# 13.7 Big-Theta {% embed url="" %} ### Formalizing Order of Growth Given some function $$Q(N)$$, we can apply our last two simplifications to get the order of growth of $$Q(N)$$. For example, if $$Q(N)=3N^3+N^2$$, the order of growth is $$N^3$$. From now onward, we will refer to order of growth as $$\Theta$$ (pronounced "big theta"). ### Order of Growth Examples The following functions have these corresponding order of growths: | Function | Order of Growth | | ----------------- | --------------- | | $$N^3+3N^4$$ | $$N^4$$ | | $$1/N + N^3$$ | $$N^3$$ | | $$1/N + 5$$ | $$1$$ | | $$Ne^N+N$$ | $$Ne^N$$ | | $$40sin(N)+4N^2$$ | $$N^2$$ | Instead of saying a function has *order of growth* \_\_\_, we say that the function *belongs to* $$\Theta (\text{\_\_})$$. In other words, it belongs to the family of functions that have that same order of growth. ### Formal Definition For some function $$R(N)$$ with order of growth $$f(N)$$, we write that: $$R(N) \in \Theta(f(N))$$ and there exists some positive constants $$k\_1$$, $$k\_2$$ such that... $$k\_1 \cdot f(N) \leq R(N) \leq k\_2\cdot f(N)$$ for all values $$N$$ greater than some $$N\_0$$ (a very large $$N$$). --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://cs61b-2.gitbook.io/cs61b-textbook/13.-asymptotics-i/13.7-big-theta.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.