This paper investigates methods of solving calculus problems in Putnam Mathematical Competition.Chapter 2 presents the methods of finding limits, and the most important theorems of continuity---Intermediate Value Theorem and Extreme Value Theorem. Chapter 3 introduces to the properties of derivatives, and the application problems change from the basic problems of derivative. It contains the tangent line and the rate and the meaning of derivative on the geometry.In this chapter also includes the most important theorem---Mean Value Theorem---in derivatives. Chapter 4 introduces to the properties of integral, and the application problems change from the basic problems of integral. There are the Fundamental Theorem of Calculus, Arc length, area, volume and the mass moment and centroid of physical. Chapter 5 investigates the integral techniques of the various forms of possible form for the integral function, to take the integral becomes relatively easy to calculate. In addition to the common variable transformation, also describes how to use the Leibniz Rule for solving integrating. In Chapter 6, it presents that how to determine terms of sequence and its limit, and introduces the infinite summation and to determine convergence or divergence of series.