Answer:
1. The first function
has the same order of growth as the second function
within a constant multiple.
2. The first
and the second
logarithmic functions have the same order of growth within a constant multiple.
3. The first function
has the same order of growth as the second function
within a constant multiple.
4. The first function
has a smaller order of growth as the second function
within a constant multiple.
Explanation:
The given functions are
1.
and ![2000n^2 + 34n](https://tex.z-dn.net/?f=2000n%5E2%20%2B%2034n)
2.
and ![log(n)](https://tex.z-dn.net/?f=log%28n%29)
3.
and ![2^n](https://tex.z-dn.net/?f=2%5En)
4.
and ![0.001n^3 - 2n](https://tex.z-dn.net/?f=0.001n%5E3%20-%202n)
The First pair:
and ![2000n^2 + 34n](https://tex.z-dn.net/?f=2000n%5E2%20%2B%2034n)
The first function can be simplified to
![n(n +1 ) \\\\(n \times n) + (n\times1)\\\\n^2 + n](https://tex.z-dn.net/?f=n%28n%20%2B1%20%29%20%20%5C%5C%5C%5C%28n%20%5Ctimes%20n%29%20%2B%20%28n%5Ctimes1%29%5C%5C%5C%5Cn%5E2%20%2B%20n)
Therefore, the first function
has the same order of growth as the second function
within a constant multiple.
The Second pair:
and ![log(n)](https://tex.z-dn.net/?f=log%28n%29)
As you can notice the difference between these two functions is of logarithm base which is given by
![log_a \: n = log_a \: b\: log_b \: n](https://tex.z-dn.net/?f=log_a%20%5C%3A%20n%20%3D%20log_a%20%5C%3A%20b%5C%3A%20log_b%20%5C%3A%20n)
Therefore, the first
and the second
logarithmic functions have the same order of growth within a constant multiple.
The Third pair:
and ![2^n](https://tex.z-dn.net/?f=2%5En)
The first function can be simplified to
![2^{n-1} \\\\\frac{2^{n}}{2} \\\\\frac{1}{2}2^{n} \\\\](https://tex.z-dn.net/?f=2%5E%7Bn-1%7D%20%5C%5C%5C%5C%5Cfrac%7B2%5E%7Bn%7D%7D%7B2%7D%20%20%5C%5C%5C%5C%5Cfrac%7B1%7D%7B2%7D2%5E%7Bn%7D%20%20%5C%5C%5C%5C)
Therefore, the first function
has the same order of growth as the second function
within a constant multiple.
The Fourth pair:
and ![0.001n^3 - 2n](https://tex.z-dn.net/?f=0.001n%5E3%20-%202n)
As you can notice the first function is quadratic and the second function is cubic.
Therefore, the first function
has a smaller order of growth as the second function
within a constant multiple.