Indicate whether the first function of each of the following pairs has a smaller, same or larger order of growth (to within a constant multiple) than the second function. Justify your answer 1. n(n+1) and 2000 n
2 + 34 n 2. In n and Ign
3. 2n-1 and 2n
4. 2 n² and 0.001 n3 - 2 n

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

2. and

3. and

4. and

The First pair:

and

The first function can be simplified to

Therefore, the first function has the same order of growth as the second function within a constant multiple.

The Second pair:

and

As you can notice the difference between these two functions is of logarithm base which is given by

Therefore, the first  and the second logarithmic functions have the same order of growth within a constant multiple.

The Third pair:

and

The first function can be simplified to

Therefore, the first function has the same order of growth as the second function within a constant multiple.

The Fourth pair:

and

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.