In this paper, some properties of continuous functions in $q$-analysis are investigated. The behavior of $q$-derivative in a neighborhood of a local extreme point is described. Two theorems are proved which are $q$-analogons of the fundamental theorems of the differential calculus. Also, two $q$-integral mean value theorems are proved and applied to estimating remainder term in $q$-Taylor formula. Finally, the previous results are used in considering some new iterative methods for equation solving.