which essentially work in the same manner as linear programming methods.
Mathematical economists can formulate many economic problems as games. Just as you can analyze a game like chess or tic-tac-toe mathematically, you can analyze economic problems mathematically. Game theory allows you to set up a situation, such as a competition between companies, a business deal or an auction, in mathematical terms. Game theory comprises the techniques that let you solve these types of problems and find an optimal strategy for each situation.